text stringlengths 6 976k | token_count float64 677 677 | cluster_id int64 1 1 |
|---|---|---|
Course:The
focus of Algebra is to shift from the arithmetic skills to the more abstract
algebraic means of representation by emphasizing topics that are current and
relevant to today's student. Students will explore the language of algebra
through verbal, graphical, and symbolic form. Each lesson is designed to
motivate students to master the content. Students will develop skills needed to
solve problems involving applications, connections, and integration.
The
Mathematics Framework for California Public Schools opens the algebraic content
standards by saying, "Symbolic reasoning and calculations with symbols are
central in algebra. Through the study of algebra, a student develops an
understanding of the symbolic language of mathematics and the sciences. In
addition, algebraic skills and concepts are developed and used is a wide
variety of problem-solving situations."
Goals and Objectives:Provide
opportunities for all students, regardless of ethnicity, gender, learning
styles, or economic status, to build and maintain mathematical power, and to be
prepared for further studies in mathematics.
Allow students to take
responsibility for their own learning and to reflect upon their work. Students
are expected to take advantage of redo opportunities for work that does not
meet the San Diego City School Standards.
Familiarize students with the
San Diego City School Standards utilizing the Principles of Learning in
designing and implementing lessons.
Basic Skills:At
Marshall we fell it is necessary and important for the student to continue
developing and refining basic skills. For that reason, we emphasize basic skill
on a regular basis. A variety of basic skill problems will be reviewed at the
beginning of the year and throughout the class as need arises. It may be
necessary for some students to put in extra effort on their own with skill
sheets in the areas they need help.
Standards:This
course contains exercises and materials that reflect the Mathematics Framework
for California Public Schools. A copy of these standards can be provided upon
request or can be found at
Textbook:The
book for this course is McDougal Littell's Algebra 1. Students will be assigned one book for which they
are responsible. There is a classroom set so students shouldn't need to bring
their book everyday.
Citizenship Standards:The
Mustang Code of Conduct is the code by which all Marshall students are to
conduct themselves. Misbehavior is not tolerated as it interferes with the
students' right to learn and the teacher's ability to teach. In general, the
citizenship guidelines are:
Please review the citizenship
standards found in the planner with your child. I do not tolerate disruptive
behavior and it will become the parent's responsibility to correct such behavior
if it persists.
Rewards for excellent
behavior are important and those students will receive praise, positive phone
calls home, homework passes, special rewards and extra treats.
Homework Policy:Late
homework will not be accepted. If there are extenuating circumstances, late
homework may be accepted with a parent signature.
In cases of absences the
student will have as many days to make up the homework as they were absent.
Homework is graded on a
five-point rubric. 5 point work is complete, shows all work and follows
directions; 4 point work is mostly complete but may be missing minor components
or is not done in pencil; 3 points is incomplete and/or shows a lack of effort;
2 points is very incomplete and shows no effort; 1 point is given for work that
is all but incomplete; 0 points will be given for missing work or work that is
unacceptable.
Homework is expected to be in
pencil at all times to facilitate corrections. Students are to maintain a
separate notebook for warm-ups, class notes, and homework. Homework is not to
be thrown away.
Assignment Schedule: Homework is assigned daily. The assignments are posted
in the classroom, on the Internet, and on the homework hotline. Students are
responsible for keeping up with assignments when absent.
Testing:Students
can retake tests when they earn a D or F. They will have two weeks from the
date they get the test back to retake a test. There will only be one retake
allowed per test and it will be harder that the original.
Grading Policy:
Warm-ups
and notes15%
Homework20%
Tests40%
Projects,
Quizzes, and Problems of the Week25%
Students who receive a low score on a project or test are
expected to meet with the teacher after school, review the material, and redo
the task in order to meet the standards. Students will be provided with
examples of good products and/or task lists to guide their work. Grading
rubrics will be discussed when the assignments are given. The student should
save work until the end of the year. | 677.169 | 1 |
In this course, students will learn to apply regular and singular perturbation methods to ordinary and partial differential equations. They will also be exposed to boundary-layer theory, long-wave asymptotic methods for partial differential equations, methods for analyzing weakly nonlinear oscillators and systems with multiple time scales, the method of moments, the Turing instability, pattern formation, and Taylor dispersion.
Aim:
This is the second of a two-part series of lectures designed to provide students the ability to formulate and extract insight from basic mathematical models. | 677.169 | 1 |
Gifts Received
Thaddeus Wert's Page
Latest Activity
" (Although they…"
"…"
"For me, it is taking a story problem (hopefully a real-world situation), teaching my students how to develop an appropriate mathematical model of the situation, and then solving it. If a student is faced with a paragraph of information, then she…""I have been using Maxima in my Calculus classes this year. If you have experience with Mathematica, it's similar. I also use JKGrapher a lot (John Kennedy has written a lot of really useful programs | 677.169 | 1 |
Algebra, Trigonometry, and Statistics
Publisher's Summary
Algebra, Trigonometry, and Statistics helps in explaining different theorems and formulas within the three branches of mathematics. Use this guide in helping one better understand the properties and rules within algebra, trigonometry, and statisticsFor the price, it is great. It might should be longer or more in depth but then it would cost more.
How would you have changed the story to make it more enjoyable?
It is a little dry but there are not many math books on here as of now.
What three words best describe James Powers's voice?
Straight-forward, easy, fast.
Do you think Algebra, Trigonometry, and Statistics needs a follow-up book? Why or why not?
Sure, hopefully as cheap again.
Any additional comments?
It seems to be the perfect book to throw on when I cannot decide which book to do next. I do not really care if I absorb it through each time; repeated usage will do the trick. Quick and painless but overall helpful. | 677.169 | 1 |
Mathematics Department
The Mathematics Department
Mathematics is a core subject throughout the school and all pupils are prepared for public examinations at both junior and leaving certificate level.
Each form has 5 lessons per week, each of 45/40 minutes length. All forms are set on ability. The top two or three sets in each year will aim to undertake higher level exams. Different styles of books and resources are used for the different ability levels.
In general, work will be set at the end of every class. This will be a small number of questions based on the work covered in the class, or revision, and usually should not take longer than 15/20 minutes in junior years and 30 minutes in senior years.
The department prides itself on having a well rounded transition year programme, with an emphasis on academic work. We are continuing to improve and refine our scheme of work, to be flexible to the needs of our pupils. The top mathematicians are also introduced to applied mathematics, to encourage them to take up the subject for leaving certificate. We also offer an advanced maths 'Q-set' at leaving certificate, which is an additional 2 periods per week, for those thinking of engineering or other maths based courses and professions.
Class tests are given at suitable intervals, with standardised tests given at the end of each term to all in each form. Records of marks and weekly effort grades are kept as an aid when writing end-of-term reports. These results also help us decide which set a pupil is best able to perform well in, and we regular adjust classes accordingly.
The department consists of five very well qualified and experienced mathematics teachers, 4 of which teach the subject fulltime. It meets at regular intervals to discuss developments within the school and department and share ideas of best practice in the classroom. Staff are sent on courses when available, to encourage development within the department and of individual teachers.
In 2016 we entered 62% of our pupils for Higher Level Leaving Certificate, 34% more than the national average. Of those sitting higher level, 32% got an A grade, compared with 11% nationally. For Junior Certificate Higher level we entered 86%, with 31% achieving an A grade, compared to 12% nationally. We are extremely proud of our outstanding results record, and continue to find ways to improve further.
The department has access to a number of digital projectors, and access to useful software packages. It has use of an ICT suite with terminals for the whole class. Various websites are used as teaching aids when appropriate.
Each year prizes are awarded as the result of examinations in Forms III and IV and a senior prize is also awarded.
In 2009 the mathematics department underwent a Department of Education Inspection. The findings of this were extremely positive and complimentary and can be read at; | 677.169 | 1 |
Compressed Zip File
Be sure that you have an application to open this file type before downloading and/or purchasing. How to unzip files.
5.21 MB | 76 pages
PRODUCT DESCRIPTION
Geometry 101. It is a must to get your students started off the right way with the building blocks of all Geometry concepts, and wouldn't it be great to have everything you need to do that all in one place? Well, here it is!
This is a GROWING unit bundle that currently contains presentation notes, student follow-along notes handouts, glossary, glossary cards, 3 practice worksheets, 6 section quizzes, and a unit test. Other items will be added over time, and the price will be modified as necessary during this process.
This is also one of the units that will be part of my complete Geometry curriculum that is currently in the works so keep checking back, and it will also work as a supplement to your current materials.
I am and have been a high school Geometry teacher for 20+ years, and all of the content has been used in my classroom.
The notes introduce each concept along with a few examples. They are in-depth enough to cover a topic well, and, at the same time, are brief enough to allow you to add related topics/examples of your choice without being overwhelming or stifling your own creativity/flexibility.
**This purchase is for one license only.**
If you are interested in purchasing multiple site licenses for a grade/department, school, or district, contact me at teacherminestore@gmail.com. Please protect the proprietary nature of this product. It should not be made available to others without purchasing the license.
***************************************************************************
How to get TPT credit to use on future purchases:
will help me to create high-quality products | 677.169 | 1 |
Math for Weld Math for Weldersis a combination text and workbook that provides numerous practical exercises designed to allow welding students to apply basic math skills. Major areas of instructional content include whole numbers, common fractions, decimalMore...
Math for Weldersis a combination text and workbook that provides numerous practical exercises designed to allow welding students to apply basic math skills. Major areas of instructional content include whole numbers, common fractions, decimal fractions, measurement, percentage, and the metric system. Answers to odd-numbered practice problems are listed in the back | 677.169 | 1 |
Precalculus Made EasySlideshows
Step by Step - Solve any System of Equations
Step by Step - Complete the Square
Step by Step - Average Rate of Change
Step by Step - Conic Analyzer
Step by Step - Find GCD
Step by Step - Compute Powers
Step by Step - Complex Numbers
Step by Step - Distance, Midpoint, Slope of 2 Points
All-in-1 Matrix Explorer
Find Domain of any Function
Exponential, Log Functions
Factoring and Distributing
Combinations, Permutations, Pascal Triangle
Probabilities
Sequences: Arithmetic, Geometric
Solve System of Equations
Trig: Solve any Triangle
View Trig Identities
Trig: View UnitCircle
Step by Step - Find Tangent Lines
Precalculus Made Easy | 677.169 | 1 |
Pre-Algebra B Course
Registration Code: MA879B Credits: 1 Price: $299.00
Two-semester preparation course for students on the verge of Algebra I concepts. Well rounded and in-depth, this course offers the following topics: rational number theory (including comparing and ordering on a number line), drawing conclusions from statistical data, ratios, proportions and percents, spatial thinking (topics include congruency, translations, and symmetry), introduction to sequences and patterns, and working with polynomials and functions. Interspersed throughout the course are numerous opportunities for the development of vital problem-solving skills at the middle school level. These problem-solving lessons are designed to be integrated into the regular curriculum or may be used as a stand-alone mini-unit course. Significant topics include: writing algebraic expressions, generating patterns and problem simulations. | 677.169 | 1 |
Polynomial Vocabulary Sorting Chart (Kagan)
Be sure that you have an application to open
this file type before downloading and/or purchasing.
35 KB|3 pages
Product Description
This is a quick activity to review polynomial vocabulary (binomial, trinomial, degree, leading coefficient). The first page (pictured) is the answer sheet. The students will receive a blank copy of the chart and the answers (in mixed up order). They need to cut out the answer, sort them out to fit in the chart correctly and paste them on. This could be done individually or with a partner. | 677.169 | 1 |
Honors Algebra 2
The student extends the skills and concepts started in Algebra I but at a higher level of difficulty and with greater emphasis on derivation and proof. Major topics included in the course are: the field properties of the real numbers; techniques of solving first and second degree equations and inequalities in one and two variables; techniques of simplifying and operating on polynomial, rational, and irrational expressions; linear and quadratic relations and functions; complex numbers; exponential functions; and logarithms. This enriched course includes graphing equations. This course requires a TI-84 Plus graphing calculator. | 677.169 | 1 |
Friday, January 25, 2008
We have reached the half way point in this academic year. In regards to this math course: Is your progress what you expected it to be? How is your progress different than your expectations? What are you doing in preparation for this class that you are proud of? What are you doing that you could do better? What is preventing you from doing as well as you could possibly do?
Make a personal growth plan that includes specific goals for the final semester and actions of how you will achieve your goals. Use the guiding questions above for assistance. Title this post "Personal Growth." This is due 2/5/08 and is worth 10 points. | 677.169 | 1 |
They not only have to learn the material, they have to still know it.
One of the constant complaints my colleagues and I have about the students is that they do not remember the prerequisite material. I teach math and math is very cumulative. For example, before a student can study quadratic equations, they should have a complete understanding of linear equations.
Sometimes students ask me if calculus is difficult, and oddly enough, it isn't. There is one concept that is difficult and it comes early: limits. Once that is understood, calculus is very logical and surprisingly easy. However, most calculus classes use a great deal of algebra and trigonometry. If a student doesn't have a firm grasp of those subjects, he will not understand calculus. This is a repeat of what happens earlier in students' careers. If they have difficulty adding, multiplication is very hard to understand.
I teach Intermediate Algebra every semester. Many of my students cannot solve simple linear equations which they should have learned, not in the prerequisite, but in the prerequisite to the prerequisite. It is hard to teach students who cannot find the common denominator for simple fractions how to find the common denominator for rational expressions. The concepts are the same. I should not have to teach grade school math.
Advertisements
Rate this:
Like this:
LikeLoading...
Related
This entry was posted on March 8, 2010 at 10:08 pm and is filed under Teaching. You can follow any responses to this entry through the RSS 2.0 feed.
You can leave a response, or trackback from your own site. | 677.169 | 1 |
Need to keep your rental past your due date? At any time before your due date you can extend or purchase your rental through your account.
Sorry, this item is currently unavailable.
Summary
This textbook shows how number theory and geometry are the essential components in the teaching and learning of mathematics for students in primary grades. The book synthesizes basic ideas that lead to an appreciation of the deeper mathematical ideas that grow from these foundations. The authors reflect their extensive experience teaching undergraduate nonscience majors, students in the Young Scholars Program, and public school K-8 teachers in the Seminars for Endorsement of Science and Mathematics Educators (SESAME). | 677.169 | 1 |
There are many topics in the BC Multiple-Choice Exam and many seem unrelated to each other. Students frequently struggle when they go through the daunting process of studying for the Advanced Placement Exam. Where to begin?
With that concept in mind, this guide will cover a good many topics that research has shown to have been part of most Advanced Placement Calculus BC multiple-choice questions in the past 25 years. I first explain the concept, and then quickly see how they are incorporated into an actual AP test type problem. Best of all, this material is FREE!
Using this 43-page manual, students will become skilled at the types of problems that most likely show up on the BC Multiple-Choice section of the AP Exam. There are 88 examples. Students will be then able to test themselves on specific topics. Unlike the free response section, an attempt is made to try and treat each topic as a separate entity instead of combining concepts (although, of course, basic differentiation and integration skills are assumed in BC). There is a great deal of attention paid to Taylor polynomials and series as well as power series. The dreaded Lagrange error (where there are so few textbook examples) is covered so students will not be afraid of seeing them in the real exam. This guide will provide an in-depth level of understanding so that students do not spend time in the actual AP Exam scratching their heads!
REVISED FOR 2017 AP EXAM
This material includes the limit comparison test, absolute and conditional convergence and alternating series error bounds, which are new to the 2017 AP exam. All questions have four choices, rather than five, in keeping with the new exam format.
FREE STUDENT VERSIONS
You may download any or all of the material below FREE OF CHARGE. Topics are combined and you can download them all in 3 downloads. They are only about 6 megs total in size.
SOLUTIONS
Detailed solutions are available for purchase. The solutions are inserted right under the problem so that you can see the problem and solution at the same time. You may either download the solutions (approximately 9 MB split into 3 PDF files) or have them mailed to you in paper format. If you purchase it in combination with Demystifying the BC Exam Free Response version, it is also available on flash drive. In that format, you also get the student versions as well as our new publication 4 AP Calculus Practice Exams | 677.169 | 1 |
Functions, Patterns, and Interpreting Graphs Tiered Worksheet
Compressed Zip File
Be sure that you have an application to open this file type before downloading and/or purchasing. How to unzip files.
0.85 MB | 20 pages
PRODUCT DESCRIPTION
This is a tiered student worksheet to review concepts of patterns, determining if a relation is a function, domain, range, creating function rules from patterns, interpreting graphs, and drawing graphs to represent a scenario. There are three levels of differentiated work for students. The first level gives the student guided material and helpful notes. The second level gives less guidance, but still gives the student some support for the problems. The highest level does not include guided notes. It also includes some extension problems. All three levels present problems from the same content. This is a great resource to use for RTI interventions. The key is included for all three levels. Teachers may choose for students to work individually or cooperatively with the activity. Ready to print, copy, and begin | 677.169 | 1 |
Welcome to High School Math at Pressly
Our mission statement for Math students at Pressly is to be successful in the classroom setting and complete the four math credits required to receive a high school diploma.
About Me
Welcome to High School Math. Math is taught from the state mandated Common Core which contains Foundations of Common Core Math, Common Core Math I, Common Core Math II, and Common Core Math III. For students on the old track of HS math, the subjects are still listed as Foundation of Math, Algebra I, Algebra II, Geometry, and Survey of Mathematics. There is a Common Core Math IV which is the higher maths such as Statistics, Pre-Calculus, and Calculus.
For more information on Common Core, please visit the website: | 677.169 | 1 |
All Math Words Dictionary
One of the difficulties many students experience in learning math skills has to do with the fact that an entire language, both spoken and written, has grown up around math. Students that acquire that language are successful in math studies. Students that do not acquire that language have serious problems with mathematics. This dictionary is designed to aid in the acquisition of the language of math. All Math Words Dictionary is written for students of pre-algebra, beginning algebra, geometry and intermediate algebra | 677.169 | 1 |
This course is a liberal arts introduction to the nature of mathematics. Topics are chosen from among logic, graph theory, number theory, symmetry (group theory), probability, statistics, infinite sets, geometry, game theory, and linear programming. These topics are independent of each other and have as prerequisite the ability to read, reason, and follow a logical argument.
Pre / Co requisites: MAT 103 requires prerequisites of a grade of C- or better in MAT Q30 or an appropriate score on the Mathematics Placement Examination.
Distance education offering may be available.
Typically offered in Fall, Spring & Summer.
MAT 104. Introduction to Applied Mathematics. 3 Credits.
The course is designed to help prepare students to understand almost any quantitative issues they will encounter in contemporary society. Topics are selected from the following: principles of reasoning, problem-solving tools, financial management, exponential growth and decay, probability, putting statistics to work, mathematics and the arts, discrete mathematics in business and society and the power of numbers.
Pre / Co requisites: MAT 104 requires prerequisites of a grade of C- or better in MAT Q30 or an appropriate score on the Mathematics Placement Examination.
Typically offered in Fall, Spring & Summer.
MAT 113. Algebra and Functions. 3 Credits.
A review of basic algebra, followed by a thorough treatment of polynomial, rational, exponential, and logarithmic functions. Successful completion of this course prepares students for MAT 143.
Pre / Co requisites: MAT 113 requires a prerequisite of a grade of C- or better in MAT Q30 or an appropriate score on the Mathematics Placement Examination.
Typically offered in Fall, Spring & Summer.
MAT 115. Algebra, Functions, and Trigonometry. 3 Credits.
Topics include polynomial, rational, exponential, logarithmic, and trigonometric functions. An emphasis is placed on using technology to understand topics of importance in the life and earth sciences. Successful completion of this course prepares students for MAT 143 or MAT 145.
Pre / Co requisites: MAT 115 requires a grade of C- or better in MAT Q30 or an appropriate score on the Mathematics Placement Examination.
Typically offered in Fall, Spring & Summer.
MAT 121. Introduction to Statistics I. 3 Credits.
Basic concepts of statistics. Frequency distributions, measures of central tendency and variability, probability and theoretical distribution, significance of differences, and hypothesis testing.
Pre / Co requisites: MAT 121 requires a prerequisite of a grade of C- or better in MAT Q30 or an appropriate score on the Mathematics Placement Examination.
Distance education offering may be available.
Typically offered in Fall, Spring & Summer.
Topics include polynomial, rational, exponential, logarithmic, and trigonometric functions. An emphasis is placed on understanding function properties and graphs without the use of technology. Successful completion of this course prepares students for MAT 161.
Pre / Co requisites: MAT 131 requires a prerequisite of a grade of C- or better in MAT Q30 or an appropriate score on the Mathematics Placement Examination.
Typically offered in Fall, Spring & Summer.
MAT 143. Brief Calculus. 3 Credits.
An intuitive approach to calculus with emphasis on conceptual understanding and applications to business. Topics include differentiation, curve-sketching, optimization, integration, and partial derivatives.
Pre / Co requisites: MAT 143 requires a prerequisite of a grade of C- or better in MAT 113, MAT 115, or MAT 131; or an appropriate score on the Mathematics Placement Examination.
Typically offered in Fall, Spring & Summer.
MAT 145. Calculus for the Life Sciences. 3 Credits.
An overview of differential and integral calculus, motivated through biological problems. Topics include mathematical modeling with functions, limits, continuity, differentiation, optimization, and integration. Graphing calculators are used as an aid in the application of calculus concepts and methods to realistic biological problems.
Pre / Co requisites: MAT 145 requires a prerequisite of a grade of C or better in MAT 115 or MAT 131; or an appropriate score on the Mathematics Placement Examination.
Typically offered in Fall, Spring & Summer.
Differential and integral calculus of real-valued functions of a single real variable with applications.
Pre / Co requisites: MAT 161 requires prerequisites of a C or better in MAT 131 or an appropriate score on the Mathematics Placement Examination.
Typically offered in Fall, Spring & Summer.
Topics announced at time of offering.
Consent: Permission of the Department required to add.
MAT 200. The Nature of Mathematics. 3 Credits.
Topics include the role of mathematics in contemporary society, career opportunities, mathematical notation and argument, structure of proofs, basic facts about logic, mathematical proofs, problem-solving techniques, and introductions to mathematical software packages.
Pre / Co requisites: MAT 200 requires a prerequisite of C or better in MAT 161. Course should be taken by the end of sophomore year.
Typically offered in Fall, Spring & Summer.
MAT 201. Elementary Functions Essential Calculus I. 3 Credits.
Elementary functions from an advanced viewpoint with detailed discussion of formal manipulations. Special emphasis on applications and the use of technology. Open only to prospective Grade 4-8 certification students.
Pre / Co requisites: MAT 201 requires prerequisite MAT 102.
MAT 202. Elementary Functions and Essential Calculus II. 3 Credits.
Elementary functions from an advanced viewpoint with detailed discussions of formal manipulations. Special emphasis on applications and the use of technology. Open only to prospective Grade 4-8 certification students.
Pre / Co requisites: MAT 202 requires prerequisite MAT 201.
The calculus of several variables. Topics include polar coordinates, vectors and three-dimensional analytic geometry, differentiation of functions of several variables, multiple integrals, and line and surface integrals.
Pre / Co requisites: MAT 261 requires a prerequisite of MAT 162 with a C or better.
Typically offered in Fall, Spring & Summer.
MAT 301. The Scientific Revolution. 3 Credits.
This course addresses how modern science began in the 17th century by examining its origins and including introductions to the heroes of science - Copernicus, Kepler, Galileo, and Newton. This course counts toward the writing emphasis requirement.
Gen Ed Attribute: Interdisciplinary Requirement, Writing Emphasis.
Typically offered in Fall & Spring.
MAT 302. Mathematics and Social Justice. 3 Credits.
In this course we will explore several social issues and we will discuss methods which can quantitatively illustrate that are taking place. By doing so, the hope is that each student will learn mathematical skills and techniques. This tool kit of basic mathematical skills is often referred to as Quantitative Literacy (QL). Moreover as attainment of QL is itself a social justice issue, we will explore ways to carry these skills to historically marginalized groups through service learning projects.
MAT 309. Topics in Math for Elementary Teachers. 3 Credits.
Introduction to programming in BASIC; computer uses for the classroom teacher; descriptive statistics with applications for teaching; and measurements of length, area, volume, and temperature that focus on the SI metric system with practice in the classroom. Additional topics in applied mathematics will be considered.
Pre / Co requisites: MAT 309 requires prerequisite of MAT 102.
Repeatable for Credit.
This course will cover simple and multiple linear regression methods and linear time series analysis with an emphasis on fitting suitable models to data and testing and evaluating models against data.
Pre / Co requisites: MAT 319 requires a prerequisite of MAT 143 or MAT 145 or MAT 161.
Typically offered in Fall & Summer.
An introduction to the use of the computer as an investigative tool in the filed of mathematics with emphasis on experimental techniques involving graphical and numerical displays.
Pre / Co requisites: MAT 325 requires a prerequisite of MAT 162 with a "C" or better.
Typically offered in Spring.
The general theory of nth order, and linear differential equations including existence and uniqueness criteria and linearity of the solution space. General solution techniques for variable coefficient equations, series solutions for variable coefficient equations, and study of systems of linear equations.
Pre / Co requisites: MAT 343 requires a prerequisite of C or better in MAT 162.
Typically offered in Fall, Spring & Summer.
Techniques for teaching children concepts such as geometry in two and three dimensions, number sentences, graphing, ratios and percentages, quantifiers, etc. Use of laboratory materials will be emphasized.
Pre / Co requisites: MAT 352 requires prerequisites of MAT 351, Field clearances and Formal Admission to Teacher Education.
Typically offered in Spring.
Methods and materials associated with the presentation of mathematics to the handicapped. Emphasis on individualization and involving thinking skills at the concrete level. Evaluative and interpretive techniques are included.
Pre / Co requisites: MAT 357 requires prerequisites of MAT 101 and MAT 102 and formal admission into teacher education.
MAT 360. Field Experiences in Middle School Mathematics. 1 Credit.
The objective of this course is to apply the skills, techniques, and dispositions required to be an effective middle and 360 requires a prerequisite of Formal Admission to Teacher Education. MAT 360 requires a co-requisite of MAT 350.
Typically offered in Fall.
MAT 362. Calculus IV. 3 Credits.
The calculus of vector-valued functions of a vector variable. Derivatives and properties of the derivative including the chain rule, fields and conservative fields, integration, and Green's, Stokes', and Gauss' theorems.
Pre / Co requisites: MAT 362 requires prerequisite of C or better in MAT 261 and C or better in MAT 311.
MAT 364. Field Experiences in Secondary School Mathematics. 1 Credit.
The objective of this course is to apply the skills, techniques, and dispositions required to be an effective 364 requires prerequisites of MAT 360 and Formal Admission to Teacher Education. MAT 364 requires a co-requisite of MAT 354.
Typically offered in Spring.
MAT 371. Mathematics of Finance. 3 Credits.
The purpose of this course is to introduce the mathematical theory behind the concepts of: measurement of interest, annuities, yield rates, amortization of loans, sinking funds, and yield rates. Understanding the fundamental concepts of financial mathematics, and how these concepts can be applied to calculate present and future values of various financial instruments, is the prevailing theme of the course.
Pre / Co requisites: MAT 371 requires prerequisite of MAT 162 with a "C" or better.
Typically offered in Fall.
MAT 381. Discrete Mathematics. 4 Credits.
This course is designed to provide a foundation for the mathematics used in the theory and application of computer science. Topics include mathematical reasoning, the notion of proof, logic, sets, relations and functions, counting techniques, algorithmic analysis, modelling, cardinality, recursions and induction, graphs, and algebra.
Pre / Co requisites: MAT 381 requires prerequisite of C or better in MAT 162.
MAT 390. Seminar in Mathematics Education. 3 Credits.
This course is the capstone course for grades 4-8 certification students completing the 30-credit mathematics certification option. Topics selected from mathematics, statistics, the history of mathematics, and mathematics education for their significance and interest. Field experience may be required.
Pre / Co requisites: MAT 390 requires prerequisite of Formal Admission to Teacher Education.
Repeatable for Credit.
MAT 400. History of Mathematics for Elementary Teachers. 3 Credits.
History and development of elementary mathematics from primitive times to the discovery of calculus. Problems of the period are considered.
Pre / Co requisites: MAT 400 requires prerequisites of MAT 212 and MAT 233.
The focus of this course is to introduce students to computer algebra packages and review important topics in algebra, calculus and linear algebra.
Pre / Co requisites: MAT 413 requires prerequisites of MAT 162 and MAT 311 with a "C" or better.
Typically offered in Fall.
A rigorous treatment of the calculus of a single real variable. Topics in several real variables and an introduction to Lebesque integration.
Pre / Co requisites: MAT 441 requires prerequisites of C or better in MAT 200 and MAT 261.
Typically offered in Fall, Spring & Summer.
MAT 442. Real Analysis II. 3 Credits.
A rigorous treatment of the calculus of a single real variable. Topics in several real variables and an introduction to Lebesque integration.
Pre / Co requisites: MAT 442 requires prerequisite of C or better in MAT 441.
Students completing this course will have a better understanding of actuarial models of life contingencies, more specifically, students will understand that life insurance payments, life annuity payments, pension payments, etc. are determined by financial random variables dependent on human life.
Pre / Co requisites: MAT 478 requires prerequisite of MAT 371 and MAT 421 with a "C" or better.
Typically offered in Spring.
MAT 479. Financial Calculus. 3 Credits.
This course aims to provide the undergraduate mathematics major with an introduction to the mathematics behind derivative pricing and portfolio management. Pricing theory is first developed through the typical binomial model and then is extended to continuous time via the Black-Scholes model. In addition, the student will be exposed to how arbitrage can be used to aid in the pricing more complicated derivatives, such as call options on dividend-paying securities and exotic options.
Pre / Co requisites: MAT 479 requires prerequisite of MAT 371 and MAT 421 with a "C" or better.
Typically offered in Spring.
MAT 491. Internship in Applied Mathematics. 2-4 Credits.
In cooperation with regional businesses and industrial companies, student will perform an internship in applied mathematics.
Repeatable for Credit.
MAT 493. Mathematical Modeling. 3 Credits.
The idea of a mathematical model of a real situation. Techniques and rationales of model building. Examples from the life, physical, and social sciences.
Pre / Co requisites: MAT 493 requires prerequisites of C or better in MAT 261 and C or better in MAT 343.
MAT 499. Independent Study in Mathematics. 1-3 Credits.
Independent investigation of an area of mathematics not covered in the department's course offerings.
Consent: Permission of the Department required to add.
Repeatable for Credit.
MAT Q20. Fundamental Skills in Arithmetic. 3 Credits.
This course is designed to strengthen basic arithmetic skills and to introduce the elements of algebra. Mathematics placement required. Credits earned in Q00-level courses do not count toward the 120 hours of credit needed for graduation.
Pre / Co requisites: MAT Q20 requires a prerequisite of an appropriate score on the Mathematics Placement Examination.
Typically offered in Fall, Spring & Summer.
MAT Q30. Fundamentals of Algebra. 3 Credits.
This course is designed to strengthen basic algebraic skills. Credits earned in Q00-level courses do not count toward the 120 hours of credit needed for graduation.
Pre / Co requisites: MAT Q30 requires a grade of C- or better in MAT Q20 or an appropriate score on the Mathematics Placement Examination.
Typically offered in Fall, Spring & Summer.
West Chester University of Pennsylvania
700 South High St, West Chester, PA 19383
610-436-1000 [Map] Mobile Site | Ethics | 677.169 | 1 |
Adventures in Group Theory is a tour through the algebra of several 'permutation puzzles'... If you like puzzles, this is a somewhat fun book. If you like algebra, this is a fun book. If you like puzzles and algebra, this is a really fun book." - MAA Online "Joyner has collated all the Rubik lore and integrated it with a self-contained introduction to group theory that equals or, more likely, exceeds what is available in typical dedicated elementary texts." - Choice "Joyner does convey some of the excitement and adventure in picking up knowledge of group theory by trying to understand Rubik's Cube. Enthusiastic students will learn a lot of mathematics from this book. | 677.169 | 1 |
Revolutionary digital learning for science, math and engineering
Energy2D recommended in computational fluid dynamics textbook
Computational fluid dynamics (CFD) is an important research method that uses numerical algorithms to solve and analyze problems that involve fluid flows. Computers are used to perform the calculations required to simulate the interaction of liquids and gases with surfaces defined by boundary conditions. Today, almost every branch of engineering rely on CFD simulations for conceptual design and product design.
"Nevertheless, first-time CFD users may wish to search the Internet to gain immediate access to an interactive CFD code. (Users may be required to register in order to freely access the interactive CFD code.) The website is provides simple CFD flow problems for first time users to solve and allows colorful graphic representation of the computed results." | 677.169 | 1 |
Essay Writing Service | Order research paper, dissertation
Simply point your camera toward a math problem and photomath will magically show the result with detailed step-by-step instructions. .
Simply point your camera toward a math problem and photomath will magically show the result with detailed step-by-step instructions. . 6 days ago. Enter your math problems and get them solved instantly with this free math problem solver. Dont become lazy though. Do your math problems. . Photomath is the worlds smartest camera calculator and math assistant! Point your camera toward a math problem and photomath will show the result with. . Solve math problems and plot functions. Math solver is a full featured scientific calculator and graphing calculator with widest range of functions available. . Free math problem solver answers your algebra homework questions with step-by-step explanations. .
Imply point your camera toward a math problem and photomath will magically show the result with detailed step-by-step instructions. . Google has quietly upgraded their search calculator to not just answer your basic math questions or graphical or 3d math questions now, it can. . Photomath promises to help solve simple linear equations and other math problems by reading questions with the help of your smartphone. .
It can take care about your math problems by just one focus such an authentic app! Worthy five star to you! Too bad it lacked functions of limitations. To download the free app photomath - camera calculator by photomath, inc. Googles calculator is designed to calculate problems without a lot of complicated math formatting, but sometimes its easier and more accurate to use some math syntax. And it gets some equations wrong for example 8 25 6. Use plenty of math operators and keep it as simple as possible if you want step by step solution after the final answer is entered, click on this will take you to the developers site where you sign in.
Simply point your camera toward a math problem and photomath will magically show the result with detailed step-by-step instructions. Work on that and gudluck thank you lowell! ) it will get better with the future updates! Thanks for suggestions android 7 update has messed the camera. I showed up a sum which showed sum with triangles if it does not for the geometry please reply me back i get more easier to solve my math problem. Its free for now and easy and most help app for people to learn and in case of you need to do a problem quickly you scan it and there tells you step by step and the answer i love this app its very useful it help me a lot because i dont have time to do my homework so i am just take a photo did not work for me. Googles calculator is more than an ordinary number cruncher.
American express security codes are 4 digits located on the front of the card and usually towards the right. I tried reinstalling the app and restarting my phone with no luck. Also the camera sometimes has trouble focusing and tapping doesnt seem to help. If itunes doesnt open, click the itunes application icon in your dock or on your windows desktop. X choose y fines the number of possible subset groups of y out of the set of x. . I need to close the app, remove it from background running apps and reopen it again so it would plot. Im expecting you to add that! I love this app who would think that you could scan any math problem and it would solve it. Trigonometric functions default input is in radian i. The morning email helps you start your workday with everything you need to know breaking news, entertainment and a dash of fun.
Work backwards - mathstories com
This app can scan and solve math equations instantly. More really hard math problems with friends a new way to prep for the sat and. . | 677.169 | 1 |
Description
Work more effectively and gauge your progress as you go along! This Student Study Guide is designed to accompany Hughes-Hallett's "Calculus: Single Variable, 4th Edition". It contains additional study aids for students that are tied directly to the text. Now in its Fourth Edition, Hughes-Hallett's "Calculus: Single Variable" reflects the strong consensus within the mathematics community for a balance between contemporary and traditional ideas. Building on previous work, it brings together the best of both new and traditional curricula in an effort to meet the needs of instructors and students alike. The text exhibits the same strengths from earlier editions including the Rule of Four, an emphasis on modeling, exposition that is easy to understand, and a flexible approach to technology.show more
Table of contents
1. A Library of Functions.2. Key Concept: The Derivative.3. Short-Cuts to Differentiation.4. Using the Derivative.5. Key Concept: The Definite Integral.6. Constructing Antiderivatives.7. Integration.8. Using the Definite Integral.9. Sequences and Series.10. Approximating Functions Using Series.11. Differential Equations.Answers to Practice Tests | 677.169 | 1 |
presents the key topics of introductory calculus through an extensive, well-chosen collection of worked examples, covering: algebraic techniques, functions and graphs, an informal discussion of limits, techniques of differentiation and integration, Maclaurin and Taylor, expansions, geometrical applications. Aimed at first-year undergraduates in mathematics and the physical sciences, the only prerequisites are basic algebra, coordinate geometry and the beginnings of differentiation as covered in school. The transition from school to university mathematics is addressed by means of a systematic development of important classes of techniques, and through careful discussion of the basic definitions and some of the theorems of calculus, with proofs where appropriate, but stopping short of the rigour involved in Real Analysis. Readers are also encouraged to practice the essential techniques through numerous exercises which are an important component of the book.
Author Biography
Keith Hirst is an experienced author and lecturer, with many years' experience teaching undergraduate courses. His research interests include Mathematics Education and Teaching Methods in Higher Education, with particular emphasis on the school / university interface. Keith also researched the area of learning problems in calculus as part of an investigation into undergraduates' conceptions of mathematical ideas. In 2000, he was awarded one of the first prestigious National Teaching Fellowships by the Institute of Learning and Teaching, to develop a database of undergraduate teaching resources. | 677.169 | 1 |
Year 12 Level 3 Certificate in Core Mathematics
Level 3 Certificate
This course is run as a one-year course, equivalent to an AS level, and contains 2 modules which result in 2 examinations in the middle of the summer term.
The compulsory Module 1 contains: Analysis and representation of data, Maths for personal finance, Estimation, Critical Analysis of given data and models.
All material is taught and applied to case studies, containing real life problems. For Module 2, there are then a number of optional modules to choose from, which can be decided upon, based the best fit for each particular group of students | 677.169 | 1 |
Why do I find engineering math textbooks much more understandable?
In many of the topics in math i hav learned so far, i just found that engin textbooks such as <advanced engin math> are much more understandable and yet covering similar depth of contents than mathematic methods book such as aftken&weber which my teacher choose as reference book for e course.
I think the latter is of so much worthless words that only makes the topic harder but not its usability.. Any way why learn anything thats not useful if I'm not a math major?
Does that mean I'm not suitable for science studies such as the physics major I'm taking now?
Different books [styles, content, motivations, etc... ] appeal to different people [possibly, at different times in their lives]. So it depends on what you are after. Some would regard all of these "engineering math" and "math methods" books as "cookbooks" mainly useful for looking things up. For an "end-user", they might be sufficient... but for real understanding, one needs to go beyond these types of books.
Echoing what robphy said, Engineering books lack proper mathematical rigor ={ Which may be why they are easier to understand sometimes, because too much rigor can cloud intuition. I've seen many times when a simple relation that can be shown easily by some algebraic manipulation, has to go through a long induction proof to prove formally. To get the best of both worlds, cross reference both books, bringing understanding to rigor is the best thing a student can have.
I think non-rigorous approach is good for developing intuition, but sometimes it just confuses people. May be the best is to read non-rigorous text until the point you feel confused and then read rigorous text to clarify things. It is like difference between proofs in physics and proofs in math. | 677.169 | 1 |
AP Calculus Limits of Rational Functions Comparing Relative Magnitude
Word Document File
Be sure that you have an application to open this file type before downloading and/or purchasing.
0.02 MB | 1 pages
PRODUCT DESCRIPTION
Students predict which functions grow quickest as x gets large. They fill out a data table using a calculator to compare the relative magnitudes of functions. Then, they order the functions by how quickly they grow. Finally, there are examples of limits of rational functions for guided practice so the class can apply what they discovered about how quickly certain functions grow as x gets large | 677.169 | 1 |
97801302384Through a clear and thorough presentation, this program fosters learning and success for students of all ability levels with extensive skills practice, real-life connections, projects, and study aids.
The accessible format helps students gain the understanding and confidence they need to improve their performance on standardized tests. Margin notes provide links to postulates and concepts previously taught; theorem boxes help students identify the big ideas in geometry. Featured lessons address calculator usage, applications, as well as paragraph proofs and constructions. Pre-taught vocabulary provides students with relevant background | 677.169 | 1 |
Thutong Exemplars Grade12
... contains important information and a detailed explanation about thutong exemplars grade12 ... which is also related with , maths literacy exampler grade12 2014, we set the standards! - exemplars, maths grade 11 exemplar.
ebook.dexcargas.com is a PDF Ebook search engine and unrelated to Adobe System Inc. No pdf files hosted in Our server. All trademarks and copyrights on this website are property of their respective owners. | 677.169 | 1 |
This textbook introduces readers to the fundamental hardware used in modern computers. The only pre-requisite is algebra, so it can be taken by college freshman or sophomore students or even used in Advanced Placement courses in high school.
Authors: LaMeres, Brock J. Written the way the material is taught, enabling a bottom-up approach to learning which culminates with a high-level of learning, with a solid foundation Emphasizes examples from which students can learn: contains a solved example for nearly every section in the book Includes more than 600 exercise problems, as well as concept check questions for each section, tied directly to specific learning outcomes | 677.169 | 1 |
You are here
Number Theory Through Inquiry
David C. Marshall, Edward Odell & Michael Starbird
TEXTBOOK*
Number Theory Through Inquiry is an innovative textbook that leads students on a guided discovery of introductory number theory. The book has two equally significant goals. One goal is to help students develop mathematical thinking skills, particularly, theorem-proving skills. The other goal is to help students understand some of the wonderfully rich ideas in the mathematical study of numbers. This book is appropriate for a proof transitions course, for an independent study experience, or for a course designed as an introduction to abstract mathematics. Mathematics or related majors, future teachers, and students or adults interested in exploring mathematical ideas on their own will enjoy Number Theory Through Inquiry.
Number Theory Through Inquiry contains a carefully arranged sequence of challenges that lead students to discover ideas about numbers and to discover methods of proof on their own. It is designed to be used with an instructional technique variously called guided discovery or Modified Moore Method or Inquiry Based Learning (IBL). The result of this approach will be that students:
Learn to think independently
Learn to depend on their own reasoning to determine right from wrong
Develop the central, important ideas of introductory number theory on their own.
From that experience, they learn that they can personally create important ideas. They develop an attitude of personal reliance and a sense that they can think effectively about difficult problems. These goals are fundamental to the educational enterprise within and beyond mathematics.
An Instructor's Manual is available to teachers who adopt Number Theory through Inquiry as a text. Contact our Service Center for details at 1-800-331-1622.
* As a textbook, Number Theory Through Inquiry does have DRM. Our DRM protected PDFs can be downloaded to three computers. iOS tablets can open secure PDFs using the AWReader app (available in the App Store). The iOS app uses the native iPad PDF reader so it is a very basic reader, no frills. Linux is not supported at this time for our secure PDFs. | 677.169 | 1 |
Affine and projective geometry, Autumn 2011
Affine and projective geometry, Autumn 2011
Instructor: Vladimir Dotsenko
This module is intended to introduce students to basic concepts of geometry and use those concepts to demonstrate how geometry and algebra can interact in a mutually beneficial way. The course will be accompanied with exercise classes; the work in class is not assessed but is extremely important as it prepares the students to approaching questions from home assignments and the final exam paper.
Syllabus
Geometric objects in two and three dimensions. Affine and projective plane: axiomatic approach.
Materials
... are available via the "Moodle" platform to which all students have access. Be warned that any handouts you might get hold of via "Moodle" or otherwise are incomplete, and by no means can they serve as a substitute for attending classes.
Assessment
There will be several home assignments of varying lengths, depending on the topics covered. However, the primary method of assessment is via a written examination. The final mark is computed as MAX(F,0.8F+0.2C), where F is the percentage earned in the final exam, C is the percentage earned by means of continuous assessment (home assignments). Thus, if a student does well in the final exam paper, their possible poor performance during the semester would not affect the final mark. However, since the final exam paper questions are modelled on questions from home assignments, students are strongly recommended to attempt as many problems from home assignments as they can.
Literature
No specific textbooks are recommended; some handouts and xerox copies of relevant literature will be disseminated in class. | 677.169 | 1 |
The book is highly popular among the student and teacher fraternity as a lot of similar problems have appeared in the various JEE Main/AIEEE and JEE Advanced/IIT-JEE exams.
Key Features:
The main features of the book are as follows:
1. The book is divided into 23 chapters. The flow of chapters has been aligned as per the NCERT books.
2. Detailed solution of each and every question has been provided for 100% conceptual clarity of the student. Well elaborated detailed solutions with user friendly language provided at the end of each chapter | 677.169 | 1 |
MATH230
Tutorial note 11
1. Introduction
Given a matrix A and a vector b, both known, we can use Gaussian Elimination for
finding the vector x such that
Ax = b .
However if the A is very large and sparse (the number of nonzero elements is a small
fraction
MATH230
Tutorial note 7
1. Numerical Integration
b
f ( x)dx ,
Given a function f which is continuous on [a,b]. If we are asked to evaluate
a
we can try to find an antiderivative of f, F ( x) , and then apply the formula
b
f ( x)dx = F (b) F (a) .
a
2
Un
MATH230
Tutorial note 6
1. Numerical differentiation
Give a function which is differentiable, one can always differentiate it ready. However,
equations with derivatives, that is differential equations, are rarely solvable. Because
of the importance of the
MATH230
Tutorial note 5
1. Data fitting
An important area in approximation is the problem of fitting a curve to experimental
data. Since the data is experimental, we must assume that it is polluted with some
degree of error, most common measurement error,
MATH230 Tutorial note 4
1. Divided differences
The Lagrange form of the interpolating polynomial gives us a very tidy construction,
but it does not lead itself well to actual computation. One of the reasons is that
whenever we decide to add a point to the
MATH230
Tutorial note 3
1. Interpolation
One of the oldest problems in mathematics is the problem of construction an
approximation to a given function f from among simple functions, typically (but not
always) polynomial. A slight variation of this problem
MATH230 Tutorial note 2
1. Newtons Method
Newtons method is the classic algorithm for finding roots of functions. It appears to
have been first used by Newton in 1669, although the ideas were known to others
beforehand.
Suppose f '( x) exists on [a,b] and | 677.169 | 1 |
NEW - FREE SHIPPING
This title is In Stock in the Booktopia Distribution Centre. We will send you a confirmation email with a Tracking Code to follow the progress of your parcel when it ships.
Excel Smartstudy Yr 8 Maths Author: Allyn Jones
ISBN: 9781741254747 Format: Paperback Published: 30 November 2013 Country of Publication: AU Description: This book serves as a structured revision program for all students undertaking Year 8 Mathematics. It has been designed to help students revise for class tests, half-yearly and end-of-year exams. It is structured to consolidate students' understanding in line with Australian Curriculum outcomes. Through concise Study Notes, Skills Checks, Intermediate/Advanced Tests, Sample Exam Papers and Worked Solutions, this book will ensure that your child is fully prepared for class exams | 677.169 | 1 |
Pages
Thursday, September 6, 2012
Math911
Are you the parent of a middle or high school student? Is Algebra the dreaded word or a struggle for your kids? Do you need a little brushing up on your Algebra skills in order to help your kids with theirs? If so continue reading as I tell you about a product I reviewed that may be what you're looking for.
Math911 is a tutorial software designed for students who struggle with mastering their math skills in Algebra. "It is intended for students who are convinced they can't do Algebra." The creator of this program is Professor Martin Weissman. He has been teaching mathematics to students since 1963. Math911 has been used in schools since 1990. The program is for students ages 12 and up who is ready to do Introductory Algebra.
Professor Weissman came up with an idea called "Algebra In A Flash" for his Math911. "It's The Mathematics Professor That Goes Wherever You Go". Algebra In A Flash is a flash drive that contains the Math911 Algebra Tutorial Software. It can be used anywhere you have access to a computer (used for Windows XP, Windows 7, or Vista only). There are different versions to choose from. There is the standard version which you can download for FREE if you are wanting to try the product out before you buy it. Just look to the left and click to get your free copy. This free copy is a complete copy of the Introductory Algebra course. If you are wanting to upgrade you can purchase the Premier version for the price of $49.95. The choices you have are Premier (one user with no password), Premier Password (for multiusers with password), and Network Password (for multiusers with passwords).
Here is a list of the Premier features:
The topics are arranged by chapter, section and levels. Within each level are all the types of problems that identify the concepts that your students must know in order to be successful in Algebra including Graphs and Word Problems.
The topics (courses) included are Introductory Algebra, Intermediate Algebra, Trigonometry, Pre Calculus, Statistics, and College Algebra.
Math911 "gives the student an unlimited amount of randomly generated problems". If the student doesn't get the answer right his or her answers won't be marked wrong. This is what Professor Weissman refers to as the Mastery Learning Approach. The student just keeps working out the problems until he or she gets it right. If he or she is having problems the program is designed where you can see step by step how the problem is worked out. There is no moving on to the next level until all the problems in that section can be done correctly. As stated before there are no grades for incorrect problems. Only correct responses are recorded.
Math911 "is what learning Algebra should be: quick, fun, and easy. It will let you absorb Algebra effortlessly". No, this is not a type of online math game that you play. No, there are no textbooks to accompany Math911. If your student needs more help understanding how to do the problems there are some PDF tutorials on the website you can download for free. Some have cartoons in it where the Professor is teaching a class and going through the steps of solving the problems with the students. There is also some exercises including the answers within the PDF that you can try out. To get a better understanding of how Math911 is set up check out these screen shots.
I used Math911 for myself since my son is no where near ready for any type of Algebra yet. We received the Premier Version for our review. I wanted to brush up on Algebra since I have not used it since I was last in school. I've shared before that math has never been my best subject, and Algebra was very hard for me to grasp and understanding of. I'm talking so hard that I couldn't get past Algebra I in high school. I switched teachers, took tutoring, and sought out help from everyone I could ask but nothing worked. I eventually was able to figure it out after meeting my husband in college. He took time with me each day and was so patient in teaching me step by step until I had it. I went from an F student to a B+ student in no time. Fast forward to the present. As I stated it's been a while since I did Algebra. I'm a little rusty with it but once I was able to get going it was coming back to me.
I will say that I started out with the Introductory Algebra course. I had lots of trouble remembering how to do Integers. I followed the steps of clicking to "see the solutions" for the problems but I kept drawing a blank. My husband saw me getting frustrated and asked if he could help me. Just like when I first met him he explained step by step how it was done. After that I was on a roll. Seeing the way the program is set up I don't believe this will work for my son when he starts Algebra. He has a learning disability and this won't fit in with his style of learning. It did not go with my style of learning either because I required more help than the program was set up to give. In my opinion I would probably recommend this program to those who just struggle somewhat with math, but I would not recommend this to someone with students that have various learning disabilities. All I can say is try out the free version first to see if it works well for your student then decide whether or not to buy from there. This is just my opinions and experience. To hear what others thought of it check out other Math911 reviews at the crew blog.
Disclaimer: The views and opinions in this review are entirely my own. As a member of the Schoolhouse Review Crew I was not paid to write this but received the Premier Version of "Math911" at no cost to me for review purposes, and to give my honest opinion. | 677.169 | 1 |
STAAR ALGEBRA 1 EOC Reporting Category 4 TEST PREP
Compressed Zip File
Be sure that you have an application to open this file type before downloading and/or purchasing. How to unzip files.
9.47 MB | 91 pages
PRODUCT DESCRIPTION
Task Cards & Around the Room with QR Codes Activities. This standardized test prep is specifically aligned with the STAAR Algebra 1 End of Course Exam but can also be used to review for most state Algebra 1 EOC exams as content standards across the U.S. are similar.
Readiness Standards: A.6(A) determine the domain and range of quadratic functions and represent the domain and range using inequalities A.7(A) graph quadratic functions on the coordinate plane and use the graph to identify key attributes, if possible, including x-intercept, y-intercept, zeros, maximum value, minimum values, vertex, and the equation of the axis of symmetry A.7(C) determine the effects on the graph of the parent function f(x) = x2 when f(x) is replaced by af(x), f(x) + d, f(x – c), f(bx) for specific values o f a, b, c, and d A.8(A) solve quadratic equations having real solutions by factoring, taking square roots, completing the square, and applying the quadratic formula
Supporting Standards: A.6(B) write equations of quadratic functions given the vertex and another point on the graph, write the equation in vertex form (f(x) = a(x – h)2 + k), and rewrite the equation from vertex form to standard form (f(x) = ax2 + bx + c) A.6(C) write quadratic functions when given real solutions and graphs of their related equations A.7(B) describe the relationship between the linear factors of quadratic expressions and the zeros of their associated quadratic functions A.8(B) write, using technology, quadratic functions that provide a reasonable fit to data to estimate solutions and make predictions for real-world problems
Each problem is coded with the specific TEKS alignment for easy lesson planning.
Includes two formats to accommodate various teaching styles and individual student | 677.169 | 1 |
A place to share my interests in the Introduction to Proofs course and to gather information about free resources for this course.
Monday, August 5, 2013
Writing ProofsAt the risk of oversimplification, doing mathematics has two
distinct stages. The first stage is to
convince yourself that you have solved the problem or proved a conjecture. This stage is a creative one and is quite
often how mathematics is actually done.
The second equally important stage is to convince other people that you
have solved the problem or proved the conjecture. This second stage often has little in common
with the first stage in the sense that it does not really communicate the
process by which you solved the problem or proved the conjecture. However, it is an important part of the
process of communicating mathematical results to a wider audience.
A mathematical proof is a convincing argument (within
the accepted standards of the mathematical community) that a certain
mathematical statement is necessarily true.
A proof generally uses
deductive reasoning and logic but also contains some amount of ordinary
language (such as English). A
mathematical proof that you write should demonstrate that you have gained a
deep understanding of the mathematical concepts involved and should convince an
appropriate audience that the result you are proving is in fact true. It should be clear from this that we consider writing to be an important part of the mathematical process. This is one reason the title of our introduction to proofs course is "Communicating in Mathematics." Issues dealing with the writing of mathematical proofs are addressed throughout the course. This was also one of the reasons that I wrote the book Mathematical Reasoning: Writing and Proof. As I search the Internet now, it is great to find many documents dealing with the writing of proofs in mathematics course. The Department of Mathematics at GVSU has also developed such a document, which can be accessed at this link. Because of such documents, I am sure that having a textbook dealing with these issues is not as important as it used to be. However, I still like the idea of having a text that supports what I try to have my students learn. Feel free to make comments describing what you feel should be part of the writing guidelines for the introduction to proofs course. One nice things about having an open-access text is that I can include revisions much more quickly than I could before. | 677.169 | 1 |
Compressed Zip File
Be sure that you have an application to open this file type before downloading and/or purchasing. How to unzip files.
4.84 MB | 3 pages
PRODUCT DESCRIPTION
Help your students visualize the solving process for one-variable, multi-step equations. Students are able to easily lead themselves or others through a problem, even if it is their first time using this method. Four versions are included: Full-sheet and half-sheet sizes with just the flowchart, a full-sheet size that also includes a section for students to revisit their notes and rework the problems to quiz themselves AND (my favorite!) a booklet version. I use the "Quiz Yourself" and booklet versions in my own interactive notebooks--they're both designed to fold up and fit perfectly | 677.169 | 1 |
solve math problems online
Category: math solverFactoring trinomials speak to the begin of the center of any algebra session. It is not so much the underlying begins of variable based math however it is the entryway to accomplishment in algebra. It denotes the start of the inside and out points. This is the reason uncommon consideration ought to be given in showing this point. Now understudies begin to settle on a choice whether they like variable based math and might want to go further into pre-analytics and after that math and after that might have a profession in building or science. Then again students may abhor algebra and arithmetic and choose to have a vocation in expressions. They pick expressions not because they like it but rather because they detest science.
The math instructor ought to utilize all his expertise in showing factoring trinomials. He ought to put extraordinary exertion in making the students comprehend the ideas and in making the subject energizing and enjoyable to understudies.
Factoring Trinomials
As a matter of first importance, the instructor needs to ensure that the students got the essentials ideal for this theme. For instance before getting into calculating the educator needs to ensure that the students comprehend and can effectively utilize the distributive property as well as factoring calculator. Furthermore, they ought to be acquainted with systems for increasing polynomials. This is on account of calculating is the same procedure in converse. Factoring transforms expansion into increase and these systems tern duplication into expansion.
The educator ought to experience the diverse factoring techniques with illustrations, including using factoring calculator. He ought, to begin with, the best basic element technique and demonstrate that it must be connected first to any algebra. Next, he ought to clarify the gathering strategy then he can swing to the distinction of two squares technique.
A few students are effective when they chip away at individual sorts of calculating yet they keep running into problems when distinctive sorts of factoring show up in one problem. This is the reason after clarifying the calculating procedures there ought to be a session particularly to disclose when to utilize every kind of factoring.
The primary thing to do is to record the trinomial and decide the components of the first and last terms. Next one ought to make a table utilizing all component mixes of the first and last terms. The table ought to have two sections, one for the primary term, and one for the last term. All variable blends of the polynomial's first term ought to be in section the whole gang component mixes of the polynomial's last term ought to be in segment two. At long last, the students ought to utilize experimentation with blends from segment one against section two till they get the craved components.
The instructor ought to give the student's word problems of genuine utilizations of factoring where the understudy can acknowledge how critical this is in our certifiable. The students can utilize the mini-computers in these problems to calculate trinomials using factoring calculator. This is to add energy to the class and to make understudies calm with the utilization of cutting edge advancements.
It is watched that understudies confront the most troubles while doing their math homework. Now and again understanding the issue itself is a test and on different events, the precarious logarithmic, trigonometric or analytics systems make it hard to get the answer effectively. The greater part of the times, every one of the understudies is discovered asking stand out an inquiry and that, "Would you help me with my math homework please?" Usually the companions, seniors or senior kin are the ones who are put to this test and not every one of them figures out how to pass it. People typically get stuck when you make this inquiry. If you are looking for a reliable and efficient math calculator, then I would suggest you try which is the best online platform to help people regarding their complex match problems.
Math Calculator
One time or the other we as a whole are discovered asking for somebody, "help me with my math homework", and now and again we don't get help. In this cutting edge age, the fastest and the most productive approach to getting help in any matter is the web, even the match homework. You can simply discover help on the web. You may not discover answers for the careful issue, but rather you can discover comparable inquiries illuminated and accessible on the web. Additionally, there are different understudy gatherings where different qualifies understudies would loan you their time and exertion and help with finishing your precarious math homework. The snappier and maybe a more proficient approach to get help is to utilize the mechanized arrangement generators.
There is different programming or online systems accessible on the sites which can be utilized to take care of scientific issues. The sites are very useful. You can confirm your answers or get complete arrangements; every progression is determined. An efficient math calculatordoes not only solve the problem, but it also offers step by step guide. These arrangements are as tenable as some other PC supported configuration. Any edge for human blunder is killed because these sites deal with widely outlined codes which have recipes as inherent, and the operations are precisely completed. The help can be general or determined it could be said that a few sites may have practical experience in polynomial math and trigonometry just, some may cook just rationale based issues, or the broad bases may give backing to all sub-disciplines including the measurable branches. Some of these aides represent considerable authority in issues just while others might be useful for charts and organizations too. You can get help with essential math, general math, polynomial math, geometry, analytics, plots and so forth. You should simply present a question, and the processor creates the complete arrangement in a matter of minutes rather seconds.
In this age, you don't have to circumvent asking everybody, "Would you be able to help me with my math homework?" All you need is a working web association. You can discover a few sources and sites inside seconds which would either furnish you with as of now assembled answers for your issues or comparative ones, or you can discover online applications which are redone to produce results continuously. You can answer to any numerical inquiry and have the regulated system also.
What is the hardest math problem to explain? The hardest thing to fathom, when it comes to getting the answer my math problem, is the one you can't settle in light of the fact that the arrangement hasn't introduced itself yet. In the endless universe of mathematics, numbers are the most imperative characters. We experience a lot of mathematical statements and recipes.
There are ideas that are exceptionally troublesome, and understudies need to battle to comprehend them. Math is one of the hardest subjects in school and it comprises of different themes that need full consideration and devotion. Taking care of math problems, for instance, is extremely dreary however if you know how to comprehend it regulated, you would discover it less demanding. There can take care of hard math problems if you know the methodologies and procedures to work out these testing problems in math.
Peruse the Problem
It is crucial to peruse the before first before whatever else particularly in noting word problems. You should choose the data that is essential for you. Attempt to separate the problem in understanding it over and over. Dissect it and attempt to break them into pieces to get the principle thought. Keep in mind that on the off chance that you are not certain with what you have to take out, break them sentence by sentence if necessary. Perusing the how to answer my math problem guidelines precisely and more than once is additionally vital since it is an instrument in discovering the answer for the problem.
Online Math Problem Solver
Apply the Formulas
It is important that you are acquainted with math formula to have the capacity to get the answer my math problem. Review the math formulas that are material in the problem. On the off chance that there are no recipes, attempt to consider comparative sample of the problem and discover the progressions in utilizing to tackle the problem or mathematical statement. Numerous math problems are like explain yet with various variables.
With Extra Special Care
Perform the operation one by one. No compelling reason to hustle the length of your answer is correct. Make your recipe and case problem an aide and complete every progression until you achieve the arrangement. A most mistake just happen at one stage, and that cause a wrong number to go through to the end; this ordinarily happens with long and complex problems. If you get stuck, backpedal to the past procedure until you answer my math problem with the formula that you feel is correct.
Check Your Answers
Check your work before hopping to another comparison. It is fundamental that you check your recipes together with plans since one oversight makes the rest off-base. In math, it is essential that you are extremely watchful in settling. In doing home work, discover an answer key if conceivable. Contrast your answer and your colleagues' works or let your different kin survey your work.
Math problems are unavoidable. Indeed, even, in actuality, circumstances, we experience the majority of them. They fluctuate contingent upon what sort of math mathematical statement or problem you are doing. Procedures are imperative since they assist understudies in coming up with answer my math problems. Obviously, you require a great deal of tolerance and commitments keeping in mind the end goal to tackle them. Like, in actuality, there are numerous questions yet knowing how to manage those variables is the thing that the best individuals on the planet do.
On the off chance that your child asks you in helping these algebra home works, and you can't do this home works, or you do not think about them where you have not done algebra in your secondary school days. This sort of circumstance is so rushed and with the assistance of some fabulous algebra, problem solver helps and your kids are all around arranged for coming up test. Right now, the web will resolve your issue; you can locate an extensive variety of polynomial math worksheets and some different instruments on the web, which helps the troublesome learning process. Be that as it may, these variable based math devices are an awesome approach to enhance your math expertise and some practice will give more focal points in up and coming math test. These worksheets contain a great many issues and mathematical statements where you can test yourself. What's more, you can discover an answer key for every one of these issues in that site.
Also, you can discover an algebra problem solver instrument which offers you some assistance with solving some troublesome algebra based math mathematical statements and this variable based math adding machine is the ideal answer you are searching for. These adding machines will help you when you when you are stuck on an issue and not able to discover the answer. These online adding machines will give some will give you point by point data and a gritty clarification of the algebra problem in an orderly process. You can locate an extensive variety of number crunchers in the on the web, which are utilizing different strategies to take care of these algebra problems. What's more, a percentage of the number cruncher programming will help you in using some different strategies, which help you in fathoming algebra problems. Furthermore, you can locate some other charting number crunchers that plot questions. This sort of algebra problem solvers will permit you to explain graphical algebra questions.
Aside from this, you can locate some other supplemental apparatus on the web is the famous variable based math solver. At the point when in correlation with adding a machine, the usefulness is the same, and this product project will give answers to hardest algebra problems too. All that you have to enter the algebra problem and the product will do whatever is left of the things. This product will give an online algebra problem solver when your kids need, and it will spare enormous cash on utilizing a mentor.
Algebra Problem Solver
Furthermore, now an algebra problem ascended in your psyche that, how to discover the algebra problem solver. So as to discover these online algebra problem solvers, you have to do a little pursuit in the web by suing some catchphrases relying upon your requirements. In prior days, this variable based math is a creature for each tyke, however with the assistance of these algebra problem solvers, they can learn polynomial math much speedier. Furthermore, the kind will appreciate the importance of having an algebra problem solver. | 677.169 | 1 |
Excel Mathematics Study 9 -10 Author: Allyn Jones
ISBN: 9781741254792 Format: Paperback Published: 30 October 2014 Country of Publication: AU Description: The Excel Mathematics Study Guide Years 9-10 is essential for all students who want to Excel at Maths. This comprehensive book both supports and challenges students by providing tools to explain important mathematical concepts and giving ample opportunity to practise at a variety of levels.
This book contains: Explanations of important concepts with Examples and Worked Solutions Checklists at the end of each chapter with page references to explanations Plenty of Exercises in the Practise Practise sections at the end of each chapter with page references to explanations; Answers and Worked Solutions are provided Two levels of timed Tests (Intermediate and Advanced) at the end of each chapter; Worked Solutions and a marking system that gives feedback are provided Two levels of timed Sample Exam Papers that cover the work for Year 9 and 10 Mathematics and provide preparation for final examinations; Worked Solutions are provided A comprehensive Index | 677.169 | 1 |
Fourier Analysis
Book Description
Fourier analysis is a subject that was born in physics but grew up in mathematics. Now it is part of the standard repertoire for mathematicians, physicists and engineers. In most books, this diversity of interest is often ignored, but here Dr Körner has provided a shop-window for some of the ideas, techniques and elegant results of Fourier analysis, and for their applications. These range from number theory, numerical analysis, control theory and statistics, to earth science, astronomy, and electrical engineering. Each application is placed in perspective by a short essay. The prerequisites are few (the reader with knowledge of second or third year undergraduate mathematics should have no difficulty following the text), and the style is lively and entertaining. In short, this stimulating account will be welcomed by all who like to read about more than the bare bones of a subject. For them this will be a meaty guide to Fourier analysis. | 677.169 | 1 |
Thanks for your interest in g(Math)! But, before installing, take a look at our new Chrome Extension, EquatIO, which takes g(Math) to the next level. Here is a link to check out EquatIO:
If after checking out EquatIO you still want to install g(Math), that's great! But, just a head's up, we'll eventually be retiring g(Math). So, we recommend making the switch to EquatIO early so that you can start exploring the future of digital math.
--
Now students at all levels - and their teachers - can include mathematical formulas and equations, graphs and statistical displays with this friendly Add-on for Google Docs, Sheets and Forms.
Write virtually any mathematical expression directly on your PC or Chromebook's keyboard or touchscreen. Input's easy: you don't need to know any math code or programming languages. g(Math) understands what you're typing, instantly turning your expressions into clear, accurate on-screen formulas.
You can save even more time by picking from a huge range of ready-made formulas and functions, including quadratic equations, matrices and more. Then insert your expression into any spreadsheet cell or doc with a simple click.
If you'd prefer to dictate or handwrite your equations or formulas, there is a Speech Input function and a Handwriting Entry tool within the Add-on. With these features, you can cater the tool to your unique learning style.
It's just as easy and rewarding to create clear, colourful graphs, from simple linear relationships to complex trigonometric expressions. Type or pick a function, choose your axes and zoom range - then click to insert.
g(Math) can create graphs from data in spreadsheet cells, or plot directly from a list of points you enter. And it can instantly visualise statistical data as histograms, bar charts and box-and-whisker plots, too.
g(Math) is ideal for students of all ages and attainment levels. Through integration with Texthelp's Read&Write literacy software, math expressions created in g(Math) can now be read aloud, just like any other content on the page.
Hi Christoph, we use the registration to give us the ability to contact you with updates and solicit feedback that is pertinent to the level you teach. If the registration bothers you, you can sign up as a student and we collect no information and the registration skips to to the end.
PROS: The equation editor portion works well and is fairly intuitive for relatively basic mathematical expressions (I haven't yet tried more involved symbols like sigma notation or definite integrals). Similarly the graph creator is simple and easy to use when plotting y as a function of x.
CONS: There seems to be little control what the equation will look like when it is inserted into the document. Often my equations are stretched and skewed such that the text is distorted and/or the size is either very large or very small. Deleting the equation and reinserting sometimes corrects the problem. In the graphing feature, there does not appear to be a way to plot x as a function of y, or to plot things that are not functions at all (like circles and other conic sections).
The built in google-doc equation editor is sorely lacking, being able to use latex with gMath is welcome, but I can't for the life of me figure out how to get the expressions to simply use the font size of my document so that I don't have to manually and imprecisely scale down my equation by hand for each new one I introduce to my document. I'm thinking of entirely ditching google docs and redoing my paper in MSWord - or switching to native LaTeX ... anyway, I may be missing it, but it's certainly not clear how to do this. Eg. 1+2+3+\ldots+n=\frac{n(n+1)}{2} appears giant in my doc compared to my doc's pt size of 11.
Hi James, we are working on the next update that will allow for better formatting of math within Docs. You could use \text{your text here} as a bad workaround currently or just type \ to create a text area in the math. If you share a Doc with me, I can help with that formatting. | 677.169 | 1 |
NEW - FREE SHIPPING
This title is In Stock in the Booktopia Distribution Centre. We will send you a confirmation email with a Tracking Code to follow the progress of your parcel when it ships.
Pearson Illustrated Maths Dictionary Author: Judith De Klerk
ISBN: 9781486009831 Format: Paperback Number Of Pages: 0 Published: 9 December 2013 Country of Publication: AU Description: The Pearson Illustrated Maths Dictionary 5th edition is a revised and improved edition of Australia's best-selling mathematics dictionary incorporating all the mathematical terms in the Australian Curriculum glossary.
The Pearson Illustrated Maths Dictionary 5th edition is an essential resource for primary and secondary students up to Year 10, teachers, student teachers and parents.
Features and Benefits: Compatible with the Australian Curriculum New, more engaging design Over 80 new entries to cater for students up to year 10 Photographs replacing some illustrations where appropriate Theme colours for letters and matching colour tabs for improved navigation An index at the back of the book Accurate definitions written in clear and simple language suitable for primary and secondary students Clear, precise and concise explanations of complex terms Examples and illustrations to support each definition A useful information section providing symbols, formulas and other supporting mathematical terms | 677.169 | 1 |
A student who has picked up basic methods of calculation quickly and is curious about why mathematics works will find this course a natural extension of personal talents and interests. Focus will be on approaches to problem solving, understanding mathematical algorithms, through preparation for algebra, and deep topics of interest to young learners.
On Saturdays during the summer, the two half-terms (the four-week July module or the three-week August module) of the 11:00am section will be the "year 1" (Math I) sections, covering basic principles of statistics during the July half-term and basic combinatorics (counting) and probability in the August half-term. Students may take either the July half-term, the August half-term, or both whether they are new to the ECAE course (as some students will be) or alumni of the school-year ECAE course (as some students will be). Problems for the Math I summer course come not just from the assigned textbook but also from other sources around the world, especially for the fourth week of the four week July half-term Problem sources are not limited to the course textbook but include problems from many other sources, including problems originally written in Russia, China, or India.
On Saturdays during the two summer half-terms (the four-week July half-term or the three-week August half-term), the 9:00am section will be the "year 2" (Math II) section, covering general problem-solving techniques such as finding patterns in problem situations, listing information as a problem-solving technique, diagramming to focus on essential details of a problem, and working backwards to generate possible solutions. Problems for the Math II summer course come not just from the assigned textbook but also from other sources around the world, especially for the fourth week of the four-week July half-term. Except for the supplementary material in the fourth week of the July half term, the year 2 half-terms for Math II in summer 2016 cover the same topics
Placement Test Instructions for Math I & II: The textbook publisher has an online placement test that shows what level is expected for starting the textbook, that is starting Year 1 of the two-year Prealgebra and Advanced Topics (PAT) course. My Course Placement Test illustrates many of the topics that are covered during ECAE's Math Year 1 class. The Year 2 section of the course assumes most of that content as background. Students can join either the Year 1 or the Year 2 section in the middle of the school year if they are ready.The Course Placement Test prints out as six pages. Calculations are to be done by thinking, or by pencil and paper, without any use of an electronic calculator, abacus, or slide rule. Encourage your child to do as much work, and to show as many steps, on the Course Placement Test as can be done in one hour, skipping ahead to problems farther along in the test if that helps complete more questions. The last question is also important. Once your child is done, please send the completed test by scanning-and-emailing to the instructor.
Prerequisite: The physics presented in this class requires pre-algebra level mathematics skills and some basic algebra. Students also need the fine motor skills and maturity to work safely with an X-acto knife.
In this class we will use model rocketry to explore the following scientific concepts: kinematics, vectors, projectile motion, Newton's laws of motion, chemistry of combustion reactions, aerodynamics and stability, and even some basic trigonometry. The physics presented in this class uses geometry to convey the fundamental ideas of calculus in order for the students to understand, not only the equations for motion, but also how they are derived. In addition, each student will build and fly a model rocket kit, and design their own model rocket from scratch. In the process they will also learn about the history of rocketry and rules to ensure a safe flight. Please note that Saturday, August 13 is a regularly-scheduled rocket launch sponsored by Tripoli Minnesota High Powered Rocket Club. Their launches are held on the second Saturday of each month near North Branch, MN. Specific directions will be given in class.
Instructor Background: Jolene Gleason has a M.S. degree in electrical engineering as well as B.S. degrees in electrical engineering and mathematics from Iowa State University. She was named ISU's Outstanding Senior in Electrical Engineering in 1987. She worked as an electrical engineer for 15 years before becoming a full-time homeschool parent. Early in her career she also taught Power Systems Analysis and Technical Calculus at Bradley University in Peoria, Illinois. Jolene's appreciation of rocket science has been fostered by her son, Scott, who is an avid rocketeer. At age 17, Scott is the real expert in rocket design and construction. He received Junior Achievement awards from the Tripoli Minnesota High Powered Rocketry Club for 2008 and 2009. | 677.169 | 1 |
Synopsis
Complete Mathematics for Cambridge Secondary 1 Homework Book 1 (Pack of 15) For Cambridge Checkpoint and Beyond by Sue Pemberton
Written by experienced examiners, this fully comprehensive course completely covers the revised Cambridge Secondary 1 curriculum, and these supportive Homework Books guarantee students get the practice they need to achieve their top potential. The sheer volume of practice will ensure all the concepts become second nature, while the focus on stretch and challenge will lay the best possible foundations for Cambridge IGCSE. Complete Mathematics is the new name for Oxford International Maths.
Reviews
Our middle school teachers have been very happy with the format and are looking forward to using this resource next academic year. Some of the things they like about it are that it does have more practice problems than our present resource and that all the topics explored at each level are in a single book which makes it easier for students to manage. Phil Bennett, Academic Leader Mathematics, International School of Luxembourg These are the best Maths resources for this level that I have ever taught from. We love the homework books which offer extra practice to consolidate core content. Carolyn Hatton, Head of Mathematics, King Richard III College, Spain | 677.169 | 1 |
3 2. INTRODUCTION 2.a COURSE CONTENT : MAC1140, Precalculus Algebra is a review of college algebra designed to prepare students for calculus. A minimum grade of C in MAC1140 satisfies three hours of the general education pure math requirement and also satisfies the math portion of the state Writing/Math requirement. Students who successfully complete this course (C or better) can directly advance to MAC2233, Survey of Calculus. For students preparing for MAC2311, Analytical Geometry and Calculus 1 , MAC1140 should be followed by MAC1114 Precalculus Trigonometry. The sequence of both MAC1140 and MAC1114 covers the same material and uses the same text as the one semester, faster paced course, MAC1147, Precalculus Algebra and Trigonometry. If you have already received credit for MAC1147, you cannot earn credit for MAC1140 or MAC1114 again. Students taking this course for general education pure math credit or the math portion of the
This
preview
has intentionally blurred sections.
Sign up to view the full version. | 677.169 | 1 |
Module 6: Summation | 677.169 | 1 |
MYTUTOR SUBJECT ANSWERS
Answers /
Maths /
IB /
What are the key elements to include in your Math assignment?
514 views
What are the key elements to include in your Math assignment?
1) Enthusiasm. The examiners want to see your interests and passion in Math exploration in your everyday life. So, do research, get inspired and look for mathematic application in your interests before you decide your topic.
2) logical and concise organazation. Make sure your exploration is in chronological order, everything clearly labled and easy to understand. Yes, logical reasoning is the key in Mathematics.
3) self- reflection. Remember your whole assignment is a self exploration, therefore you need to reflext on what you find out and learn through your journey. Basically, include everything you feel and think about your experiementaions in your exploration if relevant.
"Hello there!Who am I?I am currently an Architecture undergraduate student at the University of Edinburgh. I have just completed the International Baccalaureate Diploma last summer and that for me was an unforgettable experience in | 677.169 | 1 |
English Vocabulary for Mathematics
Free ESL in Canada English lessons for international students to study Mathematics in
Canada or USA during an exchange program. Mathematics vocabulary is necessary for exchange students to
succeed during an exchange program in the USA or Canada. Other grammar topics include vocabulary, parts
of speech, sentence structure, punctuation, tenses, verbals, conditionals and writing.
Algebraic expression. One or more variables and possibly numbers and
operation symbols. For example, 3x + 6, x, and 5x are algebraic expressions.
Algorithm. A systematic procedure for carrying out a computation. For example,
the addition algorithm is a set of rules for finding the sum of two or more
numbers.
Alternate angles. Two angles on opposite sides of a transversal when
it crosses two lines. The angles are equal when the lines are parallel. The
angles form one of these patterns
Analog clock. A timepiece that indicates the time through the position of
its hands.
Attribute. A quantitative or qualitative characteristic of an object or a
shape, for example, colour, size, thickness. bar graph. See under graph.
Bias. An emphasis on characteristics that are not typical of an
entire population.
Bisector. A line that divides a segment, an angle, a line, or a
figure into two equal halves.
Calculation method. Any of a variety of methods used for solving
problems, for example, estimation, mental calculation, pencil-and-paper
computation, the use of technology (including calculators, computer
spreadsheets).
Capacity. The greatest amount that a container can hold; usually
measured in litres or millilitres.
Census. The counting of an entire population.
Coefficient. Part of a term. In a term, the numerical factor is the
numerical coefficient, and the variable factor is the variable coefficient. For
example, in 5y, 5 is the numerical coefficient and y is the variable
coefficient.
Complementary angles. Two angles whose sum is 90º.
Composite number. A number that has factors other than itself and 1.
For example, the number 8 has four factors: 1, 2, 4, and 8.
Computer spreadsheet. Software that helps to organize information
using rows and columns.
Concrete materials. Objects that students handle and use in
constructing their own understanding of mathematical concepts and skills and in
illustrating that understanding. Some examples are base ten blocks, centicubes,
construction kits, dice, games, geoboards, geometric solids, hundreds charts,
measuring tapes, Miras, number lines, pattern blocks, spinners, and tiles. Also
called manipulatives.
Cone. A three-dimensional figure with a circular base and a curved
surface that tapers proportionately to an apex.
Congruent figures. Geometric figures that have the same size and
shape.
Conservation. The property by which something remains the same
despite changes such as physical arrangement.
Coordinate plane. A plane that contains an X-axis (horizontal) and a
Y-axis (vertical). Also called Cartesian coordinate grid or Cartesian plane.
Coordinates. An ordered pair used to describe a location on a grid or plane.
For example, the coordinates (3, 5) describe a location on a grid found by
moving 3 units horizontally from the origin (0, 0) followed by 5 units
vertically.
Data. Facts or information.
Database. An organized and sorted list of facts or information;
usually generated by a computer.
Degree. A unit for measuring angles.
Dependent variable. A variable that changes as a result of a change
in the independent variable.
Diameter. A line segment that joins two points on the circumference of
a circle and passes through the centre.
Displacement. The amount of fluid displaced by an object placed in it.
Distribution. A classification or an arrangement of statistical
information.
Equation. A mathematical statement that has equivalent terms on either
side of the equal sign.
Equivalent fractions. Fractions that represent the same part of a
whole or group, for example, 1/3 , 2/6, 3/9, 4/12.
Equivalent ratios. Ratios that represent the same fractional number
or amount, for example, 1:3, 2:6, 3:9.
Estimation strategies. Mental mathematics strategies used to obtain
an approximate answer. Students estimate when an exact answer is not required
and estimate to check the reasonableness of their mathematics work. Some
estimation strategies are: clustering. A strategy used for estimating the sum
of numbers that cluster around one particular value. For example, the numbers
42, 47, 56, 55 cluster around 50. So estimate 50 + 50 + 50 + 50 = 200.
Rounding. A process of replacing a number by an approximate value of
that number. For example, rounding to the nearest tens for 106 is 110.
Event. One of several independent probabilities.
Experimental probability. The chance of an event occurring based on
the results of an experiment.
Exponential form. A shorthand method for writing repeated
multiplication. In 53, 3, which is the exponent, indicates that 5 is to be
multiplied by itself three times. 53 is in exponential form.
Expression. A combination of numbers and variables without an equal
sign, for example, 3x + 5.
Formula. A set of ideas, words, symbols, figures, characters, or
principles used to state a general rule. For example, the formula for the area
of a rectangle is A = l x w.
Frequency. The number of times an event or item occurs.
Frequency distribution. A table or graph that shows how often each
score, event, or measurement occurred.
Graph. A representation of data in a pictorial form. Some types
of graphs are:
Bar graph. A diagram consisting of horizontal or vertical bars that
represent data. Broken-line graph. On a coordinate grid, a display of data
formed by line segments that join points representing data.
Circle graph. A graph in which a circle used to represent a whole is
divided into parts that represent parts of the whole.
Comparative bar graph. A graph consisting of two or more bar graphs placed
side by side to compare the same thing. Also called double bar graph.
concrete graph. A graph in which real objects are used to represent pieces of
information.
Coordinate graph. A grid that has data points named as ordered pairs of
numbers, for example, (4, 3).
Histogram. A type of bar graph in which each bar represents a range of
values, and the data are continuous. pictograph. A graph that illustrates
data using pictures and symbols.
Improper fraction. A fraction whose numerator is greater than its
denominator, for example, 12/5.
Independent events. Two or more events for which the occurrence or
non-occurrence of one does not change the probability of the other.
Independent variable. A variable that does not depend on another for
its value; a variable that the experimenter purposely changes. Also called cause
variable.
Inequality. A statement using symbols to show that one expression is
greater than (>), less than (<), or not equal to another expression.
Line of best fit. A line that can sometimes be determined on a
scatter plot. If a line of best fit can be found, a relationship exists between
the independent and dependent variables.
Line of symmetry. A line that divides a shape into two parts that can
be matched by folding the shape in half.
Many-to-one correspondence. The matching of elements in two sets in
such a way that more than one element in one set can be matched with one and
only one element in another set, for example, 3 pennies to each pocket.
Mass. The amount of matter in an object; usually measured in grams or
kilograms.
Mathematical communication. The use of mathematical language by
students to: respond to and describe the world around them; communicate their
attitudes about and interests in mathematics; reflect and shape their
understandings of and skills in mathematics. Students communicate by talking,
drawing pictures, drawing diagrams, writing journals, charting, dramatizing,
building with concrete materials, and using symbolic language, (e.g., 2, >,
=).
Mathematical procedures. The skills, operations, mechanics,
manipulations, and calculations that a student uses to solve problems.
Mean. The average; the sum of a set of numbers divided by the number
of numbers in the set. For example, the average of 10 + 20 + 30 is 60 ÷ 3 =
20.
Measure of central tendency. A value that can represent a set of
data, for example, mean, median, mode. Also called central measure.
Median. The middle number in a set of numbers, such that half the
numbers in the set are less and half are greater when the numbers are arranged
in order. For example, 14 is the median for the set of numbers 7, 9, 14, 21, 39.
If there is an even number of numbers, the median is the mean of the two middle
numbers. For example, 11 is the median of 5, 10, 12, and 28.
Mira. A transparent mirror used in geometry to locate reflection
lines, reflection images, and lines of symmetry, and to determine congruency and
line symmetry.
Mixed number. A number that is the sum of a whole number and a
fraction, for example, 81/4.
Mode. The number that occurs most often in a set of data. For example,
in a set of data with the values 3, 5, 6, 5, 6, 5, 4, 5, the mode is 5.
Modeling. A representation of the facts and factors of, and the
inferences to be drawn from, an entity or a situation.
Monomial. An algebraic expression with one term, for example, 2x or
5xy2.
Multiple. The product of a given number and a whole number. For
example, 4, 8, 12, . . . are multiples of 4.
Multiplication. An operation that combines numbers called factors to
give one number called a product. For example, 4 x 5 = 20; thus factor x factor
= product.
Multi-step problem. A problem whose solution requires at least two
calculations. For example, shoppers who want to find out how much money they
have left after a purchase follow these steps:
Step 1 - Add all items purchased (subtotal). Step 2 - Multiply the sum
of purchases by % of tax.
Step 3 - Add the tax to the sum of purchases (grand total).
Step 4 - Subtract the grand total from the shopper's original amount of
money.
Natural numbers. The counting numbers 1, 2, 3, 4, . . .
Net. A pattern that can be folded to make a three-dimensional figure.
Network. A set of vertices joined by paths.
Non-standard units. Measurement units used in the early development
of measurement concepts, for example, paper clips, cubes, hand spans, and so on.
Number line. A line that matches a set of numbers and a set of points one to
one.
Number operations. Mathematical processes or actions that include the
addition, subtraction, multiplication, and division of numbers.
Nth term. The last of a series of terms.
Obtuse angle. An angle that measures more than 90º and less than
180º.
One-to-one correspondence. The matching of elements in two sets in
such a way that every element in one set can be matched with one and only one
element in another set.
Ordered pair. Two numbers in order, for example, (2, 6). On a
coordinate plane, the first number is the horizontal coordinate of a point, and
the second is the vertical coordinate of the point.
Order of operations. The rules used to simplify expressions. Often
the acronym BEDMAS is used to describe this calculation process:
Brackets exponents
division or multiplication, whichever comes first addition or
subtraction, whichever comes first
Ordinal number. A number that shows relative position or place, for
example, first, second, third, fourth.
Parallel lines. Lines in the same plane that do not intersect.
Parallelogram. A quadrilateral whose opposite sides are parallel.
Perfect square. The product of an integer multiplied by itself. For
example, 9 = 3 x 3; thus 9 is a perfect square.
Perpendicular lines. Two lines that intersect at a 90º angle.
Place value. The value given to the place in which a digit appears in
a numeral. In the number 5473, 5 is in the thousands place, 4 is in the hundreds
place, 7 is in the tens place, and 3 is in the ones place.
Plane shape. A two-dimensional figure.
Polygon. A closed figure formed by three or more line segments.
Examples of polygons are triangles, quadrilaterals, pentagons, octagons.
Primary data. Information that is collected directly or first-hand.
Data from a person-on-the-street survey are primary data. Also called first-hand
data or primary-source data.
Prime factorization. An expression showing a composite number as a
product of its prime factors. The prime factorization for 42 is 2 x 3 x 7.
Prime number. A whole number greater than 1 that has only two
factors, itself and 1. For example, 7 = 1 x 7.
Prism. A three-dimensional figure with two parallel and congruent
bases. A prism is named by the shape of its bases, for example, rectangular
prism, triangular prism.
Probability. A number that shows how likely it is that an event will
happen.
Proper fraction. A fraction whose numerator is smaller than its
denominator, for example, 2/3.
Proportion. A number sentence showing that two ratios are equal, for
example, 2/3 = 6/9.
Pythagorean theorem. In a right triangle, the square of the length of
the hypotenuse is equal to the sum of the squares of the other two sides.
Quadrilateral. A polygon with four straight sides.
Radius. A line segment whose endpoints are the centre of a circle and
a point on the circle.
Range. The difference between the highest and lowest number in a
group of numbers. For example, in a data set of 8, 32, 15, 10, the range is 24,
that is, 32 - 8.
Rate. A comparison of two numbers with different units, such as
kilometres and hours, for example, 100 km/h.
Ratio. A comparison of numbers with the same units, for example, 3:4
or 3/4. rational number. A number that can be expressed as the quotient of two
integers where the divisor is not 0.
Reflection. A transformation that turns a figure over an axis. The
figure does not change size or shape, but it does change position and
orientation. A reflection image is the result of a reflection. Also called flip.
Regular polygon. A closed figure in which all sides and angles are
equal.
Rotation. A transformation that turns a figure about a fixed point.
The figure does not change size or shape, but it does change position and
orientation. A rotation image is the result of a rotation. Also called turn.
Rotational symmetry. A shape that fits onto itself after a turn less
than a full turn has rotational symmetry. For example, a square has a turn
symmetry of order 4 because it resumes its original orientation after each of 4
turns: 1/4 turn, 1/2 turn, 3/4 turn, and full turn. Also called turn symmetry.
Sample. A small, representative group chosen from a population and
examined in order to make predictions about the population. Also called
sampling.
Scale drawing. A drawing in which the lengths are a reduction or an
enlargement of actual lengths.
Scalene triangle. A triangle with three sides of different lengths.
Scatter plot. A graph that attempts to show a relationship between
two variables by means of points plotted on a coordinate grid. Also called
scatter diagram.
Scientific notation. A way of writing a number as the product of a
number between 1 and 10 and a power of 10. In scientific notation, 58 000 000 is
written 5.8 x 107.
Secondary data. Information that is not collected first-hand, for
example, data from a government document or a database. Also called second-hand
data or secondary-source data.
Sequence. A succession of things that are connected in some way, for
example, the sequence of numbers 1, 1, 2, 3, 5, . . .
Seriation line. A line used for the ordering of objects, numbers, or
ideas.
Shell. A three-dimensional figure whose interior is completely empty.
SI. The international system of measurement units, for example,
centimetre, kilogram. (From the French Système International.)
Similar figures. Geometric figures that have the same shape but not
always the same size.
Simple interest. The formula used to calculate the interest on an
investment: I = PRT where P is the principal, R is the rate of interest, and T
is the time chosen to invest the principal.
Simulation. A probability experiment to test the likelihood of an
event. For example, tossing a coin is a simulation of whether the next person
you meet is a male or a female.
Skeleton. A three-dimensional figure showing only the edges and
vertices of the figure.
Standard form. A way of writing a number in which each digit has a
place value according to its position in relation to the other digits. For
example, 7856 is in standard form.
Supplementary angles. Two angles whose sum is 180º.
Surface area. The sum of the areas of the faces of a three-dimensional
object.
Survey. A sampling of information, such as that made by asking people
questions or interviewing them.
Symbol. See under mathematical language.
Systematic counting. A process used as a check so that no event or
outcome is counted twice.
Table. An orderly arrangement of facts set out for easy reference,
for example, an arrangement of numerical values in vertical or horizontal
columns.
Tally chart. A chart that uses tally marks to count data and record
frequencies.
Tangram. An ancient Chinese puzzle made from a square cut into seven
pieces: two large triangles, one medium-sized triangle, two small triangles, one
square, and one parallelogram.
Theoretical probability. The number of favorable outcomes divided by
the number of possible outcomes.
Transformation. A change in a figure that results in a different
position, orientation, or size. The transformations include the translation
(slide), reflection (flip), rotation (turn), and dilatation (reduction or
enlargement).
Translation. A transformation that moves a figure to a new position
in the same plane. The figure does not change size, shape, or orientation; it
only changes position. A translation image is the result of a translation. Also
called slide.
Trapezoid. A quadrilateral with exactly one pair of parallel sides.
Tree diagram. A branching diagram that shows all possible
combinations or outcomes.
Variable. A letter or symbol used to represent a number.
Venn diagram. A diagram consisting of overlapping circles used to show
what two or more sets have in common.
Vertex. The common endpoint of the two segments or lines of an angle.
Volume. The amount of space occupied by an object; measured in cubic
units such as cubic centimetres. | 677.169 | 1 |
Maths
Subject Summary
Leaving Certificate Mathematics aims to develop mathematical knowledge, skills and understanding needed for continuing education, life and work. By teaching mathematics in contexts that allow learners to see connections within mathematics, between mathematics and other subjects, and between mathematics and its applications to real life, it is envisaged that learners will develop a flexible, disciplined way of thinking and the enthusiasm to search for creative solutions.
The objectives of Leaving Certificate Mathematics is to:
Develop the ability to recall relevant mathematical facts
Instrumental understanding ("knowing how")
Necessary psychomotor skills (skills of physical coordination)
Relational understanding ("knowing why")
The ability to apply their mathematical knowledge and skill to solve problems in familiar and in unfamiliar contexts | 677.169 | 1 |
General Catalog & Schedule of Classes
DISCLAIMER: Future term data are continually updated. Check the web frequently for current information.
MTH 102.
ALGEBRAIC FOUNDATIONS
(3).
This course is designed primarily for EOP students. They will use various computing technologies to explore realistic and interesting situations in which algebra is used. As they work through explorations, they will work with many of the fundamental ideas of algebra, ideas they will find important in their daily lives. | 677.169 | 1 |
Concept Readiness Tests
The Concepts Readiness Tests were developed by an MAA task force with close attention to what research indicates to be critical conceptual understandings and skills needed for success in the target course. Concept readiness tests are available for Calculus and Algebra and Precalculus.
Tests are multiple choice. There are 4 parallel versions of each test.
Reasoning Strands for Concept Readiness Tests
Quantitative Reasoning involves identifying and relating measureable attributes
of an object or situation in a problem context
Proportional Reasoning involves thinking about how two quantities change such that their ratio remains constant; attending to how one variable changes so that it is always a constant multiple of another variable
Covariational Reasoning involves thinking about how two quantities in a
functional relationship are changing together; attending to how one variable changes while imagining successive amounts of equal changes in another variable. It involves coordinating two varying quantities that change in tandem while attending to how the quantities change in relation to each other.
Variable Reasoning involves associating a letter with a numeric value, and flexibly viewing that letter as representing varying values or an unknown value as determined by the context in which the letter is defined or used.
Functional Reasoning involves either thinking of a function as a process that
accepts input and produces output or making sense of symbols used in mathematical
expressions and giving meaning to the mathematical ideas communicated by
conventional notation.
Graphical Reasoning involves making sense of graphs that represent functions, and interpreting the meaning of attributes of a graph that convey aspects of a function's behavior.
Reasoning with Representations involves representing and interpreting a
relationship between numeric values or quantities using graphs, algebraic equations, numeric values, or verbal expressions and using that relationship to change
mathematical representations into equivalent representations that reveal desired information.
Computational Abilities refers to facility with manipulations and procedures needed to evaluate functions, solve equations, compose functions, and invert linear and exponential functions, within the context of algebraic representations. | 677.169 | 1 |
Outstanding Math Guide (OMG) 9th Grade Coordinate Algebra
PDF (Acrobat) Document File
Be sure that you have an application to open this file type before downloading and/or purchasing.
13.43 MB | 141 pages
PRODUCT DESCRIPTION
The Outstanding Math Guide (OMG) contains a complete set of differentiated reproducible graphic organizers that are aligned to Common Core standards. The graphic organizers can be used to introduce or review math concepts. The graphic organizers contain simple steps, examples and vocabulary of all key concepts covered in a year. Students cut and fold the graphic organizers and then glue them into their own hand held OMG. The student's OMG is an ongoing reference that puts a year's worth of curriculum right at their fingertips! The student's OMG is a standard 3 prong folder cut and folded to contain up to 23 graphic organizers. Students keep their OMG in their Math Notebook to use as a reference in classwork, homework or to study for tests. (To see a completed student's OMG click on the pictures.)
Each book contains pictures of a completed student OMG. A Common Core alignment chart showing the graphic organizer that covers each standard. A proven school wide implementation plan. Three differentiated black line masters for each graphic organizer. A vocabulary sheet black line master to record terms, definitions and examples. A parent letter to explain the OMG program to parents.
The following differentiated graphic organizers are included: Language of Algebra, Relationships between Quantities, Inequalities, Solving Equations, Solving Literal Equations, Systems of Equations, Exponential Functions, Function Families, Functions, Function Notation/Evaluating, Intercepts, Operations with Functions, Sequences: Arithmetic, Sequences: Geometric, Slope, Slope Intercept Form, Causation vs Correlation, Data, Displaying Data, Measures of Center, Shapes of Data, Two way frequency Tables, Transforming Functions, and Distance Formula.
The downloadable file (the complete book) does not include folding videos for the graphic organizers. To obtain the folding videos a hard copy of the book would need to be purchased. The hard copy of the book can be purchased at .40.00. | 677.169 | 1 |
Differentiating Instruction with Menus: Algebra I/II Differentiating Instruction With Menus: Algebra I/II offers high school math teachers everything needed to create a student-centered learning environment based on choice. This book uses six different types of menus that students can use to selectMore...
Book details
Differentiating Instruction With Menus: Algebra I/II offers high school math teachers everything needed to create a student-centered learning environment based on choice. This book uses six different types of menus that students can use to select exciting advanced level products that they will develop so teachers can assess what has been learned-instead of using a traditional worksheet format. Topics addressed include numbers, algebra basics, exponents, graphs, functions, polynomials, and various equations typically included in the algebra I/II curriculum. Differentiating Instruction With Menus: Algebra I/II contains attractive reproducible menus, each based on the levels of Bloom's Revised taxonomy as well as incorporating different learning styles. These menus can be used to guide students in making decisions as to which products they will develop after studying a major concept or unit | 677.169 | 1 |
Welcome to Calculus! Join me on this journey through one of the great triumphs of human thought.
2 vídeos, 2 leituras
WEEK 2
Functions and Limits
Functions are the main star of our journey. Calculus isn't numbers: it's relationships between things, and how one thing changing affects something else.
13 vídeos, 5 leituras
Nota atribuída: Functions and Limits
WEEK 3
The End of Limits
People have thought about infinity for thousands of years; limits provide one way to make such ponderings precise. Continuity makes precise the idea that small changes in the input don't affect the output much.
15 vídeos, 4 leituras
Nota atribuída: The End of Limits
WEEK 4
The Beginning of Derivatives
It is time to change topics, or rather, to study change itself! When we wiggle the input, the output value changes, and that ratio of output change to input change is the derivative.
13 vídeos, 4 leituras
Nota atribuída: The Beginning of Derivatives
WEEK 5
Techniques of Differentiation
With the product rule and the quotient rule, we can differentiate products and quotients. And since the derivative is a function, we can differentiate the derivative to get the second derivative.
14 vídeos, 5 leituras
Nota atribuída: Techniques of Differentiation
WEEK 6
Chain Rule
The chain rule lets us differentiate the composition of two functions. The chain rule can be used to compute the derivative of inverse functions, too.
13 vídeos, 5 leituras
Nota atribuída: Chain Rule
WEEK 7
Derivatives of Transcendental (Trigonometric) Functions
So far, we can differentiate polynomials, exponential functions, and logarithms. Let's learn how to differentiate trigonometric functions.
Derivatives can be used to calculate limits via l'Hôpital's rule. Given a real-world equation involving two changing quantities, differentiating yields "related rates."
11 vídeos, 3 leituras
Nota atribuída: Derivatives in the Real World
WEEK 9
Optimization
In the real world, we must makes choices, and wouldn't it be great if we could make the best choice? Such optimization is made possible with calculus.
12 vídeos, 4 leituras
Nota atribuída: Optimization
WEEK 10
Linear Approximation
Replacing the curved graph by a straight line approximation helps us to estimate values and roots.
13 vídeos, 5 leituras
Nota atribuída: Linear Approximation
WEEK 11
Antidifferentiation
Antidifferentiation is the process of untaking derivatives, of finding a function whose derivatives is a given function. Since it involves working backwards, antidifferentiation feels like "unbreaking a vase" and can be just as challenging.
14 vídeos, 4 leituras
Nota atribuída: Antidifferentiation
WEEK 12
Integration
By cutting up a curved region into thin rectangles and taking a limit of the sum of the areas of those rectangles, we compute (define!) the area of a curved region.
14 vídeos, 6 leituras
Nota atribuída: Integration
WEEK 13
Fundamental Theorem of Calculus
Armed with the Fundamental Theorem of Calculus, evaluating a definite integral amounts to finding an antiderivative.
13 vídeos, 4 leituras
Nota atribuída: Fundamental Theorem of Calculus
WEEK 14
Substitution Rule
Substitution systematizes the process of using the chain rule in reverse. Considering how often we used the chain rule when differentiating, we will often want to use it in reverse to antidifferentiate.
12 vídeos, 3 leituras
Nota atribuída: Substitution Rule
WEEK 15
Techniques of Integration
Integration by parts is the product rule in reverse. Integrals of powers of trigonometric functions can be evaluated.
10 vídeos, 4 leituras
Nota atribuída: Techniques of Integration
WEEK 16
Applications of Integration
We have already used integrals to compute area; integration can also be used to compute volumes.
The Ohio State University is one of the largest universities in the United States. It's also home to a diverse group of the best and brightest people in the world: dedicated faculty, passionate students, and innovative researchers who make Ohio State one of the world's truly great universities.
Classificações e avaliações
Avaliado em 4.8 de 5 decorrente de 3,395 avaliações
Very good course. The teacher is great. Thank you.
AC
Top level, very interesting and absolutelly exciting!
BP
Great professor, but a lot of formatting issues with the problems. To avoid considerable frustration, make sure you reference this page: | 677.169 | 1 |
Be sure that you have an application to open
this file type before downloading and/or purchasing.
54 KB|6 pages
Product Description
• Create tables and coordinate graphs to represent the progression of a constant rate.
• Recognize the relationship between the dependent and independent variables and write an equation.
• Recognize linear patterns on a table or coordinate graph. | 677.169 | 1 |
Traffic Analysis and Graphing 5 th Grade Objectives Students will: organize recorded data into a chart, or something similar interpret graphs by reading.
Similar presentations
Presentation on theme: "Traffic Analysis and Graphing 5 th Grade Objectives Students will: organize recorded data into a chart, or something similar interpret graphs by reading."— Presentation transcript:
1
Traffic Analysis and Graphing 5 th Grade Objectives Students will: organize recorded data into a chart, or something similar interpret graphs by reading specific points on the graph, as well as creating points on the graph create a graph of their recorded data determine the mean, median, and mode of their data Applicable Strands and Benchmarks Strands Data Analysis and Probability Measurement Patterns, Functions and Algebra Mathematical Process Brad Hunt 1, Kate Kulesa 2 1 Norwood, OH, Norwood High School; 2 Cincinnati, OH, Xavier University 12 th Grade Calculus Objectives Students will be able to: graph a function from the graph of its derivative graph a derivative from the graph of its function graph the derivative of a function given numerically with data interpret the area under the graph as a net accumulation of a rate of change. Lesson Overview Engineering is the application of science for daily use. Transportation is one such field where decisions influence people directly. What kinds of problems occur on our highways? How long are you on the highway? Field Trip to ARTIMIS Students will collect GPS data en route to ARTIMIS Determine rates of change in the traffic patterns Analyze traffic graph and area beneath Activity Students will collect GPS data en route Determine rates of change in the traffic patterns Analyze traffic graph and area beneath Using Excel or a graphing calculator students will develop a time/velocity graph of a car trip from Norwood to the Kenwood Mall. Students will determine their own time intervals. Discussion What does the slope of each segment represent? When is the slope positive and negative? What does the area under the graph represent? I ntroduction Transportation Engineers work in a field where they must collect and analyze data and compare it to some of the data that has been collected by various organizations, such as ODOT in Ohio. After collecting the data and analyzing it, engineers think about hypothetical situations of ways to improve traffic and keep congestion to a minimum. Students will learn some basic vocabulary related to highway transportation. Lesson Overview Hands-on simulation of traffic Walking in the hallway, entering and exiting classrooms (like highway ramps Students brainstorm segmentation to measure time covered through a distance relative scale of the level of traffic collect data display in a graph in order to complete an analysis of the data. Activity Engagement Concept map, teacher gives central idea Calculation of how much time in a lifetime is spent in traffic Activity Students will segment the hallway and record the time it takes to walk each segment Analysis Analyze graphs Take recorded data and create a new graph Compare and contrast various graphs Assessment Given a velocity graph students will be able to graph the acceleration and position graphs Can students interpret the value and meaning of points on a given graph? Did students record data in a chart or organized manner? Can students create their own graphs from data? Exit slip Graphing Line graph Plot points histogram Pie chart numbers data | 677.169 | 1 |
Some people think mathematics is nothing more than adding numbers. In fact, it is more about studying all kinds of abstract structures. Granted, often in the hope that it leads to some numbers we can add! A beautiful example of such a structure is the symmetric group and this is really the backbone of this book. The object is needed to eventually introduce Bruhat cells, a subject that is still very much alive in the world of research. However, on its own, the symmetric group is very interesting as well, as it tends to pop up everywhere in mathematics and physics and is often used as a 'tool' in the various fields. Although this may all sound very vague and sketchy, this book is really aiming at the opposite; using mathematically rigorous methods to guide you every step of the way. In here you will find correct and complete proves where possible alongside many examples to help you develop the needed intuition. Although this book is originally a bachelor's thesis, you only need basic knowledge of matrices and groups to fully understand it. | 677.169 | 1 |
Description
The nine lessons in this group provide initial instruction or intervention on data analysis, probability and statistics.
The first five lessons introduce students to the terminology and tools of the branch of mathematics called statistics. Statistics deals with the collection, analysis, interpretations, and presentation of information referred to as data. The study of statistics is a mathematics course in its own right and covers many topics. These five lessons cover the basic concepts:
A19.1 Finding Mean, Median, and Mode
A19.2 Interpreting Graphs of Data
A19.3 Analyzing and Describing Graphs
A19.4 Finding a Line of Best Fit
A19.5 Solving Statistics Problems
The next section contains four lessons. The first three lay the foundation for advanced study of probability. Probability is used in a growing number of fields. Many college majors, including those in business, psychology, and education are required to take at least one probability and statistics course. The fourth lesson introduces graph theory, an essential concept in discrete mathematics, and here students solve problems.
A20.1 Finding Permutations and Combinations
A20.2 Solving Basic Probability Problems
A20.3 Solving Advanced Probability Problems
A20.4 Solving Discrete Mathematics Problems
Student print materials and Teachers' Notes are available for download at
Download the free Elevated Math app to view two complimentary lessons or buy the lessons individually | 677.169 | 1 |
Elements of Abstract and Linear Algebra by E. H. Connell
Elements of
Abstract and Linear Algebra
E. H. Connell
ii
E.H. Connell
Department of Mathematics
University of Miami
P.O. Box 249085
Coral Gables, Florida 33124 USA
ec@math.miami.edu
Mathematical Subject Classifications (1991): 12-01, 13-01, 15-01, 16-01, 20-01
c (1999 E.H. Connell
March 20, 2004
iii
Introduction
In 1965 I first taught an undergraduate course in abstract algebra. It was fun to
teach because the material was interesting and the class was outstanding. Five of
those students later earned a Ph.D. in mathematics. Since then I have taught the
course about a dozen times from various texts. Over the years I developed a set of
lecture notes and in 1985 I had them typed so they could be used as a text. They
now appear (in modified form) as the first five chapters of this book. Here were some
of my motives at the time.
1) To have something as short and inexpensive as possible. In my experience,
students like short books.
2) To avoid all innovation. To organize the material in the most simple-minded
straightforward manner.
3) To order the material linearly. To the extent possible, each section should use
the previous sections and be used in the following sections.
4) To omit as many topics as possible. This is a foundational course, not a topics
course. If a topic is not used later, it should not be included. There are three
good reasons for this. First, linear algebra has top priority. It is better to go
forward and do more linear algebra than to stop and do more group and ring
theory. Second, it is more important that students learn to organize and write
proofs themselves than to cover more subject matter. Algebra is a perfect place
to get started because there are many "easy" theorems to prove. There are
many routine theorems stated here without proofs, and they may be considered
as exercises for the students. Third, the material should be so fundamental
that it be appropriate for students in the physical sciences and in computer
science. Zillions of students take calculus and cookbook linear algebra, but few
take abstract algebra courses. Something is wrong here, and one thing wrong
is that the courses try to do too much group and ring theory and not enough
matrix theory and linear algebra.
5) To offer an alternative for computer science majors to the standard discrete
mathematics courses. Most of the material in the first four chapters of this text
is covered in various discrete mathematics courses. Computer science majors
might benefit by seeing this material organized from a purely mathematical
viewpoint.
iv
Over the years I used the five chapters that were typed as a base for my algebra
courses, supplementing them as I saw fit. In 1996 I wrote a sixth chapter, giving
enough material for a full first year graduate course. This chapter was written in the
same "style" as the previous chapters, i.e., everything was right down to the nub. It
hung together pretty well except for the last two sections on determinants and dual
spaces. These were independent topics stuck on at the end. In the academic year
1997-98 I revised all six chapters and had them typed in LaTeX. This is the personal
background of how this book came about.
It is difficult to do anything in life without help from friends, and many of my
friends have contributed to this text. My sincere gratitude goes especially to Marilyn
Gonzalez, Lourdes Robles, Marta Alpar, John Zweibel, Dmitry Gokhman, Brian
Coomes, Huseyin Kocak, and Shulim Kaliman. To these and all who contributed,
this book is fondly dedicated.
This book is a survey of abstract algebra with emphasis on linear algebra. It is
intended for students in mathematics, computer science, and the physical sciences.
The first three or four chapters can stand alone as a one semester course in abstract
algebra. However they are structured to provide the background for the chapter on
linear algebra. Chapter 2 is the most difficult part of the book because groups are
written in additive and multiplicative notation, and the concept of coset is confusing
at first. After Chapter 2 the book gets easier as you go along. Indeed, after the
first four chapters, the linear algebra follows easily. Finishing the chapter on linear
algebra gives a basic one year undergraduate course in abstract algebra. Chapter 6
continues the material to complete a first year graduate course. Classes with little
background can do the first three chapters in the first semester, and chapters 4 and 5
in the second semester. More advanced classes can do four chapters the first semester
and chapters 5 and 6 the second semester. As bare as the first four chapters are, you
still have to truck right along to finish them in one semester.
The presentation is compact and tightly organized, but still somewhat informal.
The proofs of many of the elementary theorems are omitted. These proofs are to
be provided by the professor in class or assigned as homework exercises. There is a
non-trivial theorem stated without proof in Chapter 4, namely the determinant of the
product is the product of the determinants. For the proper flow of the course, this
theorem should be assumed there without proof. The proof is contained in Chapter 6.
The Jordan form should not be considered part of Chapter 5. It is stated there only
as a reference for undergraduate courses. Finally, Chapter 6 is not written primarily
for reference, but as an additional chapter for more advanced courses.
v
This text is written with the conviction that it is more effective to teach abstract
and linear algebra as one coherent discipline rather than as two separate ones. Teach-
ing abstract algebra and linear algebra as distinct courses results in a loss of synergy
and a loss of momentum. Also with this text the professor does not extract the course
from the text, but rather builds the course upon it. I am convinced it is easier to
build a course from a base than to extract it from a big book. Because after you
extract it, you still have to build it. The bare bones nature of this book adds to its
flexibility, because you can build whatever course you want around it. Basic algebra
is a subject of incredible elegance and utility, but it requires a lot of organization.
This book is my attempt at that organization. Every effort has been extended to
make the subject move rapidly and to make the flow from one topic to the next as
seamless as possible. The student has limited time during the semester for serious
study, and this time should be allocated with care. The professor picks which topics
to assign for serious study and which ones to "wave arms at". The goal is to stay
focused and go forward, because mathematics is learned in hindsight. I would have
made the book shorter, but I did not have any more time.
When using this text, the student already has the outline of the next lecture, and
each assignment should include the study of the next few pages. Study forward, not
just back. A few minutes of preparation does wonders to leverage classroom learning,
and this book is intended to be used in that manner. The purpose of class is to
learn, not to do transcription work. When students come to class cold and spend
the period taking notes, they participate little and learn little. This leads to a dead
class and also to the bad psychology of "OK, I am here, so teach me the subject."
Mathematics is not taught, it is learned, and many students never learn how to learn.
Professors should give more direction in that regard.
Unfortunately mathematics is a difficult and heavy subject. The style and
approach of this book is to make it a little lighter. This book works best when
viewed lightly and read as a story. I hope the students and professors who try it,
enjoy it.
E. H. Connell
Department of Mathematics
University of Miami
Coral Gables, FL 33124
ec@math.miami.edu
vi
Outline
Chapter 1 Background and Fundamentals of Mathematics
Sets, Cartesian products 1
Relations, partial orderings, Hausdorff maximality principle, 3
equivalence relations
Functions, bijections, strips, solutions of equations, 5
right and left inverses, projections
Notation for the logic of mathematics 13
Integers, subgroups, unique factorization 14
Chapter 2 Groups
Groups, scalar multiplication for additive groups 19
Subgroups, order, cosets 21
Normal subgroups, quotient groups, the integers mod n 25
Homomorphisms 27
Permutations, the symmetric groups 31
Product of groups 34
Chapter 3 Rings
Rings 37
Units, domains, fields 38
The integers mod n 40
Ideals and quotient rings 41
Homomorphisms 42
Polynomial rings 45
Product of rings 49
The Chinese remainder theorem 50
Characteristic 50
Boolean rings 51
Chapter 4 Matrices and Matrix Rings
Addition and multiplication of matrices, invertible matrices 53
Transpose 56
Triangular, diagonal, and scalar matrices 56
Elementary operations and elementary matrices 57
Systems of equations 59
vii
Determinants, the classical adjoint 60
Similarity, trace, and characteristic polynomial 64
Chapter 5 Linear Algebra
Modules, submodules 68
Homomorphisms 69
Homomorphisms on R
n
71
Cosets and quotient modules 74
Products and coproducts 75
Summands 77
Independence, generating sets, and free basis 78
Characterization of free modules 79
Uniqueness of dimension 82
Change of basis 83
Vector spaces, square matrices over fields, rank of a matrix 85
Geometric interpretation of determinant 90
Linear functions approximate differentiable functions locally 91
The transpose principle 92
Nilpotent homomorphisms 93
Eigenvalues, characteristic roots 95
Jordan canonical form 96
Inner product spaces, Gram-Schmidt orthonormalization 98
Orthogonal matrices, the orthogonal group 102
Diagonalization of symmetric matrices 103
Chapter 6 Appendix
The Chinese remainder theorem 108
Prime and maximal ideals and UFD
s
109
Splitting short exact sequences 114
Euclidean domains 116
Jordan blocks 122
Jordan canonical form 123
Determinants 128
Dual spaces 130
viii
1 2 3 4
5 6 7 8
9 11 10
Abstract algebra is not only a major subject of science, but it is also
magic and fun. Abstract algebra is not all work and no play, and it is
certainly not a dull boy. See, for example, the neat card trick on page
18. This trick is based, not on sleight of hand, but rather on a theorem
in abstract algebra. Anyone can do it, but to understand it you need
some group theory. And before beginning the course, you might first try
your skills on the famous (some would say infamous) tile puzzle. In this
puzzle, a frame has 12 spaces, the first 11 with numbered tiles and the
last vacant. The last two tiles are out of order. Is it possible to slide the
tiles around to get them all in order, and end again with the last space
vacant? After giving up on this, you can study permutation groups and
learn the answer!
Chapter 1
Background and Fundamentals of
Mathematics
This chapter is fundamental, not just for algebra, but for all fields related to mathe-
matics. The basic concepts are products of sets, partial orderings, equivalence rela-
tions, functions, and the integers. An equivalence relation on a set A is shown to be
simply a partition of A into disjoint subsets. There is an emphasis on the concept
of function, and the properties of surjective, injective, and bijective. The notion of a
solution of an equation is central in mathematics, and most properties of functions
can be stated in terms of solutions of equations. In elementary courses the section
on the Hausdorff Maximality Principle should be ignored. The final section gives a
proof of the unique factorization theorem for the integers.
Notation Mathematics has its own universally accepted shorthand. The symbol
∃ means "there exists" and ∃! means "there exists a unique". The symbol ∀ means
"for each" and ⇒ means "implies". Some sets (or collections) are so basic they have
their own proprietary symbols. Five of these are listed below.
N = Z
+
= the set of positive integers = ¦1, 2, 3, ...¦
Z = the ring of integers = ¦..., −2, −1, 0, 1, 2, ...¦
Q = the field of rational numbers = ¦a/b : a, b ∈ Z, b = 0¦
R = the field of real numbers
C = the field of complex numbers = ¦a +bi : a, b ∈ R¦ (i
2
= −1)
Sets Suppose A, B, C,... are sets. We use the standard notation for intersection
and union.
A ∩ B = ¦x : x ∈ A and x ∈ B¦ = the set of all x which are elements
1
2 Background Chapter 1
of A and B.
A ∪ B = ¦x : x ∈ A or x ∈ B¦ = the set of all x which are elements of
A or B.
Any set called an index set is assumed to be non-void. Suppose T is an index set and
for each t ∈ T, A
t
is a set.
¸
t∈T
A
t
= ¦x : ∃ t ∈ T with x ∈ A
t
¦
¸
t∈T
A
t
= ¦x : if t ∈ T, x ∈ A
t
¦ = ¦x : ∀t ∈ T, x ∈ A
t
¦
Let ∅ be the null set. If A∩ B = ∅, then A and B are said to be disjoint.
Definition Suppose each of A and B is a set. The statement that A is a subset
of B (A ⊂ B) means that if a is an element of A, then a is an element of B. That
is, a ∈ A ⇒a ∈ B. If A ⊂ B we may say A is contained in B, or B contains A.
Exercise Suppose each of A and B is a set. The statement that A is not a subset
of B means .
Theorem (De Morgan's laws) Suppose S is a set. If C ⊂ S (i.e., if C is a subset
of S), let C
, the complement of C in S, be defined by C
= S−C = ¦x ∈ S : x ∈ C¦.
Then for any A, B ⊂ S,
(A ∩ B)
= A
∪ B
and
(A ∪ B)
= A
∩ B
Cartesian Products If X and Y are sets, X Y = ¦(x, y) : x ∈ X and y ∈ Y ¦.
In other words, the Cartesian product of X and Y is defined to be the set of all
ordered pairs whose first term is in X and whose second term is in Y .
Example RR = R
2
= the plane.
Chapter 1 Background 3
Definition If each of X
1
, ..., X
n
is a set, X
1
X
n
= ¦(x
1
, ..., x
n
) : x
i
∈ X
i
for 1 ≤ i ≤ n¦ = the set of all ordered n-tuples whose i-th term is in X
i
.
Example R R = R
n
= real n-space.
Question Is (RR
2
) = (R
2
R) = R
3
?
Relations
If A is a non-void set, a non-void subset R ⊂ A A is called a relation on A. If
(a, b) ∈ R we say that a is related to b, and we write this fact by the expression a ∼ b.
Here are several properties which a relation may possess.
1) If a ∈ A, then a ∼ a. (reflexive)
2) If a ∼ b, then b ∼ a. (symmetric)
2
) If a ∼ b and b ∼ a, then a = b. (anti-symmetric)
3) If a ∼ b and b ∼ c, then a ∼ c. (transitive)
Definition A relation which satisfies 1), 2
), and 3) is called a partial ordering.
In this case we write a ∼ b as a ≤ b. Then
1) If a ∈ A, then a ≤ a.
2
) If a ≤ b and b ≤ a, then a = b.
3) If a ≤ b and b ≤ c, then a ≤ c.
Definition A linear ordering is a partial ordering with the additional property
that, if a, b ∈ A, then a ≤ b or b ≤ a.
Example A = R with the ordinary ordering, is a linear ordering.
Example A = all subsets of R
2
, with a ≤ b defined by a ⊂ b, is a partial ordering.
Hausdorff Maximality Principle (HMP) Suppose S is a non-void subset of A
and ∼ is a relation on A. This defines a relation on S. If the relation satisfies any
of the properties 1), 2), 2
), or 3) on A, the relation also satisfies these properties
when restricted to S. In particular, a partial ordering on A defines a partial ordering
4 Background Chapter 1
on S. However the ordering may be linear on S but not linear on A. The HMP is
that any linearly ordered subset of a partially ordered set is contained in a maximal
linearly ordered subset.
Exercise Define a relation on A = R
2
by (a, b) ∼ (c, d) provided a ≤ c and
b ≤ d. Show this is a partial ordering which is linear on S = ¦(a, a) : a < 0¦. Find
at least two maximal linearly ordered subsets of R
2
which contain S.
One of the most useful applications of the HMP is to obtain maximal monotonic
collections of subsets.
Definition A collection of sets is said to be monotonic if, given any two sets of
the collection, one is contained in the other.
Corollary to HMP Suppose X is a non-void set and A is some non-void
collection of subsets of X, and S is a subcollection of A which is monotonic. Then ∃
a maximal monotonic subcollection of A which contains S.
Proof Define a partial ordering on A by V ≤ W iff V ⊂ W, and apply HMP.
The HMP is used twice in this book. First, to show that infinitely generated
vector spaces have free bases, and second, in the Appendix, to show that rings have
maximal ideals (see pages 87 and 109). In each of these applications, the maximal
monotonic subcollection will have a maximal element. In elementary courses, these
results may be assumed, and thus the HMP may be ignored.
Equivalence Relations A relation satisfying properties 1), 2), and 3) is called
an equivalence relation.
Exercise Define a relation on A = Z by n ∼ m iff n − m is a multiple of 3.
Show this is an equivalence relation.
Definition If ∼ is an equivalence relation on A and a ∈ A, we define the equiva-
lence class containing a by cl(a) = ¦x ∈ A : a ∼ x¦.
Chapter 1 Background 5
Theorem
1) If b ∈ cl(a) then cl(b) = cl(a). Thus we may speak of a subset of A
being an equivalence class with no mention of any element contained
in it.
2) If each of U, V ⊂ A is an equivalence class and U ∩ V = ∅, then
U = V .
3) Each element of A is an element of one and only one equivalence class.
Definition A partition of A is a collection of disjoint non-void subsets whose union
is A. In other words, a collection of non-void subsets of A is a partition of A provided
any a ∈ A is an element of one and only one subset of the collection. Note that if A
has an equivalence relation, the equivalence classes form a partition of A.
Theorem Suppose A is a non-void set with a partition. Define a relation on A by
a ∼ b iff a and b belong to the same subset of the partition. Then ∼ is an equivalence
relation, and the equivalence classes are just the subsets of the partition.
Summary There are two ways of viewing an equivalence relation — one is as a
relation on A satisfying 1), 2), and 3), and the other is as a partition of A into
disjoint subsets.
Exercise Define an equivalence relation on Z by n ∼ m iff n −m is a multiple
of 3. What are the equivalence classes?
Exercise Is there a relation on R satisfying 1), 2), 2
) and 3) ? That is, is there
an equivalence relation on R which is also a partial ordering?
Exercise Let H ⊂ R
2
be the line H = ¦(a, 2a) : a ∈ R¦. Consider the collection
of all translates of H, i.e., all lines in the plane with slope 2. Find the equivalence
relation on R
2
defined by this partition of R
2
.
Functions
Just as there are two ways of viewing an equivalence relation, there are two ways
of defining a function. One is the "intuitive" definition, and the other is the "graph"
or "ordered pairs" definition. In either case, domain and range are inherent parts of
the definition. We use the "intuitive" definition because everyone thinks that way.
6 Background Chapter 1
Definition If X and Y are (non-void) sets, a function or mapping or map with
domain X and range Y , is an ordered triple (X, Y, f) where f assigns to each x ∈ X
a well defined element f(x) ∈ Y . The statement that (X, Y, f) is a function is written
as f : X →Y or X
f
→Y .
Definition The graph of a function (X, Y, f) is the subset Γ ⊂ X Y defined
by Γ = ¦(x, f(x)) : x ∈ X¦. The connection between the "intuitive" and "graph"
viewpoints is given in the next theorem.
Theorem If f : X → Y , then the graph Γ ⊂ X Y has the property that each
x ∈ X is the first term of one and only one ordered pair in Γ. Conversely, if Γ is a
subset of X Y with the property that each x ∈ X is the first term of one and only
ordered pair in Γ, then ∃! f : X → Y whose graph is Γ. The function is defined by
"f(x) is the second term of the ordered pair in Γ whose first term is x."
Example Identity functions Here X = Y and f : X → X is defined by
f(x) = x for all x ∈ X. The identity on X is denoted by I
X
or just I : X →X.
Example Constant functions Suppose y
0
∈ Y . Define f : X → Y by f(x) =
y
0
for all x ∈ X.
Restriction Given f : X →Y and a non-void subset S of X, define f [ S : S →Y
by (f [ S)(s) = f(s) for all s ∈ S.
Inclusion If S is a non-void subset of X, define the inclusion i : S → X by
i(s) = s for all s ∈ S. Note that inclusion is a restriction of the identity.
Composition Given W
f
→X
g
→Y define g ◦ f : W →Y by
(g ◦ f)(x) = g(f(x)).
Theorem (The associative law of composition) If V
f
→ W
g
→ X
h
→ Y , then
h ◦ (g ◦ f) = (h ◦ g) ◦ f. This may be written as h ◦ g ◦ f.
Chapter 1 Background 7
Definitions Suppose f : X →Y .
1) If T ⊂ Y , the inverse image of T is a subset of X, f
−1
(T) = ¦x ∈ X :
f(x) ∈ T¦.
2) If S ⊂ X, the image of S is a subset of Y , f(S) = ¦f(s) : s ∈ S¦ =
¦y ∈ Y : ∃s ∈ S with f(s) = y¦.
3) The image of f is the image of X , i.e., image (f) = f(X) =
¦f(x) : x ∈ X¦ = ¦y ∈ Y : ∃x ∈ X with f(x) = y¦.
4) f : X →Y is surjective or onto provided image (f) = Y i.e., the image
is the range, i.e., if y ∈ Y , f
−1
(y) is a non-void subset of X.
5) f : X →Y is injective or 1-1 provided (x
1
= x
2
) ⇒f(x
1
) = f(x
2
), i.e.,
if x
1
and x
2
are distinct elements of X, then f(x
1
) and f(x
2
) are
distinct elements of Y .
6) f : X →Y is bijective or is a 1-1 correspondence provided f is surjective
and injective. In this case, there is function f
−1
: Y →X with f
−1
◦ f =
I
X
: X →X and f ◦ f
−1
= I
Y
: Y →Y . Note that f
−1
: Y →X is
also bijective and (f
−1
)
−1
= f.
Examples
1) f : R →R defined by f(x) = sin(x) is neither surjective nor injective.
2) f : R →[−1, 1] defined by f(x) = sin(x) is surjective but not injective.
3) f : [0, π/2] →R defined by f(x) = sin(x) is injective but not surjective.
4) f : [0, π/2] →[0, 1] defined by f(x) = sin(x) is bijective. (f
−1
(x) is
written as arcsin(x) or sin
−1
(x).)
5) f : R →(0, ∞) defined by f(x) = e
x
is bijective. (f
−1
(x) is written as
ln(x).)
Note There is no such thing as "the function sin(x)." A function is not defined
unless the domain and range are specified.
8 Background Chapter 1
Exercise Show there are natural bijections from (R R
2
) to (R
2
R) and
from (R
2
R) to R R R. These three sets are disjoint, but the bijections
between them are so natural that we sometimes identify them.
Exercise Suppose X is a set with 6 elements and Y is a finite set with n elements.
1) There exists an injective f : X →Y iff n .
2) There exists a surjective f : X →Y iff n .
3) There exists a bijective f : X →Y iff n .
Pigeonhole Principle Suppose X is a finite set with m elements, Y is a finite
set with n elements, and f : X →Y is a function.
1) If m = n, then f is injective iff f is surjective iff f is bijective.
2) If m > n, then f is not injective.
3) If m < n, then f is not surjective.
If you are placing 6 pigeons in 6 holes, and you run out of pigeons before you fill
the holes, then you have placed 2 pigeons in one hole. In other words, in part 1) for
m = n = 6, if f is not surjective then f is not injective. Of course, the pigeonhole
principle does not hold for infinite sets, as can be seen by the following exercise.
Exercise Show there is a function f : Z
+
→ Z
+
which is injective but not
surjective. Also show there is one which is surjective but not injective.
Exercise Suppose f : [−2, 2] → R is defined by f(x) = x
2
. Find f
−1
(f([1, 2])).
Also find f(f
−1
([3, 5])).
Exercise Suppose f : X → Y is a function, S ⊂ X and T ⊂ Y . Find the
relationship between S and f
−1
(f(S)). Show that if f is injective, S = f
−1
(f(S)).
Also find the relationship between T and f(f
−1
(T)). Show that if f is surjective,
T = f(f
−1
(T)).
Strips We now define the vertical and horizontal strips of X Y .
If x
0
∈ X, ¦(x
0
, y) : y ∈ Y ¦ = (x
0
Y ) is called a vertical strip.
If y
0
∈ Y, ¦(x, y
0
) : x ∈ X¦ = (X y
0
) is called a horizontal strip.
Chapter 1 Background 9
Theorem Suppose S ⊂ X Y . The subset S is the graph of a function with
domain X and range Y iff each vertical strip intersects S in exactly one point.
This is just a restatement of the property of a graph of a function. The purpose
of the next theorem is to restate properties of functions in terms of horizontal strips.
Theorem Suppose f : X →Y has graph Γ. Then
1) Each horizontal strip intersects Γ in at least one point iff f is .
2) Each horizontal strip intersects Γ in at most one point iff f is .
3) Each horizontal strip intersects Γ in exactly one point iff f is .
Solutions of Equations Now we restate these properties in terms of solutions of
equations. Suppose f : X → Y and y
0
∈ Y . Consider the equation f(x) = y
0
. Here
y
0
is given and x is considered to be a "variable". A solution to this equation is any
x
0
∈ X with f(x
0
) = y
0
. Note that the set of all solutions to f(x) = y
0
is f
−1
(y
0
).
Also f(x) = y
0
has a solution iff y
0
∈ image(f) iff f
−1
(y
0
) is non-void.
Theorem Suppose f : X →Y .
1) The equation f(x) = y
0
has at least one solution for each y
0
∈ Y iff
f is .
2) The equation f(x) = y
0
has at most one solution for each y
0
∈ Y iff
f is .
3) The equation f(x) = y
0
has a unique solution for each y
0
∈ Y iff
f is .
Right and Left Inverses One way to understand functions is to study right and
left inverses, which are defined after the next theorem.
Theorem Suppose X
f
→Y
g
→W are functions.
1) If g ◦ f is injective, then f is injective.
10 Background Chapter 1
2) If g ◦ f is surjective, then g is surjective.
3) If g ◦ f is bijective, then f is injective and g is surjective.
Example X = W = ¦p¦, Y = ¦p, q¦, f(p) = p, and g(p) = g(q) = p. Here
g ◦ f is the identity, but f is not surjective and g is not injective.
Definition Suppose f : X → Y is a function. A left inverse of f is a function
g : Y → X such that g ◦ f = I
X
: X → X. A right inverse of f is a function
h : Y →X such that f ◦ h = I
Y
: Y →Y .
Theorem Suppose f : X →Y is a function.
1) f has a right inverse iff f is surjective. Any such right inverse must be
injective.
2) f has a left inverse iff f is injective. Any such left inverse must be
surjective.
Corollary Suppose each of X and Y is a non-void set. Then ∃ an injective
f : X → Y iff ∃ a surjective g : Y → X. Also a function from X to Y is bijective
iff it has a left inverse and a right inverse iff it has a left and right inverse.
Note The Axiom of Choice is not discussed in this book. However, if you worked
1) of the theorem above, you unknowingly used one version of it. For completeness,
we state this part of 1) again.
The Axiom of Choice If f : X → Y is surjective, then f has a right inverse
h. That is, for each y ∈ Y , it is possible to choose an x ∈ f
−1
(y) and thus to define
h(y) = x.
Note It is a classical theorem in set theory that the Axiom of Choice and the
Hausdorff Maximality Principle are equivalent. However in this text we do not go
that deeply into set theory. For our purposes it is assumed that the Axiom of Choice
and the HMP are true.
Exercise Suppose f : X → Y is a function. Define a relation on X by a ∼ b if
f(a) = f(b). Show this is an equivalence relation. If y belongs to the image of f,
then f
−1
(y) is an equivalence class and every equivalence class is of this form. In the
next chapter where f is a group homomorphism, these equivalence classes will be
called cosets.
Chapter 1 Background 11
Projections If X
1
and X
2
are non-void sets, we define the projection maps
π
1
: X
1
X
2
→X
1
and π
2
: X
1
X
2
→X
2
by π
i
(x
1
, x
2
) = x
i
.
Theorem If Y, X
1
, and X
2
are non-void sets, there is a 1-1 correspondence
between ¦functions f: Y →X
1
X
2
¦ and ¦ordered pairs of functions (f
1
, f
2
) where
f
1
: Y →X
1
and f
2
: Y →X
2
¦.
Proof Given f, define f
1
= π
1
◦ f and f
2
= π
2
◦ f. Given f
1
and f
2
define
f : Y → X
1
X
2
by f(y) = (f
1
(y), f
2
(y)). Thus a function from Y to X
1
X
2
is
merely a pair of functions from Y to X
1
and Y to X
2
. This concept is displayed in
the diagram below. It is summarized by the equation f = (f
1
, f
2
).
X
1
X
2
X
1
X
2
Y
·
,
»
f
1
f
2
f
π
1
π
2
One nice thing about this concept is that it works fine for infinite Cartesian
products.
Definition Suppose T is an index set and for each t ∈ T, X
t
is a non-void set.
Then the product
¸
t∈T
X
t
=
¸
X
t
is the collection of all sequences ¦x
t
¦
t∈T
= ¦x
t
¦
where x
t
∈ X
t
. Formally these sequences are functions α from T to
¸
X
t
with each
α(t) in X
t
and written as α(t) = x
t
. If T = ¦1, 2, . . . , n¦ then ¦x
t
¦ is the ordered
n-tuple (x
1
, x
2
, . . . , x
n
). If T = Z
+
then ¦x
t
¦ is the sequence (x
1
, x
2
, . . .). For any T
and any s in T, the projection map π
s
:
¸
X
t
→X
s
is defined by π
s
(¦x
t
¦) = x
s
.
Theorem If Y is any non-void set, there is a 1-1 correspondence between
¦functions f : Y →
¸
X
t
¦ and ¦sequences of functions ¦f
t
¦
t∈T
where f
t
: Y → X
t
¦.
Given f, the sequence ¦f
t
¦ is defined by f
t
= π
t
◦ f. Given ¦f
t
¦, f is defined by
f(y) = ¦f
t
(y)¦.
12 Background Chapter 1
A Calculus Exercise Let A be the collection of all functions f : [0, 1] → R
which have an infinite number of derivatives. Let A
0
⊂ A be the subcollection of
those functions f with f(0) = 0. Define D : A
0
→A by D(f) = df/dx. Use the mean
value theorem to show that D is injective. Use the fundamental theorem of calculus
to show that D is surjective.
Exercise This exercise is not used elsewhere in this text and may be omitted. It
is included here for students who wish to do a little more set theory. Suppose T is a
non-void set.
1) If Y is a non-void set, define Y
T
to be the collection of all functions with domain
T and range Y . Show that if T and Y are finite sets with m and n elements, then
Y
T
has n
m
elements. In particular, when T = ¦1, 2, 3¦, Y
T
= Y Y Y has
n
3
elements. Show that if n ≥ 3, the subset of Y
{1,2,3}
of all injective functions has
n(n − 1)(n − 2) elements. These injective functions are called permutations on Y
taken 3 at a time. If T = N, then Y
T
is the infinite product Y Y . That is,
Y
N
is the set of all infinite sequences (y
1
, y
2
, . . .) where each y
i
∈ Y . For any Y and
T, let Y
t
be a copy of Y for each t ∈ T. Then Y
T
=
¸
t∈T
Y
t
.
2) Suppose each of Y
1
and Y
2
is a non-void set. Show there is a natural bijection
from (Y
1
Y
2
)
T
to Y
T
1
Y
T
2
. (This is the fundamental property of Cartesian products
presented in the two previous theorems.)
3) Define {(T), the power set of T, to be the collection of all subsets of T (including
the null set). Show that if T is a finite set with m elements, {(T) has 2
m
elements.
4) If S is any subset of T, define its characteristic function χ
S
: T → ¦0, 1¦ by
letting χ
S
(t) be 1 when t ∈ S, and be 0 when t ∈[ S. Define α : {(T) → ¦0, 1¦
T
by
α(S) = χ
S
. Define β : ¦0, 1¦
T
→ {(T) by β(f) = f
−1
(1). Show that if S ⊂ T then
β ◦ α(S) = S, and if f : T → ¦0, 1¦ then α ◦ β(f) = f. Thus α is a bijection and
β = α
−1
.
{(T) ←→¦0, 1¦
T
5) Suppose γ : T →¦0, 1¦
T
is a function and show that it cannot be surjective. If
t ∈ T, denote γ(t) by γ(t) = f
t
: T → ¦0, 1¦. Define f : T → ¦0, 1¦ by f(t) = 0 if
f
t
(t) = 1, and f(t) = 1 if f
t
(t) = 0. Show that f is not in the image of γ and thus
γ cannot be surjective. This shows that if T is an infinite set, then the set ¦0, 1¦
T
represents a "higher order of infinity than T".
6) An infinite set Y is said to be countable if there is a bijection from the positive
Chapter 1 Background 13
integers N to Y. Show Q is countable but the following three collections are not.
i) {(N), the collection of all subsets of N.
ii) ¦0, 1¦
N
, the collection of all functions f : N →¦0, 1¦.
iii) The collection of all sequences (y
1
, y
2
, . . .) where each y
i
is 0 or 1.
We know that ii) and iii) are equal and there is a natural bijection between i)
and ii). We also know there is no surjective map from N to ¦0, 1¦
N
, i.e., ¦0, 1¦
N
is
uncountable. Finally, show there is a bijection from ¦0, 1¦
N
to the real numbers R.
(This is not so easy. To start with, you have to decide what the real numbers are.)
Notation for the Logic of Mathematics
Each of the words "Lemma", "Theorem", and "Corollary" means "true state-
ment". Suppose A and B are statements. A theorem may be stated in any of the
following ways:
Theorem Hypothesis Statement A.
Conclusion Statement B.
Theorem Suppose A is true. Then B is true.
Theorem If A is true, then B is true.
Theorem A ⇒B (A implies B ).
There are two ways to prove the theorem — to suppose A is true and show B is
true, or to suppose B is false and show A is false. The expressions "A ⇔ B", "A is
equivalent to B", and "A is true iff B is true " have the same meaning (namely, that
A ⇒B and B ⇒A).
The important thing to remember is that thoughts and expressions flow through
the language. Mathematical symbols are shorthand for phrases and sentences in the
English language. For example, "x ∈ B " means "x is an element of the set B." If A
is the statement "x ∈ Z
+
" and B is the statement "x
2
∈ Z
+
", then "A ⇒ B"means
"If x is a positive integer, then x
2
is a positive integer".
Mathematical Induction is based upon the fact that if S ⊂ Z
+
is a non-void
subset, then S contains a smallest element.
14 Background Chapter 1
Theorem Suppose P(n) is a statement for each n = 1, 2, ... . Suppose P(1) is
true and for each n ≥ 1, P(n) ⇒P(n + 1). Then for each n ≥ 1, P(n) is true.
Proof If the theorem is false, then ∃ a smallest positive integer m such that
P(m) is false. Since P(m−1) is true, this is impossible.
Exercise Use induction to show that, for each n ≥ 1, 1+2+ +n = n(n+1)/2.
The Integers
In this section, lower case letters a, b, c, ... will represent integers, i.e., elements
of Z. Here we will establish the following three basic properties of the integers.
1) If G is a subgroup of Z, then ∃ n ≥ 0 such that G = nZ.
2) If a and b are integers, not both zero, and G is the collection of all linear
combinations of a and b, then G is a subgroup of Z, and its
positive generator is the greatest common divisor of a and b.
3) If n ≥ 2, then n factors uniquely as the product of primes.
All of this will follow from long division, which we now state formally.
Euclidean Algorithm Given a, b with b = 0, ∃! m and r with 0 ≤ r <[b[ and
a = bm + r. In other words, b divides a "m times with a remainder of r". For
example, if a = −17 and b = 5, then m = −4 and r = 3, −17 = 5(−4) + 3.
Definition If r = 0, we say that b divides a or a is a multiple of b. This fact is
written as b [ a. Note that b [ a ⇔ the rational number a/b is an integer ⇔ ∃! m
such that a = bm ⇔ a ∈ bZ.
Note Anything (except 0) divides 0. 0 does not divide anything.
± 1 divides anything . If n = 0, the set of integers which n divides
is nZ = ¦nm : m ∈ Z¦ = ¦..., −2n, −n, 0, n, 2n, ...¦. Also n divides
a and b with the same remainder iff n divides (a −b).
Definition A non-void subset G ⊂ Z is a subgroup provided (g ∈ G ⇒ −g ∈ G)
and (g
1
, g
2
∈ G ⇒(g
1
+g
2
) ∈ G). We say that G is closed under negation and closed
under addition.
Chapter 1 Background 15
Theorem If n ∈ Z then nZ is a subgroup. Thus if n = 0, the set of integers
which n divides is a subgroup of Z.
The next theorem states that every subgroup of Z is of this form.
Theorem Suppose G ⊂ Z is a subgroup. Then
1) 0 ∈ G.
2) If g
1
and g
2
∈ G, then (m
1
g
1
+m
2
g
2
) ∈ G for all integers m
1
, m
2
.
3) ∃! non-negative integer n such that G = nZ. In fact, if G = ¦0¦
and n is the smallest positive integer in G, then G = nZ.
Proof Since G is non-void, ∃ g ∈ G. Now (−g) ∈ G and thus 0 = g + (−g)
belongs to G, and so 1) is true. Part 2) is straightforward, so consider 3). If G = 0,
it must contain a positive element. Let n be the smallest positive integer in G. If
g ∈ G, g = nm +r where 0 ≤ r < n. Since r ∈ G, it must be 0, and g ∈ nZ.
Now suppose a, b ∈ Z and at least one of a and b is non-zero.
Theorem Let G be the set of all linear combinations of a and b, i.e., G =
¦ma +nb : m, n ∈ Z¦. Then
1) G contains a and b.
2) G is a subgroup. In fact, it is the smallest subgroup containing a and b.
It is called the subgroup generated by a and b.
3) Denote by (a, b) the smallest positive integer in G. By the previous
theorem, G = (a, b)Z, and thus (a, b) [ a and (a, b) [ b. Also note that
∃ m, n such that ma +nb = (a, b). The integer (a, b) is called
the greatest common divisor of a and b.
4) If n is an integer which divides a and b, then n also divides (a, b).
Proof of 4) Suppose n [ a and n [ b i.e., suppose a, b ∈ nZ. Since G is the
smallest subgroup containing a and b, nZ ⊃ (a, b)Z, and thus n [ (a, b).
Corollary The following are equivalent.
1) a and b have no common divisors, i.e., (n [ a and n [ b) ⇒n = ±1.
16 Background Chapter 1
2) (a, b) = 1, i.e., the subgroup generated by a and b is all of Z.
3) ∃ m, n ∈Z with ma +nb = 1.
Definition If any one of these three conditions is satisfied, we say that a and b
are relatively prime.
This next theorem is the basis for unique factorization.
Theorem If a and b are relatively prime with a not zero, then a[bc ⇒a[c.
Proof Suppose a and b are relatively prime, c ∈ Z and a [ bc. Then there exist
m, n with ma + nb = 1, and thus mac + nbc = c. Now a [ mac and a [ nbc. Thus
a [ (mac +nbc) and so a [ c.
Definition A prime is an integer p > 1 which does not factor, i.e., if p = ab then
a = ±1 or a = ±p. The first few primes are 2, 3, 5, 7, 11, 13, 17,... .
Theorem Suppose p is a prime.
1) If a is an integer which is not a multiple of p, then (p, a) = 1. In other
words, if a is any integer, (p, a) = p or (p, a) = 1.
2) If p [ ab then p [ a or p [ b.
3) If p [ a
1
a
2
a
n
then p divides some a
i
. Thus if each a
i
is a prime,
then p is equal to some a
i
.
Proof Part 1) follows immediately from the definition of prime. Now suppose
p [ ab. If p does not divide a, then by 1), (p, a) = 1 and by the previous theorem, p
must divide b. Thus 2) is true. Part 3) follows from 2) and induction on n.
The Unique Factorization Theorem Suppose a is an integer which is not 0,1,
or -1. Then a may be factored into the product of primes and, except for order, this
factorization is unique. That is, ∃ a unique collection of distinct primes p
1
, p
2
, ..., p
k
and positive integers s
1
, s
2
, ..., s
k
such that a = ±p
s
1
1
p
s
2
2
p
s
k
k
.
Proof Factorization into primes is obvious, and uniqueness follows from 3) in the
theorem above. The power of this theorem is uniqueness, not existence.
Chapter 1 Background 17
Now that we have unique factorization and part 3) above, the picture becomes
transparent. Here are some of the basic properties of the integers in this light.
Theorem (Summary)
1) Suppose [ a[> 1 has prime factorization a = ±p
s
1
1
p
s
k
k
. Then the only
divisors of a are of the form ±p
t
1
1
p
t
k
k
where 0 ≤ t
i
≤ s
i
for i = 1, ..., k.
2) If [ a [> 1 and [ b [> 1, then (a, b) = 1 iff there is no common prime in
their factorizations. Thus if there is no common prime in their
factorizations, ∃ m, n with ma +nb = 1, and also (a
2
, b
2
) = 1.
3) Suppose [ a[> 1 and [ b[> 1. Let ¦p
1
, . . . , p
k
¦ be the union of the distinct
primes of their factorizations. Thus a = ±p
s
1
1
p
s
k
k
where 0 ≤ s
i
and
b = ±p
t
1
1
p
t
k
k
where 0 ≤ t
i
. Let u
i
be the minimum of s
i
and t
i
. Then
(a, b) = p
u
1
1
p
u
k
k
. For example (2
3
5 11, 2
2
5
4
7) = 2
2
5.
3
) Let v
i
be the maximum of s
i
and t
i
. Then c = p
v
1
1
p
v
k
k
is the least
(positive) common multiple of a and b. Note that c is a multiple of
a and b, and if n is a multiple of a and b, then n is a multiple of c.
Finally, if a and b are positive, their least common multiple is
c = ab/(a, b), and if in addition a and b are relatively prime,
then their least common multiple is just their product.
4) There is an infinite number of primes. (Proof: Suppose there were only
a finite number of primes p
1
, p
2
, ..., p
k
. Then no prime would divide
(p
1
p
2
p
k
+ 1).)
5) Suppose c is an integer greater than 1. Then
√
c is rational iff
√
c is an
integer. In particular,
√
2 and
√
3 are irrational. (Proof: If
√
c is
rational, ∃ positive integers a and b with
√
c = a/b and (a, b) = 1.
If b > 1, then it is divisible by some prime, and since cb
2
= a
2
, this
prime will also appear in the prime factorization of a. This is a
contradiction and thus b = 1 and
√
c is an integer.) (See the fifth
exercise below.)
Exercise Find (180,28), i.e., find the greatest common divisor of 180 and 28,
i.e., find the positive generator of the subgroup generated by ¦180,28¦. Find integers
m and n such that 180m + 28n = (180, 28). Find the least common multiple of 180
and 28, and show that it is equal to (180 28)/(180, 28).
18 Background Chapter 1
Exercise We have defined the greatest common divisor (gcd) and the least com-
mon multiple (lcm) of a pair of integers. Now suppose n ≥ 2 and S = ¦a
1
, a
2
, .., a
n
¦
is a finite collection of integers with [a
i
[ > 1 for 1 ≤ i ≤ n. Define the gcd and the
lcm of the elements of S and develop their properties. Express the gcd and the lcm
in terms of the prime factorizations of the a
i
. When is the lcm of S equal to the
product a
1
a
2
a
n
? Show that the set of all linear combinations of the elements of
S is a subgroup of Z, and its positive generator is the gcd of the elements of S.
Exercise Show that the gcd of S = ¦90, 70, 42¦ is 2, and find integers n
1
, n
2
, n
3
such that 90n
1
+ 70n
2
+ 42n
3
= 2. Also find the lcm of the elements of S.
Exercise Show that if each of G
1
, G
2
, ..., G
m
is a subgroup of Z, then
G
1
∩ G
2
∩ ∩ G
m
is also a subgroup of Z. Now let G = (90Z) ∩ (70Z) ∩ (42Z)
and find the positive integer n with G = nZ.
Exercise Show that if the nth root of an integer is a rational number, then it
itself is an integer. That is, suppose c and n are integers greater than 1. There is a
unique positive real number x with x
n
= c. Show that if x is rational, then it is an
integer. Thus if p is a prime, its nth root is an irrational number.
Exercise Show that a positive integer is divisible by 3 iff the sum of its digits is
divisible by 3. More generally, let a = a
n
a
n−1
. . . a
0
= a
n
10
n
+ a
n−1
10
n−1
+ + a
0
where 0 ≤ a
i
≤ 9. Now let b = a
n
+a
n−1
+ +a
0
, and show that 3 divides a and b
with the same remainder. Although this is a straightforward exercise in long division,
it will be more transparent later on. In the language of the next chapter, it says that
[a] = [b] in Z
3
.
Card Trick Ask friends to pick out seven cards from a deck and then to select one
to look at without showing it to you. Take the six cards face down in your left hand
and the selected card in your right hand, and announce you will place the selected
card in with the other six, but they are not to know where. Put your hands behind
your back and place the selected card on top, and bring the seven cards in front in
your left hand. Ask your friends to give you a number between one and seven (not
allowing one). Suppose they say three. You move the top card to the bottom, then
the second card to the bottom, and then you turn over the third card, leaving it face
up on top. Then repeat the process, moving the top two cards to the bottom and
turning the third card face up on top. Continue until there is only one card face
down, and this will be the selected card. Magic? Stay tuned for Chapter 2, where it
is shown that any non-zero element of Z
7
has order 7.
Chapter 2
Groups
Groups are the central objects of algebra. In later chapters we will define rings and
modules and see that they are special cases of groups. Also ring homomorphisms and
module homomorphisms are special cases of group homomorphisms. Even though
the definition of group is simple, it leads to a rich and amazing theory. Everything
presented here is standard, except that the product of groups is given in the additive
notation. This is the notation used in later chapters for the products of rings and
modules. This chapter and the next two chapters are restricted to the most basic
topics. The approach is to do quickly the fundamentals of groups, rings, and matrices,
and to push forward to the chapter on linear algebra. This chapter is, by far and
above, the most difficult chapter in the book, because group operations may be written
as addition or multiplication, and also the concept of coset is confusing at first.
Definition Suppose G is a non-void set and φ : G G → G is a function. φ is
called a binary operation, and we will write φ(a, b) = a b or φ(a, b) = a+b. Consider
the following properties.
1) If a, b, c ∈ G then a (b c) = (a b) c. If a, b, c ∈ G then a + (b +c) = (a +b) +c.
2) ∃ e = e
G
∈ G such that if a ∈ G ∃ 0
¯
=0
¯
G
∈ G such that if a ∈ G
e a = a e = a. 0
¯
+a = a+0
¯
= a.
3) If a ∈ G, ∃b ∈ G with a b = b a = e If a ∈ G, ∃b ∈ G with a +b = b +a = 0
¯
(b is written as b = a
−1
). (b is written as b = −a).
4) If a, b ∈ G, then a b = b a. If a, b ∈ G, then a +b = b +a.
Definition If properties 1), 2), and 3) hold, (G, φ) is said to be a group. If we
write φ(a, b) = a b, we say it is a multiplicative group. If we write φ(a, b) = a + b,
19
20 Groups Chapter 2
we say it is an additive group. If in addition, property 4) holds, we say the group is
abelian or commutative.
Theorem Let (G, φ) be a multiplicative group.
(i) Suppose a, c, ¯ c ∈ G. Then a c = a ¯ c ⇒ c = ¯ c.
Also c a = ¯ c a ⇒ c = ¯ c.
In other words, if f : G →G is defined by f(c) = a c, then f is injective.
Also f is bijective with f
−1
given by f
−1
(c) = a
−1
c.
(ii) e is unique, i.e., if ¯ e ∈ G satisfies 2), then e = ¯ e. In fact,
if a, b ∈ G then (a b = a) ⇒(b = e) and (a b = b) ⇒ (a = e).
Recall that b is an identity in G provided it is a right and left
identity for any a in G. However, group structure is so rigid that if
∃ a ∈ G such that b is a right identity for a, then b = e.
Of course, this is just a special case of the cancellation law in (i).
(iii) Every right inverse is an inverse, i.e., if a b = e then b = a
−1
.
Also if b a = e then b = a
−1
. Thus inverses are unique.
(iv) If a ∈ G, then (a
−1
)
−1
= a.
(v) The multiplication a
1
a
2
a
3
= a
1
(a
2
a
3
) = (a
1
a
2
) a
3
is well-defined.
In general, a
1
a
2
a
n
is well defined.
(vi) If a, b ∈ G, (a b)
−1
= b
−1
a
−1
. Also (a
1
a
2
a
n
)
−1
=
a
−1
n
a
−1
n−1
a
−1
1
.
(vii) Suppose a ∈ G. Let a
0
= e and if n > 0, a
n
= a a (n times)
and a
−n
= a
−1
a
−1
(n times). If n
1
, n
2
, ..., n
t
∈ Z then
a
n
1
a
n
2
a
nt
= a
n
1
+···+nt
. Also (a
n
)
m
= a
nm
.
Finally, if G is abelian and a, b ∈ G, then (a b)
n
= a
n
b
n
.
Exercise. Write out the above theorem where G is an additive group. Note that
part (vii) states that G has a scalar multiplication over Z. This means that if a is in
G and n is an integer, there is defined an element an in G. This is so basic, that we
state it explicitly.
Theorem. Suppose G is an additive group. If a ∈ G, let a0 =0
¯
and if n > 0,
let an = (a + +a) where the sum is n times, and a(−n) = (−a) + (−a) +(−a),
Chapter 2 Groups 21
which we write as (−a − a − a). Then the following properties hold in general,
except the first requires that G be abelian.
(a +b)n = an +bn
a(n +m) = an +am
a(nm) = (an)m
a1 = a
Note that the plus sign is used ambiguously — sometimes for addition in G
and sometimes for addition in Z. In the language used in Chapter 5, this theorem
states that any additive abelian group is a Z-module. (See page 71.)
Exercise Suppose G is a non-void set with a binary operation φ(a, b) = a b which
satisfies 1), 2) and [ 3
) If a ∈ G, ∃b ∈ G with a b = e]. Show (G, φ) is a group,
i.e., show b a = e. In other words, the group axioms are stronger than necessary.
If every element has a right inverse, then every element has a two sided inverse.
Exercise Suppose G is the set of all functions from Z to Z with multiplication
defined by composition, i.e., f g = f ◦ g. Note that G satisfies 1) and 2) but not 3),
and thus G is not a group. Show that f has a right inverse in G iff f is surjective,
and f has a left inverse in G iff f is injective (see page 10). Also show that the set
of all bijections from Z to Z is a group under composition.
Examples G = R, G = Q, or G = Z with φ(a, b) = a +b is an additive
abelian group.
Examples G = R−0 or G = Q−0 with φ(a, b) = ab is a multiplicative
abelian group.
G = Z −0 with φ(a, b) = ab is not a group.
G = R
+
= ¦r ∈ R : r > 0¦ with φ(a, b) = ab is a multiplicative
abelian group.
Subgroups
Theorem Suppose G is a multiplicative group and H ⊂ G is a non-void subset
satisfying
1) if a, b ∈ H then a b ∈ H
and 2) if a ∈ H then a
−1
∈ H.
22 Groups Chapter 2
Then e ∈ H and H is a group under multiplication. H is called a subgroup of G.
Proof Since H is non-void, ∃a ∈ H. By 2), a
−1
∈ H and so by 1), e ∈ H. The
associative law is immediate and so H is a group.
Example G is a subgroup of G and e is a subgroup of G. These are called the
improper subgroups of G.
Example If G = Z under addition, and n ∈ Z, then H = nZ is a subgroup of
Z. By a theorem in the section on the integers in Chapter 1, every subgroup of Z
is of this form (see page 15). This is a key property of the integers.
Exercises Suppose G is a multiplicative group.
1) Let H be the center of G, i.e., H = ¦h ∈ G : g h = h g for all g ∈ G¦. Show
H is a subgroup of G.
2) Suppose H
1
and H
2
are subgroups of G. Show H
1
∩ H
2
is a subgroup of G.
3) Suppose H
1
and H
2
are subgroups of G, with neither H
1
nor H
2
contained in
the other. Show H
1
∪ H
2
is not a subgroup of G.
4) Suppose T is an index set and for each t ∈ T, H
t
is a subgroup of G.
Show
¸
t∈T
H
t
is a subgroup of G.
5) Furthermore, if ¦H
t
¦ is a monotonic collection, then
¸
t∈T
H
t
is a subgroup of G.
6) Suppose G= ¦all functions f : [0, 1] →R¦. Define an addition on G by
(f +g)(t) = f(t) +g(t) for all t ∈ [0, 1]. This makes G into an abelian group.
Let K be the subset of G composed of all differentiable functions. Let H
be the subset of G composed of all continuous functions. What theorems
in calculus show that H and K are subgroups of G? What theorem shows
that K is a subset (and thus subgroup) of H?
Order Suppose G is a multiplicative group. If G has an infinite number of
Chapter 2 Groups 23
elements, we say that o(G), the order of G, is infinite. If G has n elements, then
o(G) = n. Suppose a ∈ G and H = ¦a
i
: i ∈ Z¦. H is an abelian subgroup of G
called the subgroup generated by a. We define the order of the element a to be the
order of H, i.e., the order of the subgroup generated by a. Let f : Z → H be the
surjective function defined by f(m) = a
m
. Note that f(k + l) = f(k) f(l) where
the addition is in Z and the multiplication is in the group H. We come now to the
first real theorem in group theory. It says that the element a has finite order iff f
is not injective, and in this case, the order of a is the smallest positive integer n
with a
n
= e.
Theorem Suppose a is an element of a multiplicative group G, and
H = ¦a
i
: i ∈ Z¦. If ∃ distinct integers i and j with a
i
= a
j
, then a has some finite
order n. In this case H has n distinct elements, H = ¦a
0
, a
1
, . . . , a
n−1
¦, and a
m
= e
iff n[m. In particular, the order of a is the smallest positive integer n with a
n
= e,
and f
−1
(e) = nZ.
Proof Suppose j < i and a
i
= a
j
. Then a
i−j
= e and thus ∃ a smallest positive
integer n with a
n
= e. This implies that the elements of ¦a
0
, a
1
, ..., a
n−1
¦ are distinct,
and we must show they are all of H. If m ∈ Z, the Euclidean algorithm states that
∃ integers q and r with 0 ≤ r < n and m = nq + r. Thus a
m
= a
nq
a
r
= a
r
, and
so H = ¦a
0
, a
1
, ..., a
n−1
¦, and a
m
= e iff n[m. Later in this chapter we will see that
f is a homomorphism from an additive group to a multiplicative group and that,
in additive notation, H is isomorphic to Z or Z
n
.
Exercise Write out this theorem for G an additive group. To begin, suppose a is
an element of an additive group G, and H = ¦ai : i ∈ Z¦.
Exercise Show that if G is a finite group of even order, then G has an odd number
of elements of order 2. Note that e is the only element of order 1.
Definition A group G is cyclic if ∃ an element of G which generates G.
Theorem If G is cyclic and H is a subgroup of G, then H is cyclic.
Proof Suppose G = ¦a
i
: i ∈ Z¦ is a cyclic group and H is a subgroup
of G. If H = e, then H is cyclic, so suppose H = e. Now there is a small-
est positive integer m with a
m
∈ H. If t is an integer with a
t
∈ H, then by
the Euclidean algorithm, m divides t, and thus a
m
generates H. Note that in
the case G has finite order n, i.e., G = ¦a
0
, a
1
, . . . , a
n−1
¦, then a
n
= e ∈ H,
and thus the positive integer m divides n. In either case, we have a clear picture
of the subgroups of G. Also note that this theorem was proved on page 15 for the
additive group Z.
24 Groups Chapter 2
Cosets Suppose H is a subgroup of a group G. It will be shown below that H
partitions G into right cosets. It also partitions G into left cosets, and in general
these partitions are distinct.
Theorem If H is a subgroup of a multiplicative group G, then a ∼ b defined by
a ∼ b iff a b
−1
∈ H is an equivalence relation. If a ∈ G, cl(a) = ¦b ∈ G : a ∼ b¦ =
¦h a : h ∈ H¦ = Ha. Note that a b
−1
∈ H iff b a
−1
∈ H.
If H is a subgroup of an additive group G, then a ∼ b defined by a ∼ b iff
(a − b) ∈ H is an equivalence relation. If a ∈ G, cl(a) = ¦b ∈ G : a ∼ b¦ = ¦h + a :
h ∈ H¦ = H +a. Note that (a −b) ∈ H iff (b −a) ∈ H.
Definition These equivalence classes are called right cosets. If the relation is
defined by a ∼ b iff b
−1
a ∈ H, then the equivalence classes are cl(a) = aH and
they are called left cosets. H is a left and right coset. If G is abelian, there is no
distinction between right and left cosets. Note that b
−1
a ∈ H iff a
−1
b ∈ H.
In the theorem above, H is used to define an equivalence relation on G, and thus
a partition of G. We now do the same thing a different way. We define the right
cosets directly and show they form a partition of G. You might find this easier.
Theorem Suppose H is a subgroup of a multiplicative group G. If a ∈ G, define
the right coset containing a to be Ha = ¦h a : h ∈ H¦. Then the following hold.
1) Ha = H iff a ∈ H.
2) If b ∈ Ha, then Hb = Ha, i.e., if h ∈ H, then H(h a) = (Hh)a = Ha.
3) If Hc ∩ Ha = ∅, then Hc = Ha.
4) The right cosets form a partition of G, i.e., each a in G belongs to one and
only one right coset.
5) Elements a and b belong to the same right coset iff a b
−1
∈ H iff b a
−1
∈ H.
Proof There is no better way to develop facility with cosets than to prove this
theorem. Also write this theorem for G an additive group.
Theorem Suppose H is a subgroup of a multiplicative group G.
Chapter 2 Groups 25
1) Any two right cosets have the same number of elements. That is, if a, b ∈ G,
f : Ha →Hb defined by f(h a) = h b is a bijection. Also any two left cosets
have the same number of elements. Since H is a right and left coset, any
two cosets have the same number of elements.
2) G has the same number of right cosets as left cosets. The function F defined
by F(Ha) = a
−1
H is a bijection from the collection of right cosets to the left
cosets. The number of right (or left) cosets is called the index of H in G.
3) If G is finite, o(H) (index of H) = o(G) and so o(H) [ o(G). In other words,
o(G)/o(H) = the number of right cosets = the number of left cosets.
4) If G is finite, and a ∈ G, then o(a) [ o(G). (Proof: The order of a is the order
of the subgroup generated by a, and by 3) this divides the order of G.)
5) If G has prime order, then G is cyclic, and any element (except e) is a generator.
(Proof: Suppose o(G) = p and a ∈ G, a = e. Then o(a) [ p and thus o(a) = p.)
6) If o(G) = n and a ∈ G, then a
n
= e. (Proof: a
o(a)
= e and n = o(a) (o(G)/o(a)) .)
Exercises
i) Suppose G is a cyclic group of order 4, G = ¦e, a, a
2
, a
3
¦ with a
4
= e. Find the
order of each element of G. Find all the subgroups of G.
ii) Suppose G is the additive group Z and H = 3Z. Find the cosets of H.
iii) Think of a circle as the interval [0, 1] with end points identified. Suppose G = R
under addition and H = Z. Show that the collection of all the cosets of H
can be thought of as a circle.
iv) Let G = R
2
under addition, and H be the subgroup defined by
H = ¦(a, 2a) : a ∈ R¦. Find the cosets of H. (See the last exercise on p 5.)
Normal Subgroups
We would like to make a group out of the collection of cosets of a subgroup H. In
26 Groups Chapter 2
general, there is no natural way to do that. However, it is easy to do in case H is a
normal subgroup, which is described below.
Theorem If H is a subgroup of a group G, then the following are equivalent.
1) If a ∈ G, then aHa
−1
= H
2) If a ∈ G, then aHa
−1
⊂ H
3) If a ∈ G, then aH = Ha
4) Every right coset is a left coset, i.e., if a ∈ G, ∃ b ∈ G with Ha = bH.
Proof 1) ⇒ 2) is obvious. Suppose 2) is true and show 3). We have (aHa
−1
)a ⊂
Ha so aH ⊂ Ha. Also a(a
−1
Ha) ⊂ aH so Ha ⊂ aH. Thus aH = Ha.
3) ⇒ 4) is obvious. Suppose 4) is true and show 3). Ha = bH contains a, so
bH = aH because a coset is an equivalence class. Thus aH = Ha.
Finally, suppose 3) is true and show 1). Multiply aH = Ha on the right by a
−1
.
Definition If H satisfies any of the four conditions above, then H is said to be a
normal subgroup of G. (This concept goes back to Evariste Galois in 1831.)
Note For any group G, G and e are normal subgroups. If G is an abelian group,
then every subgroup of G is normal.
Exercise Show that if H is a subgroup of G with index 2, then H is normal.
Exercise Show the intersection of a collection of normal subgroups of G is a
normal subgroup of G. Show the union of a monotonic collection of normal subgroups
of G is a normal subgroup of G.
Exercise Let A ⊂ R
2
be the square with vertices (−1, 1), (1, 1), (1, −1), and
(−1, −1), and G be the collection of all "isometries" of A onto itself. These are
bijections of A onto itself which preserve distance and angles, i.e., which preserve dot
product. Show that with multiplication defined as composition, G is a multiplicative
group. Show that G has four rotations, two reflections about the axes, and two
reflections about the diagonals, for a total of eight elements. Show the collection of
rotations is a cyclic subgroup of order four which is a normal subgroup of G. Show
that the reflection about the x-axis together with the identity form a cyclic subgroup
of order two which is not a normal subgroup of G. Find the four right cosets of this
subgroup. Finally, find the four left cosets of this subgroup.
Chapter 2 Groups 27
Quotient Groups Suppose N is a normal subgroup of G, and C and D are
cosets. We wish to define a coset E which is the product of C and D. If c ∈ C and
d ∈ D, define E to be the coset containing c d, i.e., E = N(c d). The coset E does
not depend upon the choice of c and d. This is made precise in the next theorem,
which is quite easy.
Theorem Suppose G is a multiplicative group, N is a normal subgroup, and
G/N is the collection of all cosets. Then (Na) (Nb) = N(a b) is a well de-
fined multiplication (binary operation) on G/N, and with this multiplication, G/N
is a group. Its identity is N and (Na)
−1
= (Na
−1
). Furthermore, if G is finite,
o(G/N) = o(G)/o(N).
Proof Multiplication of elements in G/N is multiplication of subsets in G.
(Na) (Nb) = N(aN)b = N(Na)b = N(a b). Once multiplication is well defined,
the group axioms are immediate.
Exercise Write out the above theorem for G an additive group. In the additive
abelian group R/Z, determine those elements of finite order.
Example Suppose G = Z under +, n > 1, and N = nZ. Z
n
, the group of
integers mod n is defined by Z
n
= Z/nZ. If a is an integer, the coset a + nZ is
denoted by [a]. Note that [a] + [b] = [a + b], −[a] = [−a], and [a] = [a + nl] for any
integer l. Any additive abelian group has a scalar multiplication over Z, and in this
case it is just [a]m = [am]. Note that [a] = [r] where r is the remainder of a divided
by n, and thus the distinct elements of Z
n
are [0], [1], ..., [n − 1]. Also Z
n
is cyclic
because each of [1] and [−1] = [n −1] is a generator. We already know that if p is a
prime, any non-zero element of Z
p
is a generator, because Z
p
has p elements.
Theorem If n > 1 and a is any integer, then [a] is a generator of Z
n
iff (a, n) = 1.
Proof The element [a] is a generator iff the subgroup generated by [a] contains
[1] iff ∃ an integer k such that [a]k = [1] iff ∃ integers k and l such that ak +nl = 1.
Exercise Show that a positive integer is divisible by 3 iff the sum of its digits is
divisible by 3. Note that [10] = [1] in Z
3
. (See the fifth exercise on page 18.)
Homomorphisms
Homomorphisms are functions between groups that commute with the group op-
erations. It follows that they honor identities and inverses. In this section we list
28 Groups Chapter 2
the basic properties. Properties 11), 12), and 13) show the connections between coset
groups and homomorphisms, and should be considered as the cornerstones of abstract
algebra. As always, the student should rewrite the material in additive notation.
Definition If G and
¯
G are multiplicative groups, a function f : G →
¯
G is a
homomorphism if, for all a, b ∈ G, f(a b) = f(a) f(b). On the left side, the group
operation is in G, while on the right side it is in
¯
G. The kernel of f is defined by
ker(f) = f
−1
(¯ e) = ¦a ∈ G : f(a) = ¯ e¦. In other words, the kernel is the set of
solutions to the equation f(x) = ¯ e. (If
¯
G is an additive group, ker(f) = f
−1
(0
¯
).)
Examples The constant map f : G →
¯
G defined by f(a) = ¯ e is a homomorphism.
If H is a subgroup of G, the inclusion i : H → G is a homomorphism. The function
f : Z → Z defined by f(t) = 2t is a homomorphism of additive groups, while the
function defined by f(t) = t +2 is not a homomorphism. The function h : Z →R−0
defined by h(t) = 2
t
is a homomorphism from an additive group to a multiplicative
group.
We now catalog the basic properties of homomorphisms. These will be helpful
later on in the study of ring homomorphisms and module homomorphisms.
Theorem Suppose G and
¯
G are groups and f : G →
¯
G is a homomorphism.
1) f(e) = ¯ e.
2) f(a
−1
) = f(a)
−1
. The first inverse is in G, and the second is in
¯
G.
3) f is injective ⇔ ker(f) = e.
4) If H is a subgroup of G, f(H) is a subgroup of
¯
G. In particular, image(f) is
a subgroup of
¯
G.
5) If
¯
H is a subgroup of
¯
G, f
−1
(
¯
H) is a subgroup of G. Furthermore, if
¯
H is
normal in
¯
G, then f
−1
(
¯
H) is normal in G.
6) The kernel of f is a normal subgroup of G.
7) If ¯ g ∈
¯
G, f
−1
(¯ g) is void or is a coset of ker(f), i.e., if f(g) = ¯ g then
f
−1
(¯ g) = Ng where N= ker(f). In other words, if the equation f(x) = ¯ g has a
Chapter 2 Groups 29
solution, then the set of all solutions is a coset of N= ker(f). This is a key fact
which is used routinely in topics such as systems of equations and linear
differential equations.
8) The composition of homomorphisms is a homomorphism, i.e., if h :
¯
G →
=
G
is
a homomorphism, then h ◦ f : G →
=
G
is a homomorphism.
9) If f : G →
¯
G is a bijection, then the function f
−1
:
¯
G →G is a homomorphism.
In this case, f is called an isomorphism, and we write G ≈
¯
G. In the case
G =
¯
G, f is also called an automorphism.
10) Isomorphisms preserve all algebraic properties. For example, if f is an
isomorphism and H ⊂ G is a subset, then H is a subgroup of G
iff f(H) is a subgroup of
¯
G, H is normal in G iff f(H) is normal in
¯
G, G is
cyclic iff
¯
G is cyclic, etc. Of course, this is somewhat of a cop-out, because an
algebraic property is one that, by definition, is preserved under isomorphisms.
11) Suppose H is a normal subgroup of G. Then π : G →G/H defined by
π(a) = Ha is a surjective homomorphism with kernel H. Furthermore, if
f : G →
¯
G is a surjective homomorphism with kernel H, then G/H ≈
¯
G
(see below).
12) Suppose H is a normal subgroup of G. If H ⊂ ker(f), then
¯
f : G/H →
¯
G
defined by
¯
f(Ha) = f(a) is a well-defined homomorphism making
the following diagram commute.
G
¯
G
G/H
f
·
π
¯
f
Thus defining a homomorphism on a quotient group is the same as defining a
homomorphism on the numerator which sends the denominator to ¯ e. The
image of
¯
f is the image of f and the kernel of
¯
f is ker(f)/H. Thus if H = ker(f),
¯
f is injective, and thus G/H ≈ image(f).
13) Given any group homomorphism f, domain(f)/ker(f) ≈ image(f). This is
the fundamental connection between quotient groups and homomorphisms.
30 Groups Chapter 2
14) Suppose K is a group. Then K is an infinite cycle group iff K is isomorphic to
the integers under addition, i.e., K ≈ Z. K is a cyclic group of order n iff
K ≈ Z
n
.
Proof of 14) Suppose
¯
G = K is generated by some element a. Then f : Z →K
defined by f(m) = a
m
is a homomorphism from an additive group to a multiplicative
group. If o(a) is infinite, f is an isomorphism. If o(a) = n, ker(f) = nZ and
¯
f : Z
n
→K is an isomorphism.
Exercise If a is an element of a group G, there is always a homomorphism from Z
to G which sends 1 to a. When is there a homomorphism from Z
n
to G which sends [1]
to a? What are the homomorphisms from Z
2
to Z
6
? What are the homomorphisms
from Z
4
to Z
8
?
Exercise Suppose G is a group and g is an element of G, g = e.
1) Under what conditions on g is there a homomorphism f : Z
7
→G with
f([1]) = g ?
2) Under what conditions on g is there a homomorphism f : Z
15
→G with
f([1]) = g ?
3) Under what conditions on G is there an injective homomorphism f : Z
15
→G ?
4) Under what conditions on G is there a surjective homomorphism f : Z
15
→G ?
Exercise We know every finite group of prime order is cyclic and thus abelian.
Show that every group of order four is abelian.
Exercise Let G = ¦h : [0, 1] → R : h has an infinite number of derivatives¦.
Then G is a group under addition. Define f : G → G by f(h) =
dh
dt
= h
. Show f
is a homomorphism and find its kernel and image. Let g : [0, 1] → R be defined by
g(t) = t
3
−3t + 4. Find f
−1
(g) and show it is a coset of ker(f).
Exercise Let G be as above and g ∈ G. Define f : G →G by f(h) = h
+ 5h
+
6t
2
h. Then f is a group homomorphism and the differential equation h
+5h
+6t
2
h =
g has a solution iff g lies in the image of f. Now suppose this equation has a solution
and S ⊂ G is the set of all solutions. For which subgroup H of G is S an H-coset?
Chapter 2 Groups 31
Exercise Suppose G is a multiplicative group and a ∈ G. Define f : G → G to
be conjugation by a, i.e., f(g) = a
−1
g a. Show that f is a homomorphism. Also
show f is an automorphism and find its inverse.
Permutations
Suppose X is a (non-void) set. A bijection f : X → X is called a permutation
on X, and the collection of all these permutations is denoted by S = S(X). In this
setting, variables are written on the left, i.e., f = (x)f. Therefore the composition
f ◦g means "f followed by g". S(X) forms a multiplicative group under composition.
Exercise Show that if there is a bijection between X and Y , there is an iso-
morphism between S(X) and S(Y ). Thus if each of X and Y has n elements,
S(X) ≈ S(Y ), and these groups are called the symmetric groups on n elements.
They are all denoted by the one symbol S
n
.
Exercise Show that o(S
n
) = n!. Let X = ¦1, 2, ..., n¦, S
n
= S(X), and H =
¦f ∈ S
n
: (n)f = n¦. Show H is a subgroup of S
n
which is isomorphic to S
n−1
. Let
g be any permutation on X with (n)g = 1. Find g
−1
Hg.
The next theorem shows that the symmetric groups are incredibly rich and com-
plex.
Theorem (Cayley's Theorem) Suppose G is a multiplicative group with n
elements and S
n
is the group of all permutations on the set G. Then G is isomorphic
to a subgroup of S
n
.
Proof Let h : G →S
n
be the function which sends a to the bijection h
a
: G →G
defined by (g)h
a
= g a. The proof follows from the following observations.
1) For each given a, h
a
is a bijection from G to G.
2) h is a homomorphism, i.e., h
a·b
= h
a
◦ h
b
.
3) h is injective and thus G is isomorphic to image(h) ⊂ S
n
.
The Symmetric Groups Now let n ≥ 2 and let S
n
be the group of all permu-
tations on ¦1, 2, ..., n¦. The following definition shows that each element of S
n
may
32 Groups Chapter 2
be represented by a matrix.
Definition Suppose 1 < k ≤ n, ¦a
1
, a
2
, ..., a
k
¦ is a collection of distinct inte-
gers with 1 ≤ a
i
≤ n, and ¦b
1
, b
2
, ..., b
k
¦ is the same collection in some different order.
Then the matrix
a
1
a
2
... a
k
b
1
b
2
... b
k
represents f ∈ S
n
defined by (a
i
)f = b
i
for 1 ≤ i ≤ k,
and (a)f = a for all other a. The composition of two permutations is computed by
applying the matrix on the left first and the matrix on the right second.
There is a special type of permutation called a cycle. For these we have a special
notation.
Definition
a
1
a
2
...a
k−1
a
k
a
2
a
3
...a
k
a
1
is called a k-cycle, and is denoted by (a
1
, a
2
, ..., a
k
).
A 2-cycle is called a transposition. The cycles (a
1
, ..., a
k
) and (c
1
, ..., c
) are disjoint
provided a
i
= c
j
for all 1 ≤ i ≤ k and 1 ≤ j ≤ .
Listed here are eight basic properties of permutations. They are all easy except
4), which takes a little work. Properties 9) and 10) are listed solely for reference.
Theorem
1) Disjoint cycles commute. (This is obvious.)
2) Every nonidentity permutation can be written uniquely (except for order) as
the product of disjoint cycles. (This is easy.)
3) Every permutation can be written (non-uniquely) as the product of transposi-
tions. (Proof: I = (1, 2)(1, 2) and (a
1
, ..., a
k
) = (a
1
, a
2
)(a
1
, a
3
) (a
1
, a
k
). )
4) The parity of the number of these transpositions is unique. This means that if
f is the product of p transpositions and also of q transpositions, then p is
even iff q is even. In this case, f is said to be an even permutation. In the other
case, f is an odd permutation.
5) A k-cycle is even (odd) iff k is odd (even). For example (1, 2, 3) = (1, 2)(1, 3) is
an even permutation.
6) Suppose f, g ∈ S
n
. If one of f and g is even and the other is odd, then g ◦ f is
Chapter 2 Groups 33
odd. If f and g are both even or both odd, then g ◦ f is even. (Obvious.)
7) The map h : S
n
→Z
2
defined by h(even)= [0] and h(odd)= [1] is a
homomorphism from a multiplicative group to an additive group. Its kernel (the
subgroup of even permutations) is denoted by A
n
and is called the alternating
group. Thus A
n
is a normal subgroup of index 2, and S
n
/A
n
≈ Z
2
.
8) If a, b, c and d are distinct integers in ¦1, 2, . . . , n¦, then (a, b)(b, c) = (a, c, b)
and (a, b)(c, d) = (a, c, d)(a, c, b). Since I = (1, 2, 3)
3
, it follows that for
n ≥ 3, every even permutation is the product of 3-cycles.
The following parts are not included in this course. They are presented here merely
for reference.
9) For any n = 4, A
n
is simple, i.e., has no proper normal subgroups.
10) S
n
can be generated by two elements. In fact, ¦(1, 2), (1, 2, ..., n)¦ generates S
n
.
(Of course there are subgroups of S
n
which cannot be generated by two
elements).
Proof of 4) It suffices to prove if the product of t transpositions is the identity I
on ¦1, 2, . . . , n¦, then t is even. Suppose this is false and I is written as t transposi-
tions, where t is the smallest odd integer this is possible. Since t is odd, it is at least 3.
Suppose for convenience the first transposition is (a, n). We will rewrite I as a prod-
uct of transpositions σ
1
σ
2
σ
t
where (n)σ
i
= (n) for 1 ≤ i < t and (n)σ
t
= n, which
will be a contradiction. This can be done by inductively "pushing n to the right"
using the equations below. If a, b, and c are distinct integers in ¦1, 2, . . . , n − 1¦,
then (a, n)(a, n) = I, (a, n)(b, n) = (a, b)(a, n), (a, n)(a, c) = (a, c)(c, n), and
(a, n)(b, c) = (b, c)(a, n). Note that (a, n)(a, n) cannot occur here because it would
result in a shorter odd product. (Now you may solve the tile puzzle on page viii.)
Exercise
1) Write
1 2 3 4 5 6 7
6 5 4 3 1 7 2
as the product of disjoint cycles.
Write (1,5,6,7)(2,3,4)(3,7,1) as the product of disjoint cycles.
Write (3,7,1)(1,5,6,7)(2,3,4) as the product of disjoint cycles.
Which of these permutations are odd and which are even?
34 Groups Chapter 2
2) Suppose (a
1
, . . . , a
k
) and (c
1
, . . . , c
) are disjoint cycles. What is the order of
their product?
3) Suppose σ ∈ S
n
. Show that σ
−1
(1, 2, 3)σ = ((1)σ, (2)σ, (3)σ). This shows
that conjugation by σ is just a type of relabeling. Also let τ = (4, 5, 6) and
find τ
−1
(1, 2, 3, 4, 5)τ.
4) Show that H = ¦σ ∈ S
6
: (6)σ = 6¦ is a subgroup of S
6
and find its right
cosets and its left cosets.
5) Let A ⊂ R
2
be the square with vertices (−1, 1), (1, 1), (1, −1), and (−1, −1),
and G be the collection of all isometries of A onto itself. We know from a
previous exercise that G is a group with eight elements. It follows from Cayley's
theorem that G is isomorphic to a subgroup of S
8
. Show that G is isomorphic
to a subgroup of S
4
.
6) If G is a multiplicative group, define a new multiplication on the set G by
a ◦ b = b a. In other words, the new multiplication is the old multiplication
in the opposite order. This defines a new group denoted by G
op
, the opposite
group. Show that it has the same identity and the same inverses as G, and
that f : G →G
op
defined by f(a) = a
−1
is a group isomorphism. Now consider
the special case G = S
n
. The convention used in this section is that an element
of S
n
is a permutation on ¦1, 2, . . . , n¦ with the variable written on the left.
Show that an element of S
op
n
is a permutation on ¦1, 2, . . . , n¦ with the variable
written on the right. (Of course, either S
n
or S
op
n
may be called the symmetric
group, depending on personal preference or context.)
Product of Groups
The product of groups is usually presented for multiplicative groups. It is pre-
sented here for additive groups because this is the form that occurs in later chapters.
As an exercise, this section should be rewritten using multiplicative notation. The
two theorems below are transparent and easy, but quite useful. For simplicity we
first consider the product of two groups, although the case of infinite products is only
slightly more difficult. For background, read first the two theorems on page 11.
Theorem Suppose G
1
and G
2
are additive groups. Define an addition on G
1
G
2
by (a
1
, a
2
) +(b
1
, b
2
) = (a
1
+b
1
, a
2
+b
2
). This operation makes G
1
G
2
into a group.
Its "zero" is (0
¯
1
, 0
¯
2
) and −(a
1
, a
2
) = (−a
1
, −a
2
). The projections π
1
: G
1
G
2
→G
1
Chapter 2 Groups 35
and π
2
: G
1
G
2
→G
2
are group homomorphisms. Suppose G is an additive group.
We know there is a bijection from ¦functions f : G →G
1
G
2
¦ to ¦ordered pairs of
functions (f
1
, f
2
) where f
1
: G → G
1
and f
2
: G → G
2
¦. Under this bijection, f is a
group homomorphism iff each of f
1
and f
2
is a group homomorphism.
Proof It is transparent that the product of groups is a group, so let's prove
the last part. Suppose G, G
1
, and G
2
are groups and f = (f
1
, f
2
) is a function
from G to G
1
G
2
. Now f(a + b) = (f
1
(a + b), f
2
(a + b)) and f(a) + f(b) =
(f
1
(a), f
2
(a)) +(f
1
(b), f
2
(b)) = (f
1
(a) +f
1
(b), f
2
(a) +f
2
(b)). An examination of these
two equations shows that f is a group homomorphism iff each of f
1
and f
2
is a group
homomorphism.
Exercise Suppose G
1
and G
2
are groups. Show that G
1
G
2
and G
2
G
1
are
isomorphic.
Exercise If o(a
1
) = m and o(a
2
) = n, find the order of (a
1
, a
2
) in G
1
G
2
.
Exercise Show that if G is any group of order 4, G is isomorphic to Z
4
or Z
2
Z
2
.
Show Z
4
is not isomorphic to Z
2
Z
2
. Show Z
12
is isomorphic to Z
4
Z
3
. Finally,
show that Z
mn
is isomorphic to Z
m
Z
n
iff (m, n) = 1.
Exercise Suppose G
1
and G
2
are groups and i
1
: G
1
→ G
1
G
2
is defined by
i
1
(g
1
) = (g
1
, 0
¯
2
). Show i
1
is an injective group homomorphism and its image is a
normal subgroup of G
1
G
2
. Usually G
1
is identified with its image under i
1
, so G
1
may be considered to be a normal subgroup of G
1
G
2
. Let π
2
: G
1
G
2
→ G
2
be the projection map defined in the Background chapter. Show π
2
is a surjective
homomorphism with kernel G
1
. Therefore (G
1
G
2
)/G
1
≈ G
2
as you would expect.
Exercise Let R be the reals under addition. Show that the addition in the
product RR is just the usual addition in analytic geometry.
Exercise Suppose n > 2. Is S
n
isomorphic to A
n
G where G is a multiplicative
group of order 2 ?
One nice thing about the product of groups is that it works fine for any finite
number, or even any infinite number. The next theorem is stated in full generality.
36 Groups Chapter 2
Theorem Suppose T is an index set, and for any t ∈ T, G
t
is an additive
group. Define an addition on
¸
t∈T
G
t
=
¸
G
t
by ¦a
t
¦ + ¦b
t
¦ = ¦a
t
+ b
t
¦. This op-
eration makes the product into a group. Its "zero" is ¦0
¯
t
¦ and −¦a
t
¦ = ¦−a
t
¦.
Each projection π
s
:
¸
G
t
→ G
s
is a group homomorphism. Suppose G is an ad-
ditive group. Under the natural bijection from ¦functions f : G →
¸
G
t
¦ to
¦sequences of functions ¦f
t
¦
t∈T
where f
t
: G → G
t
¦, f is a group homomorphism
iff each f
t
is a group homomorphism. Finally, the scalar multiplication on
¸
G
t
by integers is given coordinatewise, i.e., ¦a
t
¦n = ¦a
t
n¦.
Proof The addition on
¸
G
t
is coordinatewise.
Exercise Suppose s is an element of T and π
s
:
¸
G
t
→G
s
is the projection map
defined in the Background chapter. Show π
s
is a surjective homomorphism and find
its kernel.
Exercise Suppose s is an element of T and i
s
: G
s
→
¸
G
t
is defined by i
s
(a) =
¦a
t
¦ where a
t
= 0
¯
if t = s and a
s
= a. Show i
s
is an injective homomorphism
and its image is a normal subgroup of
¸
G
t
. Thus each G
s
may be considered to be
a normal subgroup of
¸
G
t
.
Exercise Let f : Z → Z
30
Z
100
be the homomorphism defined by f(m) =
([4m], [3m]). Find the kernel of f. Find the order of ([4], [3]) in Z
30
Z
100
.
Exercise Let f : Z → Z
90
Z
70
Z
42
be the group homomorphism defined by
f(m) = ([m], [m], [m]). Find the kernel of f and show that f is not surjective. Let
g : Z → Z
45
Z
35
Z
21
be defined by g(m) = ([m], [m], [m]). Find the kernel of
g and determine if g is surjective. Note that the gcd of ¦45, 35, 21¦ is 1. Now let
h : Z → Z
8
Z
9
Z
35
be defined by h(m) = ([m], [m], [m]). Find the kernel of h
and show that h is surjective. Finally suppose each of b, c, and d is greater than 1
and f : Z → Z
b
Z
c
Z
d
is defined by f(m) = ([m], [m], [m]). Find necessary and
sufficient conditions for f to be surjective (see the first exercise on page 18).
Exercise Suppose T is a non-void set, G is an additive group, and G
T
is the
collection of all functions f : T →G with addition defined by (f +g)(t) = f(t) +g(t).
Show G
T
is a group. For each t ∈ T, let G
t
= G. Note that G
T
is just another way
of writing
¸
t∈T
G
t
. Also note that if T = [0, 1] and G = R, the addition defined on
G
T
is just the usual addition of functions used in calculus. (For the ring and module
versions, see exercises on pages 44 and 69.)
Chapter 3
Rings
Rings are additive abelian groups with a second operation called multiplication. The
connection between the two operations is provided by the distributive law. Assuming
the results of Chapter 2, this chapter flows smoothly. This is because ideals are also
normal subgroups and ring homomorphisms are also group homomorphisms. We do
not show that the polynomial ring F[x] is a unique factorization domain, although
with the material at hand, it would be easy to do. Also there is no mention of prime
or maximal ideals, because these concepts are unnecessary for our development of
linear algebra. These concepts are developed in the Appendix. A section on Boolean
rings is included because of their importance in logic and computer science.
Suppose R is an additive abelian group, R = 0
¯
, and R has a second binary
operation (i.e., map from R R to R) which is denoted by multiplication. Consider
the following properties.
1) If a, b, c ∈ R, (a b) c = a (b c). (The associative property
of multiplication.)
2) If a, b, c ∈ R, a (b +c) = (a b) + (a c) and (b +c) a = (b a) + (c a).
(The distributive law, which connects addition and
multiplication.)
3) R has a multiplicative identity, i.e., there is an element
1
¯
= 1
¯
R
∈ R such that if a ∈ R, a 1
¯
= 1
¯
a = a.
4) If a, b ∈ R, a b = b a. (The commutative property for
multiplication.)
Definition If 1), 2), and 3) are satisfied, R is said to be a ring. If in addition 4)
is satisfied, R is said to be a commutative ring.
Examples The basic commutative rings in mathematics are the integers Z, the
37
38 Rings Chapter 3
rational numbers Q, the real numbers R, and the complex numbers C. It will be shown
later that Z
n
, the integers mod n, has a natural multiplication under which it is a
commutative ring. Also if R is any commutative ring, we will define R[x
1
, x
2
, . . . , x
n
],
a polynomical ring in n variables. Now suppose R is any ring, n ≥ 1, and R
n
is the
collection of all nn matrices over R. In the next chapter, operations of addition and
multiplication of matrices will be defined. Under these operations, R
n
is a ring. This
is a basic example of a non-commutative ring. If n > 1, R
n
is never commutative,
even if R is commutative.
The next two theorems show that ring multiplication behaves as you would wish
it to. They should be worked as exercises.
Theorem Suppose R is a ring and a, b ∈ R.
1) a 0
¯
= 0
¯
a = 0
¯
. Since R = 0
¯
, it follows that 1
¯
= 0
¯
.
2) (−a) b = a (−b) = −(a b).
Recall that, since R is an additive abelian group, it has a scalar multiplication
over Z (page 20). This scalar multiplication can be written on the right or left, i.e.,
na = an, and the next theorem shows it relates nicely to the ring multiplication.
Theorem Suppose a, b ∈ R and n, m ∈ Z.
1) (na) (mb) = (nm)(a b). (This follows from the distributive
law and the previous theorem.)
2) Let n
¯
= n1
¯
. For example, 2
¯
= 1
¯
+ 1
¯
. Then na = n
¯
a, that is, scalar
multiplication by n is the same as ring multiplication by n
¯
.
Of course, n
¯
may be 0
¯
even though n = 0.
Units
Definition An element a of a ring R is a unit provided ∃ an element a
−1
∈ R
with a a
−1
= a
−1
a = 1
¯
.
Theorem 0
¯
can never be a unit. 1
¯
is always a unit. If a is a unit, a
−1
is also a
unit with (a
−1
)
−1
= a. The product of units is a unit with (a b)
−1
= b
−1
a
−1
. More
Chapter 3 Rings 39
generally, if a
1
, a
2
, ..., a
n
are units, then their product is a unit with (a
1
a
2
a
n
)
−1
=
a
−1
n
a
−1
n−1
a
−1
1
. The set of all units of R forms a multiplicative group denoted by
R
∗
. Finally if a is a unit, (−a) is a unit and (−a)
−1
= −(a
−1
).
In order for a to be a unit, it must have a two-sided inverse. It suffices to require
a left inverse and a right inverse, as shown in the next theorem.
Theorem Suppose a ∈ R and ∃ elements b and c with b a = a c = 1
¯
. Then
b = c and so a is a unit with a
−1
= b = c.
Proof b = b 1
¯
= b (a c) = (b a) c = 1
¯
c = c.
Corollary Inverses are unique.
Domains and Fields In order to define these two types of rings, we first consider
the concept of zero divisor.
Definition Suppose R is a commutative ring. An element a ∈ R is called a zero
divisor provided it is non-zero and ∃ a non-zero element b with a b = 0
¯
. Note that
if a is a unit, it cannot be a zero divisor.
Theorem Suppose R is a commutative ring and a ∈ (R−0
¯
) is not a zero divisor.
Then (a b = a c) ⇒b = c. In other words, multiplication by a is an injective map
from R to R. It is surjective iff a is a unit.
Definition A domain (or integral domain) is a commutative ring such that, if
a = 0
¯
, a is not a zero divisor. A field is a commutative ring such that, if a = 0
¯
, a is
a unit. In other words, R is a field if it is commutative and its non-zero elements
form a group under multiplication.
Theorem A field is a domain. A finite domain is a field.
Proof A field is a domain because a unit cannot be a zero divisor. Suppose R is
a finite domain and a = 0
¯
. Then f : R → R defined by f(b) = a b is injective and,
by the pigeonhole principle, f is surjective. Thus a is a unit and so R is a field.
40 Rings Chapter 3
Exercise Let C be the additive abelian group R
2
. Define multiplication by
(a, b) (c, d) = (ac − bd, ad + bc). Show C is a commutative ring which is a field.
Note that 1
¯
= (1, 0) and if i = (0, 1), then i
2
= −1
¯
.
Examples Z is a domain. Q, R, and C are fields.
The Integers Mod n
The concept of integers mod n is fundamental in mathematics. It leads to a neat
little theory, as seen by the theorems below. However, the basic theory cannot be
completed until the product of rings is defined. (See the Chinese Remainder Theorem
on page 50.) We know from page 27 that Z
n
is an additive abelian group.
Theorem Suppose n > 1. Define a multiplication on Z
n
by [a] [b] = [ab]. This
is a well defined binary operation which makes Z
n
into a commutative ring.
Proof Since [a +kn] [b +l n] = [ab +n(al +bk +kl n)] = [ab], the multiplication
is well-defined. The ring axioms are easily verified.
Theorem Suppose n > 1 and a ∈ Z. Then the following are equivalent.
1) [a] is a generator of the additive group Z
n
.
2) (a, n) = 1.
3) [a] is a unit of the ring Z
n
.
Proof We already know from page 27 that 1) and 2) are equivalent. Recall that
if b is an integer, [a]b = [a] [b] = [ab]. Thus 1) and 3) are equivalent, because each
says ∃ an integer b with [a]b = [1].
Corollary If n > 1, the following are equivalent.
1) Z
n
is a domain.
2) Z
n
is a field.
3) n is a prime.
Proof We already know 1) and 2) are equivalent, because Z
n
is finite. Suppose
3) is true. Then by the previous theorem, each of [1], [2],...,[n − 1] is a unit, and
thus 2) is true. Now suppose 3) is false. Then n = ab where 1 < a < n, 1 < b < n,
Chapter 3 Rings 41
[a][b] = [0], and thus [a] is a zero divisor and 1) is false.
Exercise List the units and their inverses for Z
7
and Z
12
. Show that (Z
7
)
∗
is
a cyclic group but (Z
12
)
∗
is not. Show that in Z
12
the equation x
2
= 1
¯
has four
solutions. Finally show that if R is a domain, x
2
= 1
¯
can have at most two solutions
in R (see the first theorem on page 46).
Subrings Suppose S is a subset of a ring R. The statement that S is a subring
of R means that S is a subgroup of the group R, 1
¯
∈ S , and (a, b ∈ S ⇒a b ∈ S).
Then clearly S is a ring and has the same multiplicative identity as R. Note that Z
is a subring of Q, Q is a subring of R, and R is a subring of C. Subrings do not play
a role analogous to subgroups. That role is played by ideals, and an ideal is never a
subring (unless it is the entire ring). Note that if S is a subring of R and s ∈ S, then
s may be a unit in R but not in S. Note also that Z and Z
n
have no proper subrings,
and thus occupy a special place in ring theory, as well as in group theory.
Ideals and Quotient Rings
Ideals in ring theory play a role analagous to normal subgroups in group theory.
Definition A subset I of a ring R is a
left
right
2−sided
ideal provided it is a subgroup
of the additive group R and if a ∈ R and b ∈ I, then
a b ∈ I
b a ∈ I
a b and b a ∈ I
. The
word "ideal " means "2-sided ideal". Of course, if R is commutative, every right or
left ideal is an ideal.
Theorem Suppose R is a ring.
1) R and 0
¯
are ideals of R. These are called the improper ideals.
2) If ¦I
t
¦
t∈T
is a collection of right (left, 2-sided) ideals of R, then
¸
t∈T
I
t
is a
right (left, 2-sided) ideal of R. (See page 22.)
42 Rings Chapter 3
3) Furthermore, if the collection is monotonic, then
¸
t∈T
I
t
is a right (left, 2-sided)
ideal of R.
4) If a ∈ R, I = aR is a right ideal. Thus if R is commutative, aR is an ideal,
called a principal ideal. Thus every subgroup of Z is a principal ideal,
because it is of the form nZ.
5) If R is a commutative ring and I ⊂ R is an ideal, then the following are
equivalent.
i) I = R.
ii) I contains some unit u.
iii) I contains 1
¯
.
Exercise Suppose R is a commutative ring. Show that R is a field iff R contains
no proper ideals.
The following theorem is just an observation, but it is in some sense the beginning
of ring theory.
Theorem Suppose R is a ring and I ⊂ R is an ideal, I = R. Since I is a normal
subgroup of the additive group R, R/I is an additive abelian group. Multiplication
of cosets defined by (a +I) (b +I) = (ab +I) is well-defined and makes R/I a ring.
Proof (a + I) (b + I) = a b + aI + Ib + II ⊂ a b + I. Thus multiplication
is well defined, and the ring axioms are easily verified. The multiplicative identity is
(1
¯
+I).
Observation If R = Z, n > 1, and I = nZ, the ring structure on Z
n
= Z/nZ
is the same as the one previously defined.
Homomorphisms
Definition Suppose R and
¯
R are rings. A function f : R →
¯
R is a ring homo-
morphism provided
1) f is a group homomorphism
2) f(1
¯
R
) = 1
¯
¯
R
and
3) if a, b ∈ R then f(a b) = f(a) f(b). (On the left, multiplication
Chapter 3 Rings 43
is in R, while on the right multiplication is in
¯
R.)
The kernel of f is the kernel of f considered as a group homomorphism, namely
ker(f) = f
−1
(0
¯
).
Here is a list of the basic properties of ring homomorphisms. Much of this
work has already been done by the theorem in group theory on page 28.
Theorem Suppose each of R and
¯
R is a ring.
1) The identity map I
R
: R →R is a ring homomorphism.
2) The zero map from R to
¯
R is not a ring homomorphism
(because it does not send 1
¯
R
to 1
¯
¯
R
).
3) The composition of ring homomorphisms is a ring homomorphism.
4) If f : R →
¯
R is a bijection which is a ring homomorphism,
then f
−1
:
¯
R →R is a ring homomorphism. Such an f is called
a ring isomorphism. In the case R =
¯
R, f is also called a
ring automorphism.
5) The image of a ring homomorphism is a subring of the range.
6) The kernel of a ring homomorphism is an ideal of the domain.
In fact, if f : R →
¯
R is a homomorphism and I ⊂
¯
R is an ideal,
then f
−1
(I) is an ideal of R.
7) Suppose I is an ideal of R, I = R, and π : R →R/I is the
natural projection, π(a) = (a +I). Then π is a surjective ring
homomorphism with kernel I. Furthermore, if f : R →
¯
R is a surjective
ring homomorphism with kernel I, then R/I ≈
¯
R (see below).
8) From now on the word "homomorphism" means "ring homomorphism".
Suppose f : R →
¯
R is a homomorphism and I is an ideal of R, I = R.
If I ⊂ ker(f), then
¯
f : R/I →
¯
R defined by
¯
f(a +I) = f(a)
44 Rings Chapter 3
is a well-defined homomorphism making the following diagram commute.
R
¯
R
R/I
f
·
π
¯
f
Thus defining a homomorphism on a quotient ring is the same as
defining a homomorphism on the numerator which sends the
denominator to zero. The image of
¯
f is the image of f, and
the kernel of
¯
f is ker(f)/I. Thus if I = ker(f),
¯
f is
injective, and so R/I ≈ image (f).
Proof We know all this on the group level, and it is only necessary
to check that
¯
f is a ring homomorphism, which is obvious.
9) Given any ring homomorphism f, domain(f)/ker(f) ≈ image(f).
Exercise Find a ring R with a proper ideal I and an element b such that b is not
a unit in R but (b +I) is a unit in R/I.
Exercise Show that if u is a unit in a ring R, then conjugation by u is an
automorphism on R. That is, show that f : R →R defined by f(a) = u
−1
a u is
a ring homomorphism which is an isomorphism.
Exercise Suppose T is a non-void set, R is a ring, and R
T
is the collection of
all functions f : T → R. Define addition and multiplication on R
T
point-wise. This
means if f and g are functions from T to R, then (f + g)(t) = f(t) + g(t) and
(f g)(t) = f(t)g(t). Show that under these operations R
T
is a ring. Suppose S is a
non-void set and α : S →T is a function. If f : T →R is a function, define a function
α
∗
(f) : S →R by α
∗
(f) = f ◦ α. Show α
∗
: R
T
→R
S
is a ring homomorphism.
Exercise Now consider the case T = [0, 1] and R = R. Let A ⊂ R
[0,1]
be the
collection of all C
∞
functions, i.e., A =¦f : [0, 1] →R : f has an infinite number of
derivatives¦. Show A is a ring. Notice that much of the work has been done in the
previous exercise. It is only necessary to show that A is a subring of the ring R
[0,1]
.
Chapter 3 Rings 45
Polynomial Rings
In calculus, we consider real functions f which are polynomials, f(x) = a
0
+a
1
x +
+a
n
x
n
. The sum and product of polynomials are again polynomials, and it is easy
to see that the collection of polynomial functions forms a commutative ring. We can
do the same thing formally in a purely algebraic setting.
Definition Suppose R is a commutative ring and x is a "variable" or "symbol".
The polynomial ring R[x] is the collection of all polynomials f = a
0
+a
1
x + +a
n
x
n
where a
i
∈ R. Under the obvious addition and multiplication, R[x] is a commutative
ring. The degree of a non-zero polynomial f is the largest integer n such that a
n
= 0
¯
,
and is denoted by n = deg(f). If the top term a
n
= 1
¯
, then f is said to be monic.
To be more formal, think of a polynomial a
0
+ a
1
x + as an infinite sequence
(a
0
, a
1
, ...) such that each a
i
∈ R and only a finite number are non-zero. Then
(a
0
, a
1
, ...) + (b
0
, b
1
, ...) = (a
0
+b
0
, a
1
+b
1
, ...) and
(a
0
, a
1
, ...) (b
0
, b
1
, ...) = (a
0
b
0
, a
0
b
1
+a
1
b
0
, a
0
b
2
+a
1
b
1
+a
2
b
0
, ...).
Note that on the right, the ring multiplication a b is written simply as ab, as is
often done for convenience.
Theorem If R is a domain, R[x] is also a domain.
Proof Suppose f and g are non-zero polynomials. Then deg(f)+deg(g) = deg(fg)
and thus fg is not 0
¯
. Another way to prove this theorem is to look at the bottom
terms instead of the top terms. Let a
i
x
i
and b
j
x
j
be the first non-zero terms of f
and g. Then a
i
b
j
x
i+j
is the first non-zero term of fg.
Theorem (The Division Algorithm) Suppose R is a commutative ring, f ∈
R[x] has degree ≥ 1 and its top coefficient is a unit in R. (If R is a field, the
top coefficient of f will always be a unit.) Then for any g ∈ R[x], ∃! h, r ∈ R[x]
such that g = fh +r with r = 0
¯
or deg(r) < deg(f).
Proof This theorem states the existence and uniqueness of polynomials h and
r. We outline the proof of existence and leave uniqueness as an exercise. Suppose
f = a
0
+ a
1
x + +a
m
x
m
where m ≥ 1 and a
m
is a unit in R. For any g with
deg(g) < m, set h = 0
¯
and r = g. For the general case, the idea is to divide f into g
until the remainder has degree less than m. The proof is by induction on the degree
of g. Suppose n ≥ m and the result holds for any polynomial of degree less than
46 Rings Chapter 3
n. Suppose g is a polynomial of degree n. Now ∃ a monomial bx
t
with t = n − m
and deg(g − fbx
t
) < n. By induction, ∃ h
1
and r with fh
1
+ r = (g − fbx
t
) and
deg(r) < m. The result follows from the equation f(h
1
+bx
t
) +r = g.
Note If r = 0
¯
we say that f divides g. Note that f = x − c divides g iff c is
a root of g, i.e., g(c) = 0
¯
. More generally, x −c divides g with remainder g(c).
Theorem Suppose R is a domain, n > 0, and g(x) = a
0
+ a
1
x + + a
n
x
n
is a
polynomial of degree n with at least one root in R. Then g has at most n roots. Let
c
1
, c
2
, .., c
k
be the distinct roots of g in the ring R. Then ∃ a unique sequence of
positive integers n
1
, n
2
, .., n
k
and a unique polynomial h with no root in R so that
g(x) = (x − c
1
)
n
1
(x − c
k
)
n
k
h(x). (If h has degree 0, i.e., if h = a
n
, then we say
"all the roots of g belong to R". If g = a
n
x
n
, we say "all the roots of g are 0
¯
".)
Proof Uniqueness is easy so let's prove existence. The theorem is clearly true
for n = 1. Suppose n > 1 and the theorem is true for any polynomial of degree less
than n. Now suppose g is a polynomial of degree n and c
1
is a root of g. Then ∃
a polynomial h
1
with g(x) = (x − c
1
)h
1
. Since h
1
has degree less than n, the result
follows by induction.
Note If g is any non-constant polynomial in C[x], all the roots of g belong to C,
i.e., C is an algebraically closed field. This is called The Fundamental Theorem of
Algebra, and it is assumed without proof for this textbook.
Exercise Suppose g is a non-constant polynomial in R[x]. Show that if g has
odd degree then it has a real root. Also show that if g(x) = x
2
+ bx + c, then it has
a real root iff b
2
≥ 4c, and in that case both roots belong to R.
Definition A domain T is a principal ideal domain (PID) if, given any ideal I,
∃ t ∈ T such that I = tT. Note that Z is a PID and any field is PID.
Theorem Suppose F is a field, I is a proper ideal of F[x], and n is the smallest
positive integer such that I contains a polynomial of degree n. Then I contains a
unique polynomial of the form f = a
0
+ a
1
x + +a
n−1
x
n−1
+ x
n
and it has the
property that I = fF[x]. Thus F[x] is a PID. Furthermore, each coset of I can be
written uniquely in the form (c
0
+c
1
x + +c
n−1
x
n−1
+I).
Proof. This is a good exercise in the use of the division algorithm. Note this is
similar to showing that a subgroup of Z is generated by one element (see page 15).
Chapter 3 Rings 47
Theorem. Suppose R is a subring of a commutative ring C and c ∈ C. Then
∃! homomorphism h : R[x] → C with h(x) = c and h(r) = r for all r ∈ R. It is
defined by h(a
0
+ a
1
x + +a
n
x
n
) = a
0
+ a
1
c + +a
n
c
n
, i.e., h sends f(x) to f(c).
The image of h is the smallest subring of C containing R and c.
This map h is called an evaluation map. The theorem says that adding two
polynomials in R[x] and evaluating is the same as evaluating and then adding in C.
Also multiplying two polynomials in R[x] and evaluating is the same as evaluating
and then multiplying in C. In street language the theorem says you are free to send
x wherever you wish and extend to a ring homomorphism on R[x].
Exercise Let C = ¦a + bi : a, b ∈ R¦. Since R is a subring of C, there exists a
homomorphism h : R[x] → C which sends x to i, and this h is surjective. Show
ker(h) = (x
2
+ 1)R[x] and thus R[x]/(x
2
+ 1) ≈ C. This is a good way to look
at the complex numbers, i.e., to obtain C, adjoin x to R and set x
2
= −1.
Exercise Z
2
[x]/(x
2
+ x + 1) has 4 elements. Write out the multiplication table
for this ring and show that it is a field.
Exercise Show that, if R is a domain, the units of R[x] are just the units of R.
Thus if F is a field, the units of F[x] are the non-zero constants. Show that [1] +[2]x
is a unit in Z
4
[x].
In this chapter we do not prove F[x] is a unique factorization domain, nor do
we even define unique factorization domain. The next definition and theorem are
included merely for reference, and should not be studied at this stage.
Definition Suppose F is a field and f ∈ F[x] has degree ≥ 1. The statement
that g is an associate of f means ∃ a unit u ∈ F[x] such that g = uf. The statement
that f is irreducible means that if h is a non-constant polynomial which divides f,
then h is an associate of f.
We do not develop the theory of F[x] here. However, the development is easy
because it corresponds to the development of Z in Chapter 1. The Division Algo-
rithm corresponds to the Euclidean Algorithm. Irreducible polynomials correspond
to prime integers. The degree function corresponds to the absolute value function.
One difference is that the units of F[x] are non-zero constants, while the units of Z
48 Rings Chapter 3
are just ±1. Thus the associates of f are all cf with c = 0
¯
while the associates of an
integer n are just ±n. Here is the basic theorem. (This theory is developed in full in
the Appendix under the topic of Euclidean domains.)
Theorem Suppose F is a field and f ∈ F[x] has degree ≥ 1. Then f factors as the
product of irreducibles, and this factorization is unique up to order and associates.
Also the following are equivalent.
1) F[x]/(f) is a domain.
2) F[x]/(f) is a field.
3) f is irreducible.
Definition Now suppose x and y are "variables". If a ∈ R and n, m ≥ 0, then
ax
n
y
m
= ay
m
x
n
is called a monomial. Define an element of R[x, y] to be any finite
sum of monomials.
Theorem R[x, y] is a commutative ring and (R[x])[y] ≈ R[x, y] ≈ (R[y])[x]. In
other words, any polynomial in x and y with coefficients in R may be written as a
polynomial in y with coefficients in R[x], or as a polynomial in x with coefficients in
R[y].
Side Comment It is true that if F is a field, each f ∈ F[x, y] factors as the
product of irreducibles. However F[x, y] is not a PID. For example, the ideal
I = xF[x, y] +yF[x, y] = ¦f ∈ F[x, y] : f(0
¯
, 0
¯
) = 0
¯
¦ is not principal.
If R is a commutative ring and n ≥ 2, the concept of a polynomial ring in
n variables works fine without a hitch. If a ∈ R and v
1
, v
2
, ..., v
n
are non-negative
integers, then ax
v
1
1
x
v
2
2
x
vn
n
is called a monomial. Order does not matter here.
Define an element of R[x
1
, x
2
, ..., x
n
] to be any finite sum of monomials. This
gives a commutative ring and there is canonical isomorphism R[x
1
, x
2
, ..., x
n
] ≈
(R[x
1
, x
2
, ..., x
n−1
])[x
n
]. Using this and induction on n, it is easy to prove the fol-
lowing theorem.
Theorem If R is a domain, R[x
1
, x
2
, ..., x
n
] is a domain and its units are just the
units of R.
Chapter 3 Rings 49
Exercise Suppose R is a commutative ring and f : R[x, y] → R[x] is the eval-
uation map which sends y to 0
¯
. This means f(p(x, y)) = p(x, 0
¯
). Show f is a ring
homomorphism whose kernel is the ideal (y) = yR[x, y]. Use the fact that "the do-
main mod the kernel is isomorphic to the image" to show R[x, y]/(y) is isomorphic
to R[x]. That is, if you adjoin y to R[x] and then factor it out, you get R[x] back.
Product of Rings
The product of rings works fine, just as does the product of groups.
Theorem Suppose T is an index set and for each t ∈ T, R
t
is a ring. On the
additive abelian group
¸
t∈T
R
t
=
¸
R
t
, define multiplication by ¦r
t
¦ ¦s
t
¦ = ¦r
t
s
t
¦.
Then
¸
R
t
is a ring and each projection π
s
:
¸
R
t
→ R
s
is a ring homomorphism.
Suppose R is a ring. Under the natural bijection from ¦functions f : R →
¸
R
t
¦
to ¦sequences of functions ¦f
t
¦
t∈T
where f
t
: R → R
t
¦, f is a ring homomorphism
iff each f
t
is a ring homomorphism.
Proof We already know f is a group homomorphism iff each f
t
is a group homo-
morphism (see page 36). Note that ¦1
¯
t
¦ is the multiplicative identity of
¸
R
t
, and
f(1
¯
R
) = ¦1
¯
t
¦ iff f
t
(1
¯
R
) = 1
¯
t
for each t ∈ T. Finally, since multiplication is defined
coordinatewise, f is a ring homomorphism iff each f
t
is a ring homomorphism.
Exercise Suppose R and S are rings. Note that R 0 is not a subring of R S
because it does not contain (1
¯
R
, 1
¯
S
). Show R0
¯
is an ideal and (RS/R0
¯
) ≈ S.
Suppose I ⊂ R and J ⊂ S are ideals. Show I J is an ideal of RS and every
ideal of R S is of this form.
Exercise Suppose R and S are commutative rings. Show T = R S is not a
domain. Let e = (1, 0) ∈ RS and show e
2
= e, (1 −e)
2
= (1 −e), R0 = eT,
and 0 S = (1 −e)T.
Exercise If T is any ring, an element e of T is called an idempotent provided
e
2
= e. The elements 0 and 1 are idempotents called the trivial idempotents. Suppose
T is a commutative ring and e ∈ T is an idempotent with 0 = e = 1. Let R = eT
and S = (1 − e)T. Show each of the ideals R and S is a ring with identity, and
f : T →RS defined by f(t) = (et, (1−e)t) is a ring isomorphism. This shows that
a commutative ring T splits as the product of two rings iff it contains a non-trivial
idempotent.
50 Rings Chapter 3
The Chinese Remainder Theorem
The natural map from Z to Z
m
Z
n
is a group homomorphism and also a ring
homomorphism. If m and n are relatively prime, this map is surjective with kernel
mnZ, and thus Z
mn
and Z
m
Z
n
are isomorphic as groups and as rings. The next
theorem is a classical generalization of this. (See exercise three on page 35.)
Theorem Suppose n
1
, ..., n
t
are integers, each n
i
> 1, and (n
i
, n
j
) = 1 for all
i = j. Let f
i
: Z → Z
n
i
be defined by f
i
(a) = [a]. (Note that the bracket symbol is
used ambiguously.) Then the ring homomorphism f = (f
1
, .., f
t
) : Z →Z
n
1
Z
nt
is surjective. Furthermore, the kernel of f is nZ, where n = n
1
n
2
n
t
. Thus Z
n
and Z
n
1
Z
nt
are isomorphic as rings, and thus also as groups.
Proof We wish to show that the order of f(1) is n, and thus f(1) is a group
generator, and thus f is surjective. The element f(1)m = ([1], .., [1])m = ([m], .., [m])
is zero iff m is a multiple of each of n
1
, .., n
t
. Since their least common multiple is n,
the order of f(1) is n. (See the fourth exercise on page 36 for the case t = 3.)
Exercise Show that if a is an integer and p is a prime, then [a] = [a
p
] in Z
p
(Fermat's Little Theorem). Use this and the Chinese Remainder Theorem to show
that if b is a positive integer, it has the same last digit as b
5
.
Characteristic
The following theorem is just an observation, but it shows that in ring theory, the
ring of integers is a "cornerstone".
Theorem If R is a ring, there is one and only one ring homomorphism f : Z →R.
It is given by f(m) = m1
¯
= m
¯
. Thus the subgroup of R generated by 1
¯
is a subring
of R isomorphic to Z or isomorphic to Z
n
for some positive integer n.
Definition Suppose R is a ring and f : Z → R is the natural ring homomor-
phism f(m) = m1
¯
= m
¯
. The non-negative integer n with ker(f) = nZ is called the
characteristic of R. Thus f is injective iff R has characteristic 0 iff 1
¯
has infinite
order. If f is not injective, the characteristic of R is the order of 1
¯
.
It is an interesting fact that, if R is a domain, all the non-zero elements of R
have the same order. (See page 23 for the definition of order.)
Chapter 3 Rings 51
Theorem Suppose R is a domain. If R has characteristic 0, then each non-zero
a ∈ R has infinite order. If R has finite characteristic n, then n is a prime and each
non-zero a ∈ R has order n.
Proof Suppose R has characteristic 0, a is a non-zero element of R, and m is a
positive integer. Then ma = m
¯
a cannot be 0
¯
because m
¯
, a = 0
¯
and R is a domain.
Thus o(a) = ∞. Now suppose R has characteristic n. Then R contains Z
n
as a
subring, and thus Z
n
is a domain and n is a prime. If a is a non-zero element of R,
na = n
¯
a = 0
¯
a = 0
¯
and thus o(a)[n and thus o(a) = n.
Exercise Show that if F is a field of characteristic 0, F contains Q as a subring.
That is, show that the injective homomorphism f : Z → F extends to an injective
homomorphism
¯
f : Q →F.
Boolean Rings
This section is not used elsewhere in this book. However it fits easily here, and is
included for reference.
Definition A ring R is a Boolean ring if for each a ∈ R, a
2
= a, i.e., each
element of R is an idempotent.
Theorem Suppose R is a Boolean ring.
1) R has characteristic 2. If a ∈ R, 2a = a +a = 0
¯
, and so a = −a.
Proof (a +a) = (a +a)
2
= a
2
+ 2a
2
+a
2
= 4a. Thus 2a = 0
¯
.
2) R is commutative.
Proof (a +b) = (a +b)
2
= a
2
+ (a b) + (b a) +b
2
= a + (a b) −(b a) +b. Thus a b = b a.
3) If R is a domain, R ≈ Z
2
.
Proof Suppose a = 0
¯
. Then a (1
¯
−a) = 0
¯
and so a = 1
¯
.
4) The image of a Boolean ring is a Boolean ring. That is, if I is an ideal
of R with I = R, then every element of R/I is idempotent and thus
R/I is a Boolean ring. It follows from 3) that R/I is a domain iff R/I
is a field iff R/I ≈ Z
2
. (In the language of Chapter 6, I is a prime
ideal iff I is a maximal ideal iff R/I ≈ Z
2
).
52 Rings Chapter 3
Suppose X is a non-void set. If a is a subset of X, let a
= (X−a) be a complement
of a in X. Now suppose R is a non-void collection of subsets of X. Consider the
following properties which the collection R may possess.
1) a ∈ R ⇒ a
belongs to
R and so 3) is true. Since R is non-void, it contains some element a. Then ∅ = a ∩a
and X = a ∪ a
belong to R, and so 4) is true.
Theorem Suppose R is a Boolean algebra of sets. Define an addition on R by
a +b = (a ∪ b) −(a ∩ b). Under this addition, R is an abelian group with 0
¯
= ∅ and
a = −a. Define a multiplication on R by a b = a ∩ b. Under this multiplication R
becomes a Boolean ring with 1
¯
= X.
Exercise Let X = ¦1, 2, ..., n¦ and let R be the Boolean ring of all subsets of
X. Note that o(R) = 2
n
. Define f
i
: R → Z
2
by f
i
(a) = [1] iff i ∈ a. Show each
f
i
is a homomorphism and thus f = (f
1
, ..., f
n
) : R → Z
2
Z
2
Z
2
is a ring
homomorphism. Show f is an isomorphism. (See exercises 1) and 4) on page 12.)
Exercise Use the last exercise on page 49 to show that any finite Boolean ring is
isomorphic to Z
2
Z
2
Z
2
, and thus also to the Boolean ring of subsets above.
Note Suppose R is a Boolean ring. It is a classical theorem that ∃ a Boolean
algebra of sets whose Boolean ring is isomorphic to R. So let's just suppose R is
a Boolean algebra of sets which is a Boolean ring with addition and multiplication
defined as above. Now define a ∨ b = a ∪ b and a ∧ b = a ∩ b. These operations cup
and cap are associative, commutative, have identity elements, and each distributes
over the other. With these two operations (along with complement), R is called a
Boolean algebra. R is not a group under cup or cap. Anyway, it is a classical fact
that, if you have a Boolean ring (algebra), you have a Boolean algebra (ring). The
advantage of the algebra is that it is symmetric in cup and cap. The advantage of
the ring viewpoint is that you can draw from the rich theory of commutative rings.
Chapter 4
Matrices and Matrix Rings
We first consider matrices in full generality, i.e., over an arbitrary ring R. However,
after the first few pages, it will be assumed that R is commutative. The topics,
such as invertible matrices, transpose, elementary matrices, systems of equations,
and determinant, are all classical. The highlight of the chapter is the theorem that a
square matrix is a unit in the matrix ring iff its determinant is a unit in the ring.
This chapter concludes with the theorem that similar matrices have the same deter-
minant, trace, and characteristic polynomial. This will be used in the next chapter
to show that an endomorphism on a finitely generated vector space has a well-defined
determinant, trace, and characteristic polynomial.
Definition Suppose R is a ring and m and n are positive integers. Let R
m,n
be
the collection of all mn matrices
A = (a
i,j
) =
¸
¸
¸
a
1,1
. . . a
1,n
.
.
.
.
.
.
a
m,1
. . . a
m,n
¸
where each entry a
i,j
∈ R.
A matrix may be viewed as m n-dimensional row vectors or as n m-dimensional
column vectors. A matrix is said to be square if it has the same number of rows
as columns. Square matrices are so important that they have a special notation,
R
n
= R
n,n
. R
n
is defined to be the additive abelian group R R R.
To emphasize that R
n
does not have a ring structure, we use the "sum" notation,
R
n
= R⊕R⊕ ⊕R. Our convention is to write elements of R
n
as column vectors,
i.e., to identify R
n
with R
n,1
. If the elements of R
n
are written as row vectors, R
n
is
identified with R
1,n
.
53
54 Matrices Chapter 4
Addition of matrices To "add" two matrices, they must have the same number
of rows and the same number of columns, i.e., addition is a binary operation R
m,n
R
m,n
→R
m,n
. The addition is defined by (a
i,j
) +(b
i,j
) = (a
i,j
+b
i,j
), i.e., the i, j term
of the sum is the sum of the i, j terms. The following theorem is just an observation.
Theorem R
m,n
is an additive abelian group. Its "zero" is the matrix 0 = 0
m,n
all of whose terms are zero. Also −(a
i,j
) = (−a
i,j
). Furthermore, as additive groups,
R
m,n
≈ R
mn
.
Scalar multiplication An element of R is called a scalar. A matrix may be
"multiplied" on the right or left by a scalar. Right scalar multiplication is defined
by (a
i,j
)c = (a
i,j
c). It is a function R
m,n
R → R
m,n
. Note in particular that
scalar multiplication is defined on R
n
. Of course, if R is commutative, there is no
distinction between right and left scalar multiplication.
Theorem Suppose A, B ∈ R
m,n
and c, d ∈ R. Then
(A+B)c = Ac +Bc
A(c +d) = Ac +Ad
A(cd) = (Ac)d
and A1 = A
This theorem is entirely transparent. In the language of the next chapter, it merely
states that R
m,n
is a right module over the ring R.
Multiplication of Matrices The matrix product AB is defined iff the number
of columns of A is equal to the number of rows of B. The matrix AB will have the
same number of rows as A and the same number of columns as B, i.e., multiplication
is a function R
m,n
R
n,p
→R
m,p
. The product (a
i,j
)(b
i,j
) is defined to be the matrix
whose (s, t) term is a
s,1
b
1,t
+ + a
s,n
b
n,t
, i.e., the dot product of row s of A
with column t of B.
Exercise Consider real matrices A =
=
¸
j
a
s,j
y
j,t
which is the (s, t) term of A(BC).
Theorem For each ring R and integer n ≥ 1, R
n
is a ring.
Proof This elegant little theorem is immediate from the theorems above. The
units of R
n
are called invertible or non-singular matrices. They form a group under
multiplication called the general linear group and denoted by GL
n
(R) = (R
n
)
∗
.
Exercise Recall that if A is a ring and a ∈ A, then aA is right ideal of A. Let
A = R
2
and a = (a
i,j
) where a
1,1
= 1 and the other entries are 0. Find aR
2
and R
2
a.
Show that the only ideal of R
2
containing a is R
2
itself.
Multiplication by blocks Suppose A, E ∈ R
n
, B, F ∈ R
n,m
, C, G ∈ R
m,n
, and
D, H ∈ R
m
. Then multiplication in R
n+m
is given by
A B
C D
E F
G H
=
AE +BG AF +BH
CE +DG CF +DH
.
56 Matrices Chapter 4
Transpose
Notation For the remainder of this chapter on matrices, suppose R is a commu-
tative ring. Of course, for n > 1, R
n
is non-commutative.
Transpose is a function from R
m,n
to R
n,m
. If A ∈ R
m,n
, A
t
∈ R
n,m
is the matrix
whose (i, j) term is the (j, i) term of A. So row i (column i) of A becomes column
i (row i) of A
t
. If A is an n-dimensional row vector, then A
t
is an n-dimensional
column vector. If A is a square matrix, A
t
is also square.
Theorem 1) (A
t
)
t
= A
2) (A+B)
t
= A
t
+B
t
3) If c ∈ R, (Ac)
t
= A
t
c
4) (AB)
t
= B
t
A
t
5) If A ∈ R
n
, then A is invertible iff A
t
is invertible.
In this case (A
−1
)
t
= (A
t
)
−1
.
Proof of 5) Suppose A is invertible. Then I = I
t
= (AA
−1
)
t
= (A
−1
)
t
A
t
.
Exercise Characterize those invertible matrices A ∈ R
2
which have A
−1
= A
t
.
Show that they form a subgroup of GL
2
(R).
Triangular Matrices
If A ∈ R
n
, then A is upper (lower) triangular provided a
i,j
= 0 for all i > j (all
j > i). A is strictly upper (lower) triangular provided a
i,j
= 0 for all i ≥ j (all j ≥ i).
A is diagonal if it is upper and lower triangular, i.e., a
i,j
= 0 for all i = j. Note
that if A is upper (lower) triangular, then A
t
is lower (upper) triangular.
Theorem If A ∈ R
n
is strictly upper (or lower) triangular, then A
n
= 0.
Proof The way to understand this is just multiply it out for n = 2 and n = 3.
The geometry of this theorem will become transparent later in Chapter 5 when the
matrix A defines an R-module endomorphism on R
n
(see page 93).
Definition If T is any ring, an element t ∈ T is said to be nilpotent provided ∃n
such that t
n
= 0. In this case, (1 − t) is a unit with inverse 1 + t + t
2
+ + t
n−1
.
Thus if T = R
n
and B is a nilpotent matrix, I −B is invertible.
Chapter 4 Matrices 57
Exercise Let R = Z. Find the inverse of
¸
¸
1 2 −3
0 1 4
0 0 1
¸
.
Exercise Suppose A =
¸
¸
¸
¸
¸
¸
¸
a
1
a
2
0
0
a
n
¸
is a diagonal matrix, B ∈ R
m,n
,
and C ∈ R
n,p
. Show that BA is obtained from B by multiplying column i of B
by a
i
. Show AC is obtained from C by multiplying row i of C by a
i
. Show A is a
unit in R
n
iff each a
i
is a unit in R.
Scalar matrices A scalar matrix is a diagonal matrix for which all the diagonal
terms are equal, i.e., a matrix of the form cI
n
. The map R → R
n
which sends c to
cI
n
is an injective ring homomorphism, and thus we may consider R to be a subring
of R
n
. Multiplying by a scalar is the same as multiplying by a scalar matrix, and
thus scalar matrices commute with everything, i.e., if B ∈ R
n
, (cI
n
)B = cB = Bc =
B(cI
n
). Recall we are assuming R is a commutative ring.
Exercise Suppose A ∈ R
n
and for each B ∈ R
n
, AB = BA. Show A is a scalar
matrix. For n > 1, this shows how non-commutative R
n
is.
Elementary Operations and Elementary Matrices
Suppose R is a commutative ring and A is a matrix over R. There are 3 types of
elementary row and column operations on the matrix A. A need not be square.
Type 1 Multiply row i by some Multiply column i by some
unit a ∈ R. unit a ∈ R.
Type 2 Interchange row i and row j. Interchange column i and column j.
Type 3 Add a times row j Add a times column i
to row i where i = j and a to column j where i = j and a
is any element of R. is any element of R.
58 Matrices Chapter 4
Elementary Matrices Elementary matrices are square and invertible. There
are three types. They are obtained by performing row or column operations on the
identity matrix.
Type 1 B =
¸
¸
¸
¸
¸
¸
¸
¸
¸
1
1 0
a
1
0 1
1
¸
where a is a unit in R.
Type 2 B =
¸
¸
¸
¸
¸
¸
¸
¸
¸
1
0 1
1
1
1 0
1
¸
Type 3 B =
¸
¸
¸
¸
¸
¸
¸
¸
¸
1
1 a
i,j
1
1
0 1
1
¸
where i = j and a
i,j
is
any element of R.
In type 1, all the off-diagonal elements are zero. In type 2, there are two non-zero
off-diagonal elements. In type 3, there is at most one non-zero off-diagonal element,
and it may be above or below the diagonal.
Exercise Show that if B is an elementary matrix of type 1,2, or 3, then B is
invertible and B
−1
is an elementary matrix of the same type.
The following theorem is handy when working with matrices.
Theorem Suppose A is a matrix. It need not be square. To perform an elemen-
tary row (column) operation on A, perform the operation on an identity matrix to
obtain an elementary matrix B, and multiply on the left (right). That is, BA = row
operation on A and AB = column operation on A. (See the exercise on page 54.)
Chapter 4 Matrices 59
Exercise Suppose F is a field and A ∈ F
m,n
.
1) Show ∃ invertible matrices B ∈ F
m
and C ∈ F
n
such that BAC = (d
i,j
)
where d
1,1
= = d
t,t
= 1 and all other entries are 0. The integer t is
called the rank of A. (See page 89 of Chapter 5.)
2) Suppose A ∈ F
n
is invertible. Show A is the product of elementary
matrices.
3) A matrix T is said to be in row echelon form if, for each 1 ≤ i < m, the
first non-zero term of row (i + 1) is to the right of the first non-zero
term of row i. Show ∃ an invertible matrix B ∈ F
m
such that BA is in
row echelon form.
4) Let A =
3 11
0 4
and D =
3 11
1 4
. Write A and D as products
of elementary matrices over Q. Is it possible to write them as products
of elementary matrices over Z?
For 1), perform row and column operations on A to reach the desired form. This
shows the matrices B and C may be selected as products of elementary matrices.
Part 2) also follows from this procedure. For part 3), use only row operations. Notice
that if T is in row-echelon form, the number of non-zero rows is the rank of T.
Systems of Equations
Suppose A = (a
i,j
) ∈ R
m,n
and C =
or AX = C.
60 Matrices Chapter 4
Define f : R
n
→R
m
by f(D) = AD. Then f is a group homomorphism and also
f(Dc) = f(D)c for any c ∈ R. In the language of the next chapter, this says that
f is an R-module homomorphism. The next theorem summarizes what we already
know about solutions of linear equations in this setting.
Theorem
1) AX = 0 is called the homogeneous equation. Its solution set is ker(f).
2) AX = C has a solution iff C ∈ image(f). If D ∈ R
n
is one
solution, the solution set f
−1
(C) is the coset D + ker(f) in R
n
.
(See part 7 of the theorem on homomorphisms in Chapter 2, page 28.)
3) Suppose B ∈ R
m
is invertible. Then AX = C and (BA)X = BC have
the same set of solutions. Thus we may perform any row operation
on both sides of the equation and not change the solution set.
4) If m = n and A ∈ R
m
is invertible, then AX = C has the unique
solution X = A
−1
C.
The geometry of systems of equations over a field will not become really trans-
parent until the development of linear algebra in Chapter 5.
Determinants
The concept of determinant is one of the most amazing in all of mathematics.
The proper development of this concept requires a study of multilinear forms, which
is given in Chapter 6. In this section we simply present the basic properties.
For each n ≥ 1 and each commutative ring R, determinant is a function from R
n
to R. For n = 1, [ (a) [ = a. For n = 2,
a b
c d
= ad −bc.
Definition Let A = (a
i,j
) ∈ R
n
. If σ is a permutation on ¦1, 2, ..., n¦, let sign(σ) =
1 if σ is an even permutation, and sign(σ) = −1 if σ is an odd permutation. The
determinant is defined by [ A [=
¸
all σ
sign(σ) a
1,σ(1)
a
2,σ(2)
a
n,σ(n)
. Check that for
n = 2, this agrees with the definition above. (Note that here we are writing the
permutation functions as σ(i) and not as (i)σ.)
Chapter 4 Matrices 61
For each σ, a
1,σ(1)
a
2,σ(2)
a
n,σ(n)
contains exactly one factor from each row and
one factor from each column. Since R is commutative, we may rearrange the factors
so that the first comes from the first column, the second from the second column, etc.
This means that there is a permutation τ on ¦1, 2, . . . , n¦ such that a
1,σ(1)
a
n,σ(n)
=
a
τ(1),1
a
τ(n),n
. We wish to show that τ = σ
−1
and thus sign(σ) = sign(τ). To
reduce the abstraction, suppose σ(2) = 5. Then the first expression will contain
the factor a
2,5
. In the second expression, it will appear as a
τ(5),5
, and so τ(5) = 2.
Anyway, τ is the inverse of σ and thus there are two ways to define determinant. It
follows that the determinant of a matrix is equal to the determinant of its transpose.
Theorem [A[ =
¸
all σ
sign(σ)a
1,σ(1)
a
2,σ(2)
a
n,σ(n)
=
¸
all τ
sign(τ)a
τ(1),1
a
τ(2),2
a
τ(n),n
.
Corollary [A[ = [A
t
[.
You may view an n n matrix A as a sequence of n column vectors or as a
sequence of n row vectors. Here we will use column vectors. This means we write the
matrix A as A = (A
1
, A
2
, . . . , A
n
) where each A
i
∈ R
n,1
= R
n
.
Theorem If two columns of A are equal, then [A[ = 0
¯
.
Proof For simplicity, assume the first two columns are equal, i.e., A
1
= A
2
.
Now [A[ =
¸
all τ
sign(τ)a
τ(1),1
a
τ(2),2
a
τ(n),n
and this summation has n! terms and
n! is an even number. Let γ be the transposition which interchanges one and two.
Then for any τ, a
τ(1),1
a
τ(2),2
a
τ(n),n
= a
τγ(1),1
a
τγ(2),2
a
τγ(n),n
. This pairs up
the n! terms of the summation, and since sign(τ)=−sign(τγ), these pairs cancel
in the summation. Therefore [A[ = 0
¯
.
Theorem Suppose 1 ≤ r ≤ n, C
r
∈ R
n,1
, and a, c ∈ R. Then [(A
1
, . . . , A
r−1
,
aA
r
+cC
r
, A
r+1
, . . . , A
n
)[ = a[(A
1
, . . . , A
n
)[ + c[(A
1
, . . . , A
r−1
, C
r
, A
r+1
, . . . , A
n
)[
Proof This is immediate from the definition of determinant and the distributive
law of multiplication in the ring R.
Summary Determinant is a function d : R
n
→ R. In the language used in the
Appendix, the two previous theorems say that d is an alternating multilinear form.
The next two theorems show that alternating implies skew-symmetric (see page 129).
62 Matrices Chapter 4
Theorem Interchanging two columns of A multiplies the determinant by minus
one.
Proof For simplicity, show that [(A
2
, A
1
, A
3
, . . . , A
n
)[ = −[A[. We know 0
¯
=
[(A
1
+ A
2
, A
1
+ A
2
, A
3
, . . . , A
n
)[ = [(A
1
, A
1
, A
3
, . . . , A
n
)[ + [(A
1
, A
2
, A
3
, . . . , A
n
)[ +
[(A
2
, A
1
, A
3
, . . . , A
n
)[ + [(A
2
, A
2
, A
3
, . . . , A
n
)[. Since the first and last of these four
terms are zero, the result follows.
Theorem If τ is a permutation of (1, 2, . . . , n), then
[A[ = sign(τ)[(A
τ(1)
, A
τ(2)
, . . . , A
τ(n)
)[.
Proof The permutation τ is the finite product of transpositions.
Exercise Rewrite the four preceding theorems using rows instead of columns.
The following theorem is just a summary of some of the work done so far.
Theorem Multiplying any row or column of matrix by a scalar c ∈ R, multiplies
the determinant by c. Interchanging two rows or two columns multiplies the determi-
nant by −1. Adding c times one row to another row, or adding c times one column
to another column, does not change the determinant. If a matrix has two rows equal
or two columns equal, its determinant is zero. More generally, if one row is c times
another row, or one column is c times another column, then the determinant is zero.
There are 2n ways to compute [ A[; expansion by any row or expansion by any
column. Let M
i,j
be the determinant of the (n − 1) (n − 1) matrix obtained by
removing row i and column j from A. Let C
i,j
= (−1)
i+j
M
i,j
. M
i,j
and C
i,j
are
called the (i, j) minor and cofactor of A. The following theorem is useful but the
proof is a little tedious and should not be done as an exercise.
Theorem For any 1 ≤ i ≤ n, [ A[= a
i,1
C
i,1
+ a
i,2
C
i,2
+ + a
i,n
C
i,n
. For any
1 ≤ j ≤ n, [ A[= a
1,j
C
1,j
+a
2,j
C
2,j
+ +a
n,j
C
n,j
. Thus if any row or any column is
zero, the determinant is zero.
Exercise Let A =
¸
¸
a
1
a
2
a
3
b
1
b
2
b
3
c
1
c
2
c
3
¸
. The determinant of A is the sum of six terms.
Chapter 4 Matrices 63
Write out the determinant of A expanding by the first column and also expanding by
the second row.
Theorem If A is an upper or lower triangular matrix, [ A[ is the product of the
diagonal elements. If A is an elementary matrix of type 2, [ A [= −1. If A is an
elementary matrix of type 3, [ A[= 1.
Proof We will prove the first statement for upper triangular matrices. If A ∈ R
2
is an upper triangular matrix, then its determinant is the product of the diagonal
elements. Suppose n > 2 and the theorem is true for matrices in R
n−1
. Suppose
A ∈ R
n
is upper triangular. The result follows by expanding by the first column.
An elementary matrix of type 3 is a special type of upper or lower triangular
matrix, so its determinant is 1. An elementary matrix of type 2 is obtained from the
identity matrix by interchanging two rows or columns, and thus has determinant −1.
Theorem (Determinant by blocks) Suppose A ∈ R
n
, B ∈ R
n,m
, and D ∈ R
m
.
Then the determinant of
A B
O D
is [ A[[ D[.
Proof Expand by the first column and use induction on n.
The following remarkable theorem takes some work to prove. We assume it here
without proof. (For the proof, see page 130 of the Appendix.)
Theorem The determinant of the product is the product of the determinants,
i.e., if A, B ∈ R
n
, [ AB[ = [ A[[ B[. Thus [ AB[ = [ BA[ and if C is invertible,
[ C
−1
AC [ = [ACC
−1
[ = [ A[.
Corollary If A is a unit in R
n
, then [ A[ is a unit in R and [ A
−1
[ = [ A[
−1
.
Proof 1 = [ I [ = [ AA
−1
[ = [ A[[ A
−1
[ .
One of the major goals of this chapter is to prove the converse of the preceding
corollary.
Classical adjoint Suppose R is a commutative ring and A ∈ R
n
. The classical
adjoint of A is (C
i,j
)
t
, i.e., the matrix whose (j, i) term is the (i, j) cofactor. Before
64 Matrices Chapter 4
we consider the general case, let's examine 2 2 matrices.
If A =
.
Here is the general case.
Theorem If R is commutative and A ∈ R
n
, then A(C
i,j
)
t
= (C
i,j
)
t
A = [ A[ I.
Proof We must show that the diagonal elements of the product A(C
i,j
)
t
are all
[ A[ and the other elements are 0. The (s, s) term is the dot product of row s of A
with row s of (C
i,j
) and is thus [ A[ (computed by expansion by row s). For s = t,
the (s, t) term is the dot product of row s of A with row t of (C
i,j
). Since this is the
determinant of a matrix with row s = row t, the (s, t) term is 0. The proof that
(C
i,j
)
t
A = [A[I is similar and is left as an exercise.
We are now ready for one of the most beautiful and useful theorems in all of
mathematics.
Theorem Suppose R is a commutative ring and A ∈ R
n
. Then A is a unit in
R
n
iff [ A[ is a unit in R. (Thus if R is a field, A is invertible iff [ A[ = 0.) If A is
invertible, then A
−1
= [ A[
−1
(C
i,j
)
t
. Thus if [ A[ = 1, A
−1
= (C
i,j
)
t
, the classical
adjoint of A.
Proof This follows immediately from the preceding theorem.
Exercise Show that any right inverse of A is also a left inverse. That is, suppose
A, B ∈ R
n
and AB = I. Show A is invertible with A
−1
= B, and thus BA = I.
Similarity
Suppose A, B ∈ R
n
. B is said to be similar to A if ∃ an invertible C ∈ R
n
such
that B = C
−1
AC, i.e., B is similar to A iff B is a conjugate of A.
Theorem B is similar to B.
Chapter 4 Matrices 65
B is similar to A iff A is similar to B.
If D is similar to B and B is similar to A, then D is similar to A.
"Similarity" is an equivalence relation on R
n
.
Proof This is a good exercise using the definition.
Theorem Suppose A and B are similar. Then [ A[ = [ B[ and thus A is invertible
iff B is invertible.
Proof Suppose B = C
−1
AC. Then [ B[ = [ C
−1
AC [ = [ACC
−1
[ = [ A[.
Trace Suppose A = (a
i,j
) ∈ R
n
. Then the trace is defined by trace(A) = a
1,1
+
a
2,2
+ +a
n,n
. That is, the trace of A is the sum of its diagonal terms.
One of the most useful properties of trace is trace(AB) = trace(BA) whenever AB
and BA are defined. For example, suppose A = (a
1
, a
2
, ..., a
n
) and B = (b
1
, b
2
, ..., b
n
)
t
.
Then AB is the scalar a
1
b
1
+ + a
n
b
n
while BA is the n n matrix (b
i
a
j
). Note
that trace(AB) = trace(BA). Here is the theorem in full generality.
Theorem Suppose A ∈ R
m,n
and B ∈ R
n,m
. Then AB and BA are square
matrices with trace(AB) = trace(BA).
Proof This proof involves a change in the order of summation. By definition,
trace(AB) =
¸
1≤i≤m
a
i,1
b
1,i
+ +a
i,n
b
n,i
=
¸
1≤i≤m
1≤j≤n
a
i,j
b
j,i
=
¸
1≤j≤n
b
j,1
a
1,j
+ +b
j,m
a
m,j
=
trace(BA).
Theorem If A, B ∈ R
n
, trace(A+B) = trace(A) + trace(B) and
trace(AB) = trace(BA).
Proof The first part of the theorem is immediate, and the second part is a special
case of the previous theorem.
Theorem If A and B are similar, then trace(A) = trace(B).
Proof trace(B) = trace(C
−1
AC) = trace(ACC
−1
) = trace(A).
66 Matrices Chapter 4
Summary Determinant and trace are functions from R
n
to R. Determinant is a
multiplicative homomorphism and trace is an additive homomorphism. Furthermore
[ AB[ = [ BA[ and trace(AB) = trace(BA). If A and B are similar, [ A[ = [ B[ and
trace(A) = trace(B).
Exercise Suppose A ∈ R
n
and a ∈ R. Find [aA[ and trace(aA).
Characteristic polynomials If A ∈ R
n
, the characteristic polynomial CP
A
(x) ∈
R[x] is defined by CP
A
(x) = [ (xI − A) [. Any λ ∈ R which is a root of CP
A
(x) is
called a characteristic root of A.
Theorem CP
A
(x) = a
0
+ a
1
x + + a
n−1
x
n−1
+ x
n
where trace(A) = −a
n−1
and [ A[ = (−1)
n
a
0
.
Proof This follows from a direct computation of the determinant.
Theorem If A and B are similar, then they have the same characteristic polyno-
mials.
Proof Suppose B = C
−1
AC. CP
B
(x) = [ (xI −C
−1
AC) [ = [ C
−1
(xI −A)C[ =
[ (xI −A)[ = CP
A
(x).
Exercise Suppose R is a commutative ring, A =
a b
c d
is a matrix in R
2
, and
CP
A
(x) = a
0
+ a
1
x + x
2
. Find a
0
and a
1
and show that a
0
I + a
1
A + A
2
= 0, i.e.,
show A satisfies its characteristic polynomial. In other words, CP
A
(A) = 0.
Exercise Suppose F is a field and A ∈ F
2
. Show the following are equivalent.
1) A
2
= 0.
2) [ A [= trace(A) = 0.
3) CP
A
(x) = x
2
.
4) ∃ an elementary matrix C such that C
−1
AC is strictly upper triangular.
Note This exercise is a special case of a more general theorem. A square matrix
over a field is nilpotent iff all its characteristic roots are 0
¯
iff it is similar to a strictly
upper triangular matrix. This remarkable result cannot be proved by matrix theory
alone, but depends on linear algebra (see pages 93, 94, and 98).
Chapter 5
Linear Algebra
The exalted position held by linear algebra is based upon the subject's ubiquitous
utility and ease of application. The basic theory is developed here in full generality,
i.e., modules are defined over an arbitrary ring R and not just over a field. The
elementary facts about cosets, quotients, and homomorphisms follow the same pat-
tern as in the chapters on groups and rings. We give a simple proof that if R is a
commutative ring and f : R
n
→ R
n
is a surjective R-module homomorphism, then
f is an isomorphism. This shows that finitely generated free R-modules have a well
defined dimension, and simplifies some of the development of linear algebra. It is in
this chapter that the concepts about functions, solutions of equations, matrices, and
generating sets come together in one unified theory.
After the general theory, we restrict our attention to vector spaces, i.e., modules
over a field. The key theorem is that any vector space V has a free basis, and thus
if V is finitely generated, it has a well defined dimension, and incredible as it may
seem, this single integer determines V up to isomorphism. Also any endomorphism
f : V →V may be represented by a matrix, and any change of basis corresponds to
conjugation of that matrix. One of the goals in linear algebra is to select a basis so
that the matrix representing f has a simple form. For example, if f is not injective,
then f may be represented by a matrix whose first column is zero. As another
example, if f is nilpotent, then f may be represented by a strictly upper triangular
matrix. The theorem on Jordan canonical form is not proved in this chapter, and
should not be considered part of this chapter. It is stated here in full generality only
for reference and completeness. The proof is given in the Appendix. This chapter
concludes with the study of real inner product spaces, and with the beautiful theory
relating orthogonal matrices and symmetric matrices.
67
68 Linear Algebra Chapter 5
Definition Suppose R is a ring and M is an additive abelian group. The state-
ment that M is a right R-module means there is a scalar multiplication
M R → M satisfying (a
1
+a
2
)r = a
1
r +a
2
r
(m, r) → mr a(r
1
+r
2
) = ar
1
+ar
2
a(r
1
r
2
) = (ar
1
)r
2
a1
¯
= a
for all a, a
1
, a
2
∈ M and r, r
1
, r
2
∈ R.
The statement that M is a left R-module means there is a scalar multiplication
R M → M satisfying r(a
1
+a
2
) = ra
1
+ra
2
(r, m) → rm (r
1
+r
2
)a = r
1
a +r
2
a
(r
1
r
2
)a = r
1
(r
2
a)
1
¯
a = a
Note that the plus sign is used ambiguously, as addition in M and as addition in R.
Notation The fact that M is a right (left) R-module will be denoted by M = M
R
(M =
R
M). If R is commutative and M = M
R
then left scalar multiplication defined
by ra = ar makes M into a left R-module. Thus for commutative rings, we may write
the scalars on either side. In this text we stick to right R-modules.
Convention Unless otherwise stated, it is assumed that R is a ring and the word
"R-module" (or sometimes just "module") means "right R-module".
Theorem Suppose M is an R-module.
1) If r ∈ R, then f : M →M defined by f(a) = ar is a homomorphism of
additive groups. In particular (0
¯
M
)r = 0
¯
M
.
2) If a ∈ M, a0
¯
R
= 0
¯
M
.
3) If a ∈ M and r ∈ R, then (−a)r = −(ar) = a(−r).
Proof This is a good exercise in using the axioms for an R-module.
Chapter 5 Linear Algebra 69
Submodules If M is an R-module, the statement that a subset N ⊂ M is a
submodule means it is a subgroup which is closed under scalar multiplication, i.e., if
a ∈ N and r ∈ R, then ar ∈ N. In this case N will be an R-module because the
axioms will automatically be satisfied. Note that 0
¯
and M are submodules, called the
improper submodules of M.
Theorem Suppose M is an R-module, T is an index set, and for each t ∈ T,
N
t
is a submodule of M.
1)
¸
t∈T
N
t
is a submodule of M.
2) If ¦N
t
¦ is a monotonic collection,
¸
t∈T
N
t
is a submodule.
3) +
t∈T
N
t
= ¦all finite sums a
1
+ +a
m
: each a
i
belongs
to some N
t
¦ is a submodule. If T = ¦1, 2, .., n¦,
then this submodule may be written as
N
1
+N
2
+ +N
n
= ¦a
1
+a
2
+ +a
n
: each a
i
∈ N
i
¦.
Proof We know from page 22 that versions of 1) and 2) hold for subgroups, and
in particular for subgroups of additive abelian groups. To finish the proofs it is only
necessary to check scalar multiplication, which is immediate. Also the proof of 3) is
immediate. Note that if N
1
and N
2
are submodules of M, N
1
+ N
2
is the smallest
submodule of M containing N
1
∪ N
2
.
Exercise Suppose T is a non-void set, N is an R-module, and N
T
is the collection
of all functions f : T →N with addition defined by (f +g)(t) = f(t)+g(t), and scalar
multiplication defined by (fr)(t) = f(t)r. Show N
T
is an R-module. (We know from
the last exercise in Chapter 2 that N
T
is a group, and so it is only necessary to check
scalar multiplication.) This simple fact is quite useful in linear algebra. For example,
in 5) of the theorem below, it is stated that Hom
R
(M, N) forms an abelian group.
So it is only necessary to show that Hom
R
(M, N) is a subgroup of N
M
. Also in 8) it
is only necessary to show that Hom
R
(M, N) is a submodule of N
M
.
Homomorphisms
Suppose M and N are R-modules. A function f : M → N is a homomorphism
(i.e., an R-module homomorphism) provided it is a group homomorphism and if
a ∈ M and r ∈ R, f(ar) = f(a)r. On the left, scalar multiplication is in M and on
the right it is in N. The basic facts about homomorphisms are listed below. Much
70 Linear Algebra Chapter 5
of this work has already been done in the chapter on groups (see page 28).
Theorem
1) The zero map M →N is a homomorphism.
2) The identity map I : M →M is a homomorphism.
3) The composition of homomorphisms is a homomorphism.
4) The sum of homomorphisms is a homomorphism. If f, g : M →N are
homomorphisms, define (f +g) : M →N by (f +g)(a) = f(a) +g(a).
Then f +g is a homomorphism. Also (−f) defined by (−f)(a) = −f(a)
is a homomorphism. If h : N →P is a homomorphism,
h ◦ (f +g) = (h ◦ f) + (h ◦ g). If k : P →M is a homomorphism,
(f +g ) ◦ k = (f ◦ k) + (g ◦ k).
5) Hom
R
(M, N) = Hom(M
R
, N
R
), the set of all homomorphisms from M
to N, forms an abelian group under addition. Hom
R
(M, M), with
multiplication defined to be composition, is a ring.
6) If a bijection f : M →N is a homomorphism, then f
−1
: N →M is also
a homomorphism. In this case f and f
−1
are called isomorphisms. A
homomorphism f : M →M is called an endomorphism. An isomorphism
f : M →M is called an automorphism. The units of the endomorphism
ring Hom
R
(M, M) are the automorphisms. Thus the automorphisms on
M form a group under composition. We will see later that if M = R
n
,
Hom
R
(R
n
, R
n
) is just the matrix ring R
n
and the automorphisms
are merely the invertible matrices.
7) If R is commutative and r ∈ R, then g : M →M defined by g(a) = ar
is a homomorphism. Furthermore, if f : M →N is a homomorphism,
fr defined by (fr)(a) = f(ar) = f(a)r is a homomorphism.
8) If R is commutative, Hom
R
(M, N) is an R-module.
9) Suppose f : M →N is a homomorphism, G ⊂ M is a submodule,
and H ⊂ N is a submodule. Then f(G) is a submodule of N
and f
−1
(H) is a submodule of M. In particular, image(f) is a
submodule of N and ker(f) = f
−1
(0
¯
) is a submodule of M.
Proof This is just a series of observations.
Chapter 5 Linear Algebra 71
Abelian groups are Z-modules On page 21, it is shown that any additive
group M admits a scalar multiplication by integers, and if M is abelian, the properties
are satisfied to make M a Z-module. Note that this is the only way M can be a Z-
module, because a1 = a, a2 = a + a, etc. Furthermore, if f : M → N is a group
homomorphism of abelian groups, then f is also a Z-module homomorphism.
Summary Additive abelian groups are "the same things" as Z-modules. While
group theory in general is quite separate from linear algebra, the study of additive
abelian groups is a special case of the study of R-modules.
Exercise R-modules are also Z-modules and R-module homomorphisms are also
Z-module homomorphisms. If M and N are Q-modules and f : M → N is a
Z-module homomorphism, must it also be a Q-module homomorphism?
Homomorphisms on R
n
R
n
as an R-module On page 54 it was shown that the additive abelian
group R
m,n
admits a scalar multiplication by elements in R. The properties listed
there were exactly those needed to make R
m,n
an R-module. Of particular importance
is the case R
n
= R ⊕ ⊕R = R
n,1
(see page 53). We begin with the case n = 1.
R as a right R-module Let M = R and define scalar multiplication on the right
by ar = a r. That is, scalar multiplication is just ring multiplication. This makes
R a right R-module denoted by R
R
(or just R). This is the same as the definition
before for R
n
when n = 1.
Theorem Suppose R is a ring and N is a subset of R. Then N is a submodule
of R
R
(
R
R) iff N is a right (left) ideal of R.
Proof The definitions are the same except expressed in different language.
Theorem Suppose M = M
R
and f, g : R →M are homomorphisms with f(1
¯
) =
g(1
¯
). Then f = g. Furthermore, if m ∈ M, ∃! homomorphism h : R → M with
h(1
¯
) = m. In other words, Hom
R
(R, M) ≈ M.
Proof Suppose f(1
¯
) = g(1
¯
). Then f(r) = f(1
¯
r) = f(1
¯
)r = g(1
¯
)r = g(1
¯
r) =
g(r). Given m ∈ M, h : R → M defined by h(r) = mr is a homomorphism. Thus
72 Linear Algebra Chapter 5
evaluation at 1
¯
gives a bijection from Hom
R
(R, M) to M, and this bijection is clearly
a group isomorphism. If R is commutative, it is an isomorphism of R-modules.
In the case M = R, the above theorem states that multiplication on left by some
m ∈ R defines a right R-module homomorphism from R to R, and every module
homomorphism is of this form. The element m should be thought of as a 1 1
matrix. We now consider the case where the domain is R
n
.
Homomorphisms on R
n
Define e
i
∈ R
n
by e
i
=
¸
¸
¸
¸
¸
¸
¸
0
¯
1
¯
i
0
¯
¸
. Note that any
¸
¸
¸
¸
¸
¸
¸
r
1
r
n
¸
can be written uniquely as e
1
r
1
+ +e
n
r
n
. The sequence ¦e
1
, .., e
n
¦ is called the
canonical free basis or standard basis for R
n
.
Theorem Suppose M = M
R
and f, g : R
n
→ M are homomorphisms with
f(e
i
) = g(e
i
) for 1 ≤ i ≤ n. Then f = g. Furthermore, if m
1
, m
2
, ..., m
n
∈ M, ∃!
homomorphism h : R
n
→ M with h(e
i
) = m
i
for 1 ≤ i ≤ m. The homomorphism
h is defined by h(e
1
r
1
+ +e
n
r
n
) = m
1
r
1
+ +m
n
r
n
.
Proof The proof is straightforward. Note this theorem gives a bijection from
Hom
R
(R
n
, M) to M
n
= M M M and this bijection is a group isomorphism.
We will see later that the product M
n
is an R-module with scalar multiplication
defined by (m
1
, m
2
, .., m
n
)r = (m
1
r, m
2
r, .., m
n
r). If R is commutative so that
Hom
R
(R
n
, M) is an R-module, this theorem gives an R-module isomorphism from
Hom
R
(R
n
, M) to M
n
.
This theorem reveals some of the great simplicity of linear algebra. It does not
matter how complicated the ring R is, or which R-module M is selected. Any
R-module homomorphism from R
n
to M is determined by its values on the basis,
and any function from that basis to M extends uniquely to a homomorphism from
R
n
to M.
Exercise Suppose R is a field and f : R
R
→ M is a non-zero homomorphism.
Show f is injective.
Chapter 5 Linear Algebra 73
Now let's examine the special case M = R
m
and show Hom
R
(R
n
, R
m
) ≈ R
m,n
.
Theorem Suppose A = (a
i,j
) ∈ R
m,n
. Then f : R
n
→R
m
defined by f(B) = AB
is a homomorphism with f(e
i
) = column i of A. Conversely, if v
1
, . . . , v
n
∈ R
m
, define
A ∈ R
m,n
to be the matrix with column i = v
i
. Then f defined by f(B) = AB is
the unique homomorphism from R
n
to R
m
with f(e
i
) = v
i
.
Even though this follows easily from the previous theorem and properties of ma-
trices, it is one of the great classical facts of linear algebra. Matrices over R give
R-module homomorphisms! Furthermore, addition of matrices corresponds to addi-
tion of homomorphisms, and multiplication of matrices corresponds to composition
of homomorphisms. These properties are made explicit in the next two theorems.
Theorem If f, g : R
n
→ R
m
are given by matrices A, C ∈ R
m,n
, then f + g is
given by the matrix A+C. Thus Hom
R
(R
n
, R
m
) and R
m,n
are isomorphic as additive
groups. If R is commutative, they are isomorphic as R-modules.
Theorem If f : R
n
→ R
m
is the homomorphism given by A ∈ R
m,n
and g :
R
m
→R
p
is the homomorphism given by C ∈ R
p,m
, then g ◦ f : R
n
→R
p
is given by
CA ∈ R
p,n
. That is, composition of homomorphisms corresponds to multiplication
of matrices.
Proof This is just the associative law of matrix multiplication, C(AB) = (CA)B.
The previous theorem reveals where matrix multiplication comes from. It is the
matrix which represents the composition of the functions. In the case where the
domain and range are the same, we have the following elegant corollary.
Corollary Hom
R
(R
n
, R
n
) and R
n
are isomorphic as rings. The automorphisms
correspond to the invertible matrices.
This corollary shows one way non-commutative rings arise, namely as endomor-
phism rings. Even if R is commutative, R
n
is never commutative unless n = 1.
We now return to the general theory of modules (over some given ring R).
74 Linear Algebra Chapter 5
Cosets and Quotient Modules
After seeing quotient groups and quotient rings, quotient modules go through
without a hitch. As before, R is a ring and module means R-module.
Theorem Suppose M is a module and N ⊂ M is a submodule. Since N is a
normal subgroup of M, the additive abelian quotient group M/N is defined. Scalar
multiplication defined by (a + N)r = (ar + N) is well-defined and gives M/N the
structure of an R-module. The natural projection π : M → M/N is a surjective
homomorphism with kernel N. Furthermore, if f : M →
¯
M is a surjective homomor-
phism with ker(f) = N, then M/N ≈
¯
M (see below).
Proof On the group level, this is all known from Chapter 2 (see pages 27 and 29).
It is only necessary to check the scalar multiplication, which is obvious.
The relationship between quotients and homomorphisms for modules is the same
as for groups and rings, as shown by the next theorem.
Theorem Suppose f : M →
¯
M is a homomorphism and N is a submodule of M.
If N ⊂ ker(f), then
¯
f : (M/N) →
¯
M defined by
¯
f(a + N) = f(a) is a well-defined
homomorphism making the following diagram commute.
M
¯
M
M/N
f
·
π
¯
f
Thus defining a homomorphism on a quotient module is the same as defining a homo-
morphism on the numerator that sends the denominator to 0
¯
. The image of
¯
f is the
image of f, and the kernel of
¯
f is ker(f)/N. Thus if N = ker(f),
¯
f is injective, and
thus (M/N) ≈image(f). Therefore for any homomorphism f, (domain(f)/ker(f)) ≈
image(f).
Proof On the group level this is all known from Chapter 2 (see page 29). It is
only necessary to check that
¯
f is a module homomorphism, and this is immediate.
Chapter 5 Linear Algebra 75
Theorem Suppose M is an R-module and K and L are submodules of M.
i) The natural homomorphism K →(K + L)/L is surjective with kernel
K ∩ L. Thus (K/K ∩ L)
≈
→(K +L)/L is an isomorphism.
ii) Suppose K ⊂ L. The natural homomorphism M/K →M/L is surjective
with kernel L/K. Thus (M/K)/(L/K)
≈
→M/L is an isomorphism.
Examples These two examples are for the case R = Z, i.e., for abelian groups.
1) M = Z K = 3Z L = 5Z K ∩ L = 15Z K +L = Z
K/K ∩ L = 3Z/15Z ≈ Z/5Z = (K +L)/L
2) M = Z K = 6Z L = 3Z (K ⊂ L)
(M/K)/(L/K) = (Z/6Z)/(3Z/6Z) ≈ Z/3Z = M/L
Products and Coproducts
Infinite products work fine for modules, just as they do for groups and rings.
This is stated below in full generality, although the student should think of the finite
case. In the finite case something important holds for modules that does not hold
for non-abelian groups or rings – namely, the finite product is also a coproduct. This
makes the structure of module homomorphisms much more simple. For the finite
case we may use either the product or sum notation, i.e., M
1
M
2
M
n
=
M
1
⊕M
2
⊕ ⊕M
n
.
Theorem Suppose T is an index set and for each t ∈ T, M
t
is an R-module. On
the additive abelian group
¸
t∈T
M
t
=
¸
M
t
define scalar multiplication by ¦m
t
¦r =
¦m
t
r¦. Then
¸
M
t
is an R-module and, for each s ∈ T, the natural projection
π
s
:
¸
M
t
→M
s
is a homomorphism. Suppose M is a module. Under the natural 1-1
correspondence from ¦functions f : M →
¸
M
t
¦ to ¦sequence of functions ¦f
t
¦
t∈T
where f
t
: M →M
t
¦, f is a homomorphism iff each f
t
is a homomorphism.
Proof We already know from Chapter 2 that f is a group homomorphism iff each
f
t
is a group homomorphism. Since scalar multiplication is defined coordinatewise,
f is a module homomorphism iff each f
t
is a module homomorphism.
76 Linear Algebra Chapter 5
Definition If T is finite, the coproduct and product are the same module. If T
is infinite, the coproduct or sum
¸
t∈T
M
t
=
´
Theorem For finite T, the 1-1 correspondences in the above theorems actually
produce group isomorphisms. If R is commutative, they give isomorphisms of R-
modules.
Hom
R
(M, M
1
⊕ ⊕M
n
) ≈ Hom
R
(M, M
1
) ⊕ ⊕Hom
R
(M, M
n
) and
Hom
R
(M
1
⊕ ⊕M
n
, M) ≈ Hom
R
(M
1
, M) ⊕ ⊕Hom
R
(M
n
, M)
Proof Let's look at this theorem for products with n = 2. All it says is that if f =
(f
1
, f
2
) and h = (h
1
, h
2
), then f +h = (f
1
+h
1
, f
2
+h
2
). If R is commutative, so that
the objects are R-modules and not merely additive groups, then the isomorphisms
are module isomorphisms. This says merely that fr = (f
1
, f
2
)r = (f
1
r, f
2
r).
Chapter 5 Linear Algebra 77
Exercise Suppose M and N are R-modules. Show that M ⊕N is isomorphic to
N ⊕ M. Now suppose A ⊂ M, B ⊂ N are submodules and show (M ⊕ N)/(A ⊕B)
is isomorphic to (M/A) ⊕ (N/B). In particular, if a ∈ R and b ∈ R, then
(R ⊕ R)/(aR ⊕ bR) is isomorphic to (R/aR) ⊕ (R/bR). For example, the abelian
group (Z ⊕ Z)/(2Z ⊕ 3Z) is isomorphic to Z
2
⊕ Z
3
. These isomorphisms are trans-
parent and are used routinely in algebra without comment (see Th 4, page 118).
Exercise Suppose R is a commutative ring, M is an R-module, and n ≥ 1. Define
a function α : Hom
R
(R
n
, M) →M
n
which is a R-module isomorphism.
Summands
One basic question in algebra is "When does a module split as the sum of two
modules?". Before defining summand, here are two theorems for background.
Theorem Consider M
1
= M
1
⊕0
¯
as a submodule of M
1
⊕M
2
. Then the projection
map π
2
: M
1
⊕ M
2
→ M
2
is a surjective homomorphism with kernel M
1
. Thus
(M
1
⊕M
2
)/M
1
is isomorphic to M
2
. (See page 35 for the group version.)
This is exactly what you would expect, and the next theorem is almost as intuitive.
Theorem Suppose K and L are submodules of M and f : K ⊕ L → M is the
natural homomorphism, f(k, l) = k + l. Then the image of f is K + L and the
kernel of f is ¦(a, −a) : a ∈ K ∩ L¦. Thus f is an isomorphism iff K + L = M and
K ∩ L = 0
¯
. In this case we write K ⊕ L = M. This abuse of notation allows us to
avoid talking about "internal" and "external" direct sums.
Definition Suppose K is a submodule of M. The statement that K is a summand
of M means ∃ a submodule L of M with K ⊕ L = M. According to the previous
theorem, this is the same as there exists a submodule L with K + L = M and
K ∩ L = 0
¯
. If such an L exists, it need not be unique, but it will be unique up to
isomorphism, because L ≈ M/K. Of course, M and 0
¯
are always summands of M.
Exercise Suppose M is a module and K = ¦(m, m) : m ∈ M¦ ⊂ M ⊕M. Show
K is a submodule of M ⊕M which is a summand.
Exercise R is a module over Q, and Q ⊂ R is a submodule. Is Q a summand of
R? With the material at hand, this is not an easy question. Later on, it will be easy.
78 Linear Algebra Chapter 5
Exercise Answer the following questions about abelian groups, i.e., Z-modules.
(See the third exercise on page 35.)
1) Is 2Z a summand of Z?
2) Is 2Z
4
a summand of Z
4
?
3) Is 3Z
12
a summand of Z
12
?
4) Suppose m, n > 1. When is nZ
mn
a summand of Z
mn
?
Exercise If T is a ring, define the center of T to be the subring ¦t : ts =
st for all s ∈ T¦. Let R be a commutative ring and T = R
n
. There is a exercise
on page 57 to show that the center of T is the subring of scalar matrices. Show R
n
is a left T-module and find Hom
T
(R
n
, R
n
).
Independence, Generating Sets, and Free Basis
This section is a generalization and abstraction of the brief section Homomor-
phisms on R
n
. These concepts work fine for an infinite index set T because linear
combination means finite linear combination. However, to avoid dizziness, the student
should first consider the case where T is finite.
Definition Suppose M is an R-module, T is an index set, and for each t ∈ T,
s
t
∈ M. Let S be the sequence ¦s
t
¦
t∈T
= ¦s
t
¦. The statement that S is dependent
means ∃ a finite number of distinct elements t
1
, ..., t
n
in T, and elements r
1
, .., r
n
in
R, not all zero, such that the linear combination s
t
1
r
1
+ +s
tn
r
n
= 0
¯
. Otherwise,
S is independent. Note that if some s
t
= 0
¯
, then S is dependent. Also if ∃ distinct
elements t
1
and t
2
in T with s
t
1
= s
t
2
, then S is dependent.
Let SR be the set of all linear combinations s
t
1
r
1
+ +s
tn
r
n
. SR is a submodule
of M called the submodule generated by S. If S is independent and generates M,
then S is said to be a basis or free basis for M. In this case any v ∈ M can be written
uniquely as a linear combination of elements in S. An R-module M is said to be a
free R-module if it is zero or if it has a basis. The next two theorems are obvious,
except for the confusing notation. You might try first the case T = ¦1, 2, ..., n¦ and
⊕R
t
= R
n
(see p 72).
Theorem For each t ∈ T, let R
t
= R
R
and for each c ∈ T, let e
c
∈ ⊕R
t
=
t∈T
R
t
be e
c
= ¦r
t
¦ where r
c
= l
¯
and r
t
= 0
¯
if t = c. Then ¦e
c
¦
c∈T
is a basis for ⊕R
t
called
the canonical basis or standard basis.
Chapter 5 Linear Algebra 79
Theorem Suppose N is an R-module and M is a free R-module with a basis
¦s
t
¦. Then ∃ a 1-1 correspondence from the set of all functions g : ¦s
t
¦ →N and the
set of all homomorphisms f : M → N. Given g, define f by f(s
t
1
r
1
+ +s
tn
r
n
) =
g(s
t
1
)r
1
+ +g(s
tn
)r
n
. Given f, define g by g(s
t
) = f(s
t
). In other words, f is
completely determined by what it does on the basis S, and you are "free" to send the
basis any place you wish and extend to a homomorphism.
Recall that we have already had the preceding theorem in the case S is the canon-
ical basis for M = R
n
(p 72). The next theorem is so basic in linear algebra that it
is used without comment. Although the proof is easy, it should be worked carefully.
Theorem Suppose N is a module, M is a free module with basis S = ¦s
t
¦, and
f : M →N is a homomorphism. Let f(S) be the sequence ¦f(s
t
)¦ in N.
1) f(S) generates N iff f is surjective.
2) f(S) is independent in N iff f is injective.
3) f(S) is a basis for N iff f is an isomorphism.
4) If h : M →N is a homomorphism, then f = h iff f [ S = h [ S.
Exercise Let (A
1
, .., A
n
) be a sequence of n vectors with each A
i
∈ Z
n
.
Show this sequence is linearly independent over Z iff it is linearly independent over Q.
Is it true the sequence is linearly independent over Z iff it is linearly independent
over R? This question is difficult until we learn more linear algebra.
Characterization of Free Modules
Any free R-module is isomorphic to one of the canonical free R-modules ⊕R
t
.
This is just an observation, but it is a central fact in linear algebra.
Theorem A non-zero R-module M is free iff ∃ an index set T such that
M ≈
t∈T
R
t
. In particular, M has a finite free basis of n elements iff M ≈ R
n
.
Proof If M is isomorphic to ⊕R
t
then M is certainly free. So now suppose M
has a free basis ¦s
t
¦. Then the homomorphism f : M → ⊕R
t
with f(s
t
) = e
t
sends
the basis for M to the canonical basis for ⊕R
t
. By 3) in the preceding theorem, f is
an isomorphism.
80 Linear Algebra Chapter 5
Exercise Suppose R is a commutative ring, A ∈ R
n
, and the homomorphism
f : R
n
→ R
n
defined by f(B) = AB is surjective. Show f is an isomorphism, i.e.,
show A is invertible. This is a key theorem in linear algebra, although it is usually
stated only for the case where R is a field. Use the fact that ¦e
1
, .., e
n
¦ is a free basis
for R
n
.
The next exercise is routine, but still informative.
Exercise Let R = Z, A =
2 1 0
3 2 −5
and f: Z
3
→ Z
2
be the group homo-
morphism defined by A. Find a non-trivial linear combination of the columns of A
which is 0
¯
. Also find a non-zero element of kernel(f).
If R is a commutative ring, you can relate properties of R as an R-module to
properties of R as a ring.
Exercise Suppose R is a commutative ring and v ∈ R, v = 0
¯
.
1) v is independent iff v is .
2) v is a basis for R iff v generates R iff v is .
Note that 2) here is essentially the first exercise for the case n = 1. That is, if
f : R →R is a surjective R-module homomorphism, then f is an isomorphism.
Relating these concepts to matrices
The theorem stated below gives a summary of results we have already had. It
shows that certain concepts about matrices, linear independence, injective homo-
morphisms, and solutions of equations, are all the same — they are merely stated in
different language. Suppose A ∈ R
m,n
and f : R
n
→R
m
is the homomorphism associ-
ated with A, i.e., f(B) = AB. Let v
1
, .., v
n
∈ R
m
be the columns of A, i.e., f(e
i
) = v
i
= column i of A. Let B =
¸
¸
b
1
.
b
n
¸
represent an element of R
n
and C =
¸
¸
c
1
.
c
m
¸
Chapter 5 Linear Algebra 81
represent an element of R
m
.
Theorem
1) The element f(B) is a linear combination of the columns of A, that is
f(B) = f(e
1
b
1
+ +e
n
b
n
) = v
1
b
1
+ +v
n
b
n
. Thus the image of f
is generated by the columns of A. (See bottom of page 89.)
2) ¦v
1
, .., v
n
¦ generates R
m
iff f is surjective iff (for any C ∈ R
m
, AX = C
has a solution).
3) ¦v
1
, .., v
n
¦ is independent iff f is injective iff AX = 0
¯
has a unique
solution iff (∃ C ∈ R
m
such that AX = C has a unique solution).
4) ¦v
1
, .., v
n
¦ is a basis for R
m
iff f is an isomorphism iff (for any C ∈ R
m
,
AX = C has a unique solution).
Relating these concepts to square matrices
We now look at the preceding theorem in the special case where n = m and R
is a commutative ring. So far in this chapter we have just been cataloging. Now we
prove something more substantial, namely that if f : R
n
→ R
n
is surjective, then f
is injective. Later on we will prove that if R is a field, injective implies surjective.
Theorem Suppose R is a commutative ring, A ∈ R
n
, and f : R
n
→R
n
is defined
by f(B) = AB. Let v
1
, .., v
n
∈ R
n
be the columns of A, and w
1
, .., w
n
∈ R
n
= R
1,n
be the rows of A. Then the following are equivalent.
1) f is an automorphism.
2) A is invertible, i.e., [ A [ is a unit in R.
3) ¦v
1
, .., v
n
¦ is a basis for R
n
.
4) ¦v
1
, .., v
n
¦ generates R
n
.
5) f is surjective.
2
t
) A
t
is invertible, i.e., [ A
t
[ is a unit in R.
3
t
) ¦w
1
, .., w
n
¦ is a basis for R
n
.
82 Linear Algebra Chapter 5
4
t
) ¦w
1
, .., w
n
¦ generates R
n
.
Proof Suppose 5) is true and show 2). Since f is onto, ∃ u
1
, ..., u
n
∈ R
n
with
f(u
i
) = e
i
. Let g : R
n
→R
n
be the homomorphism satisfying g(e
i
) = u
i
. Then f ◦ g
is the identity. Now g comes from some matrix D and thus AD = I. This shows that
A has a right inverse and is thus invertible. Recall that the proof of this fact uses
determinant, which requires that R be commutative (see the exercise on page 64).
We already know the first three properties are equivalent, 4) and 5) are equivalent,
and 3) implies 4). Thus the first five are equivalent. Furthermore, applying this
result to A
t
shows that the last three properties are equivalent to each other. Since
[ A [=[ A
t
[, 2) and 2
t
) are equivalent.
Uniqueness of Dimension
There exists a ring R with R
2
≈ R
3
as R-modules, but this is of little interest. If
R is commutative, this is impossible, as shown below. First we make a convention.
Convention For the remainder of this chapter, R will be a commutative ring.
Theorem If f : R
m
→R
n
is a surjective R-module homomorphism, then m ≥ n.
Proof Suppose k = n − m is positive. Define h : (R
m
⊕ R
k
= R
n
) → R
n
by
h(u, v) = f(u). Then h is a surjective homomorphism, and by the previous section,
also injective. This is a contradiction and thus m ≥ n.
Corollary If f : R
m
→R
n
is an isomorphism, then m = n.
Proof Each of f and f
−1
is surjective, so m = n by the previous theorem.
Corollary If ¦v
1
, .., v
m
¦ generates R
n
, then m ≥ n.
Proof The hypothesis implies there is a surjective homomorphism R
m
→R
n
. So
this follows from the first theorem.
Lemma Suppose M is a f.g. module (i.e., a finite generated R-module). Then
if M has a basis, that basis is finite.
Chapter 5 Linear Algebra 83
Proof Suppose U ⊂ M is a finite generating set and S is a basis. Then any
element of U is a finite linear combination of elements of S, and thus S is finite.
Theorem Suppose M is a f.g. module. If M has a basis, that basis is finite
and any other basis has the same number of elements. This number is denoted by
dim(M), the dimension of M. (By convention, 0
¯
is a free module of dimension 0.)
Proof By the previous lemma, any basis for M must be finite. M has a basis of
n elements iff M ≈ R
n
. The result follows because R
n
≈ R
m
iff n = m.
Change of Basis
Before changing basis, we recall what a basis is. Previously we defined generat-
ing, independence, and basis for sequences, not for collections. For the concept of
generating it matters not whether you use sequences or collections, but for indepen-
dence and basis, you must use sequences. Consider the columns of the real matrix
A =
2 3 2
1 4 1
. If we consider the column vectors of A as a collection, there are
only two of them, yet we certainly don't wish to say the columns of A form a basis for
R
2
. In a set or collection, there is no concept of repetition. In order to make sense,
we must consider the columns of A as an ordered triple of vectors, and this sequence
is dependent. In the definition of basis on page 78, basis is defined for sequences, not
for sets or collections.
Two sequences cannot begin to be equal unless they have the same index set.
Here we follow the classical convention that an index set with n elements will be
¦1, 2, .., n¦, and thus a basis for M with n elements is a sequence S = ¦u
1
, .., u
n
¦
or if you wish, S = (u
1
, .., u
n
) ∈ M
n
. Suppose M is an R-module with a basis of
n elements. Recall there is a bijection α : Hom
R
(R
n
, M) → M
n
defined by α(h) =
(h(e
1
), .., h(e
n
)). Now h : R
n
→M is an isomorphism iff α(h) is a basis for M.
Summary The point of all this is that selecting a basis of n elements for M
is the same as selecting an isomorphism from R
n
to M, and from this viewpoint,
change of basis can be displayed by the diagram below.
Endomorphisms on R
n
are represented by square matrices, and thus have a de-
terminant and trace. Now suppose M is a f.g. free module and f : M → M is a
homomorphism. In order to represent f by a matrix, we must select a basis for M
(i.e., an isomorphism with R
n
). We will show that this matrix is well defined up to
similarity, and thus the determinant, trace, and characteristic polynomial of f are
well-defined.
84 Linear Algebra Chapter 5
Definition Suppose M is a free module, S = ¦u
1
, .., u
n
¦ is a basis for M, and
f : M → M is a homomorphism. The matrix A = (a
i,j
) ∈ R
n
of f w.r.t. the basis
S is defined by f(u
i
) = u
1
a
1,i
+ +u
n
a
n,i
. (Note that if M = R
n
and u
i
= e
i
, A is
the usual matrix associated with f).
Theorem Suppose T = ¦v
1
, .., v
n
¦ is another basis for M and B ∈ R
n
is the
matrix of f w.r.t. T. Define C = (c
i,j
) ∈ R
n
by v
i
= u
1
c
1,i
+ +u
n
c
n,i
. Then C is
invertible and B = C
−1
AC, i.e., A and B are similar. Therefore [A[ = [B[,
trace(A)=trace(B), and A and B have the same characteristic polynomial (see page
66 of chapter 4).
Conversely, suppose C = (c
i,j
) ∈ R
n
is invertible. Define T = ¦v
1
, .., v
n
¦ by
v
i
= u
1
c
1,i
+ +u
n
c
n,i
. Then T is a basis for M and the matrix of f w.r.t. T is
B = C
−1
AC. In other words, conjugation of matrices corresponds to change of basis.
Proof The proof follows by seeing that the following diagram is commutative.
R
n
R
n
R
n
R
n
M M C C
A
B
≈ ≈
≈ ≈
≈ ≈
e
i
v
i
e
i
u
i
v
i
e
i
u
i
e
i
f
· ·
´
»
»
´
»
´
´
»
The diagram also explains what it means for A to be the matrix of f w.r.t. the
basis S. Let h : R
n
→ M be the isomorphism with h(e
i
) = u
i
for 1 ≤ i ≤ n. Then
the matrix A ∈ R
n
is the one determined by the endomorphism h
−1
◦f ◦h : R
n
→R
n
.
In other words, column i of A is h
−1
(f(h(e
i
))).
An important special case is where M = R
n
and f : R
n
→ R
n
is given by some
matrix W. Then h is given by the matrix U whose i
th
column is u
i
and A =
U
−1
WU. In other words, W represents f w.r.t. the standard basis, and U
−1
WU
represents f w.r.t. the basis ¦u
1
, .., u
n
¦.
Definition Suppose M is a f.g. free module and f : M →M is a homomorphism.
Define [f[ to be [A[, trace(f) to be trace(A), and CP
f
(x) to be CP
A
(x), where A is
Chapter 5 Linear Algebra 85
the matrix of f w.r.t. some basis. By the previous theorem, all three are well-defined,
i.e., do not depend upon the choice of basis.
Exercise Let R = Z and f : Z
2
→ Z
2
be defined by f(D) =
3 3
0 −1
D.
Find the matrix of f w.r.t. the basis
2
1
,
3
1
¸
.
Exercise Let L ⊂ R
2
be the line L = ¦(r, 2r)
t
: r ∈ R¦. Show there is one
and only one homomorphism f : R
2
→ R
2
which is the identity on L and has
f((−1, 1)
t
) = (1, −1)
t
. Find the matrix A ∈ R
2
which represents f with respect
to the basis ¦(1, 2)
t
, (−1, 1)
t
¦. Find the determinant, trace, and characteristic
polynomial of f. Also find the matrix B ∈ R
2
which represents f with respect to
the standard basis. Finally, find an invertible matrix C ∈ R
2
with B = C
−1
AC.
Vector Spaces
So far in this chapter we have been developing the theory of linear algebra in
general. The previous theorem, for example, holds for any commutative ring R, but
it must be assumed that the module M is free. Endomorphisms in general will not
have a determinant, trace, or characteristic polynomial. We now focus on the case
where R is a field F, and show that in this case, every F-module is free. Thus any
finitely generated F-module will have a well-defined dimension, and endomorphisms
on it will have well-defined determinant, trace, and characteristic polynomial.
In this section, F is a field. F-modules may also be called vector spaces and
F-module homomorphisms may also be called linear transformations.
Theorem Suppose M is an F-module and v ∈ M. Then v = 0
¯
iff v is independent.
That is, if v ∈ V and r ∈ F, vr = 0
¯
implies v = 0
¯
in M or r = 0
¯
in F.
Proof Suppose vr = 0
¯
and r = 0
¯
. Then 0
¯
= (vr)r
−1
= v1
¯
= v.
Theorem Suppose M = 0
¯
is an F-module and v ∈ M. Then v generates M iff v
is a basis for M. Furthermore, if these conditions hold, then M ≈ F
F
, any non-zero
element of M is a basis, and any two elements of M are dependent.
86 Linear Algebra Chapter 5
Proof Suppose v generates M. Then v = 0
¯
and is thus independent by the
previous theorem. In this case M ≈ F, and any non-zero element of F is a basis, and
any two elements of F are dependent.
Theorem Suppose M = 0
¯
is a finitely generated F-module. If S = ¦v
1
, .., v
m
¦
generates M, then any maximal independent subsequence of S is a basis for M. Thus
any finite independent sequence can be extended to a basis. In particular, M has a
finite free basis, and thus is a free F-module.
Proof Suppose, for notational convenience, that ¦v
1
, .., v
n
¦ is a maximal inde-
pendent subsequence of S, and n < i ≤ m. It must be shown that v
i
is a linear
combination of ¦v
1
, .., v
n
¦. Since ¦v
1
, .., v
n
, v
i
¦ is dependent, ∃ r
1
, ..., r
n
, r
i
not all
zero, such that v
1
r
1
+ +v
n
r
n
+v
i
r
i
= 0
¯
. Then r
i
= 0
¯
and v
i
= −(v
1
r
1
+ +v
n
r
n
)r
−1
i
.
Thus ¦v
1
, .., v
n
¦ generates S and thus all of M. Now suppose T is a finite indepen-
dent sequence. T may be extended to a finite generating sequence, and inside that
sequence it may be extended to a maximal independent sequence. Thus T extends
to a basis.
After so many routine theorems, it is nice to have one with real power. It not
only says any finite independent sequence can be extended to a basis, but it can be
extended to a basis inside any finite generating set containing it. This is one of the
theorems that makes linear algebra tick. The key hypothesis here is that the ring
is a field. If R = Z, then Z is a free module over itself, and the element 2 of Z is
independent. However it certainly cannot be extended to a basis. Also the finiteness
hypothesis in this theorem is only for convenience, as will be seen momentarily.
Since F is a commutative ring, any two bases of M must have the same number
of elements, and thus the dimension of M is well defined (see theorem on page 83).
Theorem Suppose M is an F-module of dimension n, and ¦v
1
, ..., v
m
¦ is an
independent sequence in M. Then m ≤ n and if m = n, ¦v
1
, .., v
m
¦ is a basis.
Proof ¦v
1
, .., v
m
¦ extends to a basis with n elements.
The next theorem is just a collection of observations.
Theorem Suppose M and N are finitely generated F-modules.
Chapter 5 Linear Algebra 87
1) M ≈ F
n
iff dim(M) = n.
2) M ≈ N iff dim(M) = dim(N).
3) F
m
≈ F
n
iff n = m.
4) dim(M ⊕N) = dim(M) + dim(N).
Here is the basic theorem for vector spaces in full generality.
Theorem Suppose M = 0
¯
is an F-module and S = ¦v
t
¦
t∈T
generates M.
1) Any maximal independent subsequence of S is a basis for M.
2) Any independent subsequence of S may be extended to a maximal
independent subsequence of S, and thus to a basis for M.
3) Any independent subsequence of M can be extended to a basis for M.
In particular, M has a free basis, and thus is a free F-module.
Proof The proof of 1) is the same as in the case where S is finite. Part 2) will
follow from the Hausdorff Maximality Principle. An independent subsequence of S is
contained in a maximal monotonic tower of independent subsequences. The union of
these independent subsequences is still independent, and so the result follows. Part
3) follows from 2) because an independent sequence can always be extended to a
generating sequence.
Theorem Suppose M is an F-module and K ⊂ M is a submodule.
1) K is a summand of M, i.e., ∃ a submodule L of M with K ⊕L = M.
2) If M is f.g., then dim(K) ≤ dim(M) and K = M iff dim(K) = dim(M).
Proof Let T be a basis for K. Extend T to a basis S for M. Then S−T generates
a submodule L with K ⊕L = M. Part 2) follows from 1).
Corollary Q is a summand of R. In other words, ∃ a Q-submodule V ⊂ R
with Q⊕V = R as Q-modules. (See exercise on page 77.)
Proof Q is a field, R is a Q-module, and Q is a submodule of R.
Corollary Suppose M is a f.g. F-module, N is an F-module, and f : M → N
is a homomorphism. Then dim(M) = dim(ker(f)) + dim(image(f)).
88 Linear Algebra Chapter 5
Proof Let K = ker(f) and L ⊂ M be a submodule with K ⊕ L = M. Then
f [ L : L →image(f) is an isomorphism.
Exercise Suppose R is a domain with the property that, for R-modules, every
submodule is a summand. Show R is a field.
Exercise Find a free Z-module which has a generating set containing no basis.
Exercise The real vector space R
2
is generated by the sequence S =
¦(π, 0), (2, 1), (3, 2)¦. Show there are three maximal independent subsequences of
S, and each is a basis for R
2
. (Row vectors are used here just for convenience.)
The real vector space R
3
is generated by S = ¦(1, 1, 2), (1, 2, 1), (3, 4, 5), (1, 2, 0)¦.
Show there are three maximal independent subsequences of S and each is a basis
for R
3
. You may use determinant.
Square matrices over fields
This theorem is just a summary of what we have for square matrices over fields.
Theorem Suppose A ∈ F
n
and f : F
n
→ F
n
is defined by f(B) = AB. Let
v
1
, .., v
n
∈ F
n
be the columns of A, and w
1
, .., w
n
∈ F
n
= F
1,n
be the rows of A.
Then the following are equivalent.
1) ¦v
1
, .., v
n
¦ is independent, i.e., f is injective.
2) ¦v
1
, .., v
n
¦ is a basis for F
n
, i.e., f is an automorphism, i.e., A is
invertible, i.e., [ A [ = 0
¯
.
3) ¦v
1
, .., v
n
¦ generates F
n
, i.e., f is surjective.
1
t
) ¦w
1
, .., w
n
¦ is independent.
2
t
) ¦w
1
, .., w
n
¦ is a basis for F
n
, i.e., A
t
is invertible, i.e., [ A
t
[ = 0
¯
.
3
t
) ¦w
1
, .., w
n
¦ generates F
n
.
Chapter 5 Linear Algebra 89
Proof Except for 1) and 1
t
), this theorem holds for any commutative ring R.
(See the section Relating these concepts to square matrices, pages 81 and 82.)
Parts 1) and 1
t
) follow from the preceding section.
Exercise Add to this theorem more equivalent statements in terms of solutions
of n equations in n unknowns.
Overview Suppose each of X and Y is a set with n elements and f : X →Y is a
function. By the pigeonhole principle, f is injective iff f is bijective iff f is surjective.
Now suppose each of U and V is a vector space of dimension n and f : U → V is
a linear transformation. It follows from the work done so far that f is injective iff
f is bijective iff f is surjective. This shows some of the simple and definitive nature
of linear algebra.
Exercise Let A = (A
1
, .., A
n
) be an nn matrix over Z with column i = A
i
∈
Z
n
. Let f : Z
n
→ Z
n
be defined by f(B) = AB and
¯
f : R
n
→ R
n
be defined by
¯
f(C) = AC. Show the following are equivalent. (See the exercise on page 79.)
1) f : Z
n
→Z
n
is injective.
2) The sequence (A
1
, .., A
n
) is linearly independent over Z.
3) [A[ = 0.
4)
¯
f : R
n
→R
n
is injective.
5) The sequence (A
1
, .., A
n
) is linearly independent over R.
Rank of a matrix Suppose A ∈ F
m,n
. The row (column) rank of A is defined
to be the dimension of the submodule of F
n
(F
m
) generated by the rows (columns)
of A.
Theorem If C ∈ F
m
and D ∈ F
n
are invertible, then the row (column) rank of
A is the same as the row (column) rank of CAD.
Proof Suppose f : F
n
→ F
m
is defined by f(B) = AB. Each column of A
is a vector in the range F
m
, and we know from page 81 that each f(B) is a linear
90 Linear Algebra Chapter 5
combination of those vectors. Thus the image of f is the submodule of F
m
generated
by the columns of A, and its dimension is the column rank of A. This dimension
is the same as the dimension of the image of g ◦ f ◦ h : F
n
→ F
m
, where h is any
automorphism on F
n
and g is any automorphism on F
m
. This proves the theorem
for column rank. The theorem for row rank follows using transpose.
Theorem If A ∈ F
m,n
, the row rank and the column rank of A are equal. This
number is called the rank of A and is ≤ min¦m, n¦.
Proof By the theorem above, elementary row and column operations change
neither the row rank nor the column rank. By row and column operations, A may be
changed to a matrix H where h
1,1
= = h
t,t
= 1
¯
and all other entries are 0
¯
(see the
first exercise on page 59). Thus row rank = t = column rank.
Exercise Suppose A has rank t. Show that it is possible to select t rows and t
columns of A such that the determined t t matrix is invertible. Show that the rank
of A is the largest integer t such that this is possible.
Exercise Suppose A ∈ F
m,n
has rank t. What is the dimension of the solution
set of AX = 0
¯
?
Definition If N and M are finite dimensional vector spaces and f : N →M is a
linear transformation, the rank of f is the dimension of the image of f. If f : F
n
→F
m
is given by a matrix A, then the rank of f is the same as the rank of the matrix A.
Geometric Interpretation of Determinant
Suppose V ⊂ R
n
is some nice subset. For example, if n = 2, V might be the
interior of a square or circle. There is a concept of the n-dimensional volume of V .
For n = 1, it is length. For n = 2, it is area, and for n = 3 it is "ordinary volume".
Suppose A ∈ R
n
and f : R
n
→R
n
is the homomorphism given by A. The volume of
V does not change under translation, i.e., V and V +p have the same volume. Thus
f(V ) and f(V +p) = f(V ) +f(p) have the same volume. In street language, the next
theorem says that "f multiplies volume by the absolute value of its determinant".
Theorem The n-dimensional volume of f(V ) is ±[A[(the n-dimensional volume
of V ). Thus if [A[ = ±1, f preserves volume.
Chapter 5 Linear Algebra 91
Proof If [A[ = 0, image(f) has dimension < n and thus f(V ) has n-dimensional
volume 0. If [A[ = 0 then A is the product of elementary matrices (see page 59)
and for elementary matrices, the theorem is obvious. The result follows because the
determinant of the composition is the product of the determinants.
Corollary If P is the n-dimensional parallelepiped determined by the columns
v
1
, .. , v
n
of A, then the n-dimensional volume of P is ±[A[.
Proof Let V = [0, 1] [0, 1] = ¦e
1
t
1
+ +e
n
t
n
: 0 ≤ t
i
≤ 1¦. Then
P = f(V ) = ¦v
1
t
1
+ +v
n
t
n
: 0 ≤ t
i
≤ 1¦.
Linear functions approximate differentiable functions locally
We continue with the special case F = R. Linear functions arise naturally in
business, science, and mathematics. However this is not the only reason that linear
algebra is so useful. It is a central fact that smooth phenomena may be approx-
imated locally by linear phenomena. Without this great simplification, the world
of technology as we know it today would not exist. Of course, linear transforma-
tions send the origin to the origin, so they must be adjusted by a translation. As
a simple example, suppose h : R → R is differentiable and p is a real number. Let
f : R →R be the linear transformation f(x) = h
V
1dxdy. From the previous section we know that
any homomorphism f multiplies area by [ f [. The student may now understand
the following theorem from calculus. (Note that if h is the restriction of a linear
transformation from R
2
to R
2
, this theorem is immediate from the previous section.)
Theorem Suppose the determinant of J(h)(x, y) is non-negative for each
(x, y) ∈ V . Then the area of h(V ) is
V
[ J(h) [ dxdy.
92 Linear Algebra Chapter 5
The Transpose Principle
We now return to the case where F is a field (of arbitrary characteristic). F-
modules may also be called vector spaces and submodules may be called subspaces.
The study of R-modules in general is important and complex. However the study of
F-modules is short and simple – every vector space is free and every subspace is a
summand. The core of classical linear algebra is not the study of vector spaces, but
the study of homomorphisms, and in particular, of endomorphisms. One goal is to
show that if f : V → V is a homomorphism with some given property, there exists
a basis of V so that the matrix representing f displays that property in a prominent
manner. The next theorem is an illustration of this.
Theorem Let F be a field and n be a positive integer.
1) Suppose V is an n-dimensional vector space and f : V →V is a
homomorphism with [f[ = 0
¯
. Then ∃ a basis of V such that the matrix
representing f has its first row zero.
2) Suppose A ∈ F
n
has [A[ = 0
¯
. Then ∃ an invertible matrix C such that
C
−1
AC has its first row zero.
3) Suppose V is an n-dimensional vector space and f : V →V is a
homomorphism with [f[ = 0. Then ∃ a basis of V such that the matrix
representing f has its first column zero.
4) Suppose A ∈ F
n
has [A[ = 0
¯
. Then ∃ an invertible matrix D such that
D
−1
AD has its first column zero.
We first wish to show that these 4 statements are equivalent. We know that
1) and 2) are equivalent and also that 3) and 4) are equivalent because change of
basis corresponds to conjugation of the matrix. Now suppose 2) is true and show
4) is true. Suppose [A[ = 0
¯
. Then [A
t
[ = 0
¯
and by 2) ∃ C such that C
−1
A
t
C has
first row zero. Thus (C
−1
A
t
C)
t
= C
t
A(C
t
)
−1
has first row column zero. The result
follows by defining D = (C
t
)
−1
. Also 4) implies 2).
This is an example of the transpose principle. Loosely stated, it is that theorems
about change of basis correspond to theorems about conjugation of matrices and
theorems about the rows of a matrix correspond to theorems about the columns of a
matrix, using transpose. In the remainder of this chapter, this will be used without
further comment.
Chapter 5 Linear Algebra 93
Proof of the theorem We are free to select any of the 4 parts, and we select
part 3). Since [ f [= 0, f is not injective and ∃ a non-zero v
1
∈ V with f(v
1
) = 0
¯
.
Now v
1
is independent and extends to a basis ¦v
1
, .., v
n
¦. Then the matrix of f w.r.t
this basis has first column zero.
Exercise Let A =
and find an invertible matrix D ∈ R
3
so that D
−1
AD has first column zero.
Exercise Suppose M is an n-dimensional vector space over a field F, k is an
integer with 0 < k < n, and f : M → M is an endomorphism of rank k. Show
there is a basis for M so that the matrix representing f has its first n −k rows zero.
Also show there is a basis for M so that the matrix representing f has its first n −k
columns zero. Work these out directly without using the transpose principle.
Nilpotent Homomorphisms
In this section it is shown that an endomorphism f is nilpotent iff all of its char-
acteristic roots are 0
¯
iff it may be represented by a strictly upper triangular matrix.
Definition An endomorphism f : V →V is nilpotent if ∃ m with f
m
= 0
¯
. Any
f represented by a strictly upper triangular matrix is nilpotent (see page 56).
Theorem Suppose V is an n-dimensional vector space and f : V → V is a
nilpotent homomorphism. Then f
n
= 0
¯
and ∃ a basis of V such that the matrix
representing f w.r.t. this basis is strictly upper triangular. Thus the characteristic
polynomial of f is CP
f
(x) = x
n
.
Proof Suppose f = 0
¯
is nilpotent. Let t be the largest positive integer with
f
t
= 0
¯
. Then f
t
(V ) ⊂ f
t−1
(V ) ⊂ ⊂ f(V ) ⊂ V . Since f is nilpotent, all of these
inclusions are proper. Therefore t < n and f
n
= 0
¯
. Construct a basis for V by
starting with a basis for f
t
(V ), extending it to a basis for f
t−1
(V ), etc. Then the
matrix of f w.r.t. this basis is strictly upper triangular.
Note To obtain a matrix which is strictly lower triangular, reverse the order of
the basis.
94 Linear Algebra Chapter 5
Exercise Use the transpose principle to write 3 other versions of this theorem.
Theorem Suppose V is an n-dimensional vector space and f : V → V is a
homomorphism. Then f is nilpotent iff CP
f
(x) = x
n
. (See the exercise at the end
of Chapter 4 for the case n = 2.)
Proof Suppose CP
f
(x) = x
n
. For n = 1 this implies f = 0
¯
, so suppose n > 1.
Since the constant term of CP
f
(x) is 0
¯
, the determinant of f is 0
¯
. Thus ∃ a basis
of V such that the matrix A representing f has its first column zero. Let B ∈ F
n−1
be the matrix obtained from A by removing its first row and first column. Now
CP
A
(x) = x
n
= xCP
B
(x). Thus CP
B
(x) = x
n−1
and by induction on n, B is
nilpotent and so ∃ C such that C
−1
BC is strictly upper triangular. Then
¸
¸
¸
¸
¸
¸
¸
1 0 0
0
C
−1
0
¸
¸
¸
¸
¸
¸
¸
¸
0 ∗ ∗
B
0
¸
¸
¸
¸
¸
¸
¸
¸
1 0 0
0
C
0
¸
=
¸
¸
¸
¸
¸
¸
¸
0 ∗ ∗
0
C
−1
BC
0
¸
is strictly upper triangular.
Exercise Suppose F is a field, A ∈ F
3
is a strictly lower triangular matrix of
rank 2, and B =
¸
¸
0 0 0
1 0 0
0 1 0
¸
. Using conjugation by elementary matrices, show there
is an invertible matrix C so that C
−1
AC = B. Now suppose V is a 3-dimensional
vector space and f : V →V is a nilpotent endomorphism of rank 2. We know f can
be represented by a strictly lower triangular matrix. Show there is a basis ¦v
1
, v
2
, v
3
¦
for V so that B is the matrix representing f. Also show that f(v
1
) = v
2
, f(v
2
) = v
3
,
and f(v
3
) = 0
¯
. In other words, there is a basis for V of the form ¦v, f(v), f
2
(v)¦
with f
3
(v) = 0
¯
.
Exercise Suppose V is a 3-dimensional vector space and f : V →V is a nilpotent
endomorphism of rank 1. Show there is a basis for V so that the matrix representing
f is
¸
¸
0 0 0
1 0 0
0 0 0
¸
.
Chapter 5 Linear Algebra 95
Eigenvalues
Our standing hypothesis is that V is an n-dimensional vector space over a field F
and f : V →V is a homomorphism.
Definition An element λ ∈ F is an eigenvalue of f if ∃ a non-zero v ∈ V with
f(v) = λv. Any such v is called an eigenvector. E
λ
⊂ V is defined to be the set of
all eigenvectors for λ (plus 0
¯
). Note that E
λ
= ker(λI − f) is a subspace of V . The
next theorem shows the eigenvalues of f are just the characteristic roots of f.
Theorem If λ ∈ F then the following are equivalent.
1) λ is an eigenvalue of f, i.e., (λI −f) : V →V is not injective.
2) [ (λI −f) [= 0
¯
.
3) λ is a characteristic root of f, i.e., a root of the characteristic
polynomial CP
f
(x) =[ (xI −A) [, where A is any matrix representing f.
Proof It is immediate that 1) and 2) are equivalent, so let's show 2) and 3)
are equivalent. The evaluation map F[x] → F which sends h(x) to h(λ) is a ring
homomorphism (see theorem on page 47). So evaluating (xI − A) at x = λ and
taking determinant gives the same result as taking the determinant of (xI − A) and
evaluating at x = λ. Thus 2) and 3) are equivalent.
The nicest thing you can say about a matrix is that it is similar to a diagonal
matrix. Here is one case where that happens.
Theorem Suppose λ
1
, .., λ
k
are distinct eigenvalues of f, and v
i
is an eigenvector
of λ
i
for 1 ≤ i ≤ k. Then the following hold.
1) ¦v
1
, .., v
k
¦ is independent.
2) If k = n, i.e., if CP
f
(x) = (x −λ
1
) (x −λ
n
), then ¦v
1
, .., v
n
¦ is a
basis for V . The matrix of f w.r.t. this basis is the diagonal matrix whose
(i, i) term is λ
i
.
Proof Suppose ¦v
1
, .., v
k
¦ is dependent. Suppose t is the smallest positive integer
such that ¦v
1
, .., v
t
¦ is dependent, and v
1
r
1
+ +v
t
r
t
= 0
¯
is a non-trivial linear
combination. Note that at least two of the coefficients must be non-zero. Now
(f −λ
t
)(v
1
r
1
+ +v
t
r
t
) = v
1
(λ
1
−λ
t
)r
1
+ +v
t−1
(λ
t−1
−λ
t
)r
t−1
+0
¯
= 0
¯
is a shorter
96 Linear Algebra Chapter 5
non-trivial linear combination. This is a contradiction and proves 1). Part 2) follows
from 1) because dim(V ) = n.
Exercise Let A =
0 1
−1 0
∈ R
2
. Find an invertible C ∈ C
2
such that
C
−1
AC is diagonal. Show that C cannot be selected in R
2
. Find the characteristic
polynomial of A.
Exercise Suppose V is a 3-dimensional vector space and f : V →V is an endo-
morphism with CP
f
(x) = (x−λ)
3
. Show that (f −λI) has characteristic polynomial
x
3
and is thus a nilpotent endomorphism. Show there is a basis for V so that the
matrix representing f is
¸
¸
λ 0 0
1 λ 0
0 1 λ
¸
,
¸
¸
λ 0 0
1 λ 0
0 0 λ
¸
or
¸
¸
λ 0 0
0 λ 0
0 0 λ
¸
.
We could continue and finally give an ad hoc proof of the Jordan canonical form,
but in this chapter we prefer to press on to inner product spaces. The Jordan form
will be developed in Chapter 6 as part of the general theory of finitely generated
modules over Euclidean domains. The next section is included only as a convenient
reference.
Jordan Canonical Form
This section should be just skimmed or omitted entirely. It is unnecessary for the
rest of this chapter, and is not properly part of the flow of the chapter. The basic
facts of Jordan form are summarized here simply for reference.
The statement that a square matrix B over a field F is a Jordan block means that
∃ λ ∈ F such that B is a lower triangular matrix of the form
B =
∈ R
n
has f(C) = b. Then f
−1
(b) is the set of all solutions to a
1
x
1
+ +a
n
x
n
= b which
is the coset L+C, and this the set of all solutions to a
1
(x
1
−c
1
) + +a
n
(x
n
−c
n
) = 0.
Gram-Schmidt orthonormalization
Theorem (Fourier series) Suppose W is an IPS with an orthonormal basis
¦w
1
, .., w
n
¦. Then if v ∈ W, v = w
1
(v w
1
) + +w
n
(v w
n
).
Proof v = w
1
r
1
+ +w
n
r
n
and v w
i
= (w
1
r
1
+ +w
n
r
n
) w
i
= r
i
Theorem Suppose W is an IPS, Y ⊂ W is a subspace with an orthonormal basis
¦w
1
, .., w
k
¦, and v ∈ W−Y . Define the projection of v onto Y by p(v) = w
1
(v w
1
)+
+w
k
(vw
k
), and let w = v−p(v). Then (ww
i
) = (v−w
1
(vw
1
)−w
k
(vw
k
))w
i
= 0.
Thus if w
k+1
=
w
|w|
, then ¦w
1
, .., w
k+1
¦ is an orthonormal basis for the subspace
generated by ¦w
1
, .., w
k
, v¦. If ¦w
1
, .., w
k
, v¦ is already orthonormal, w
k+1
= v.
Theorem (Gram-Schmidt) Suppose W is an IPS with a basis ¦v
1
, .., v
n
¦.
Then W has an orthonormal basis ¦w
1
, .., w
n
¦. Moreover, any orthonormal sequence
in W extends to an orthonormal basis of W.
Proof Let w
1
=
v
1
|v
1
|
. Suppose inductively that ¦w
1
, .., w
k
¦ is an orthonormal
basis for Y , the subspace generated by ¦v
1
, .., v
k
¦. Let w = v
k+1
− p(v
k+1
) and
Chapter 5 Linear Algebra 101
w
k+1
=
w
|w|
. Then by the previous theorem, ¦w
1
, .., w
k+1
¦ is an orthonormal basis
for the subspace generated by ¦w
1
, .., w
k
, v
k+1
¦. In this manner an orthonormal basis
for W is constructed. Notice that this construction defines a function h which sends
a basis for W to an orthonormal basis for W (see topology exercise on page 103).
Now suppose W has dimension n and ¦w
1
, .., w
k
¦ is an orthonormal sequence in
W. Since this sequence is independent, it extends to a basis ¦w
1
, .., w
k
, v
k+1
, .., v
n
¦.
The process above may be used to modify this to an orthonormal basis ¦w
1
, .., w
n
¦.
Exercise Let f : R
3
→ R be the homomorphism defined by the matrix (2,1,3).
Find an orthonormal basis for the kernel of f. Find the projection of (e
1
+ e
2
) onto
ker(f). Find the angle between e
1
+e
2
and the plane ker(f).
Exercise Let W = R
3
have the standard inner product and Y ⊂ W be the
subspace generated by ¦w
1
, w
2
¦ where w
1
= (1, 0, 0)
t
and w
2
= (0, 1, 0)
t
. W is
generated by the sequence ¦w
1
, w
2
, v¦ where v = (1, 2, 3)
t
. As in the first theorem
of this section, let w = v − p(v), where p(v) is the projection of v onto Y , and set
w
3
=
w
|w|
. Find w
3
and show that for any t with 0 ≤ t ≤ 1, ¦w
1
, w
2
, (1 −t)v +tw
3
¦
is a basis for W. This is a key observation for an exercise on page 103 showing O(n)
is a deformation retract of GL
n
(R).
Isometries Suppose each of U and V is an IPS. A homomorphism f : U → V
is said to be an isometry provided it is an isomorphism and for any u
1
, u
2
in U,
(u
1
u
2
)
U
= (f(u
1
) f(u
2
))
V
.
Theorem Suppose each of U and V is an n-dimensional IPS, ¦u
1
, .., u
n
¦ is an
orthonormal basis for U, and f : U →V is a homomorphism. Then f is an isometry
iff ¦f(u
1
), .., f(u
n
)¦ is an orthonormal sequence in V .
Proof Isometries certainly preserve orthonormal sequences. So suppose T =
¦f(u
1
), .., f(u
n
)¦ is an orthonormal sequence in V . Then T is independent and thus
T is a basis for V and thus f is an isomorphism (see the second theorem on page 79).
It is easy to check that f preserves inner products.
We now come to one of the definitive theorems in linear algebra. It is that, up to
isometry, there is only one inner product space for each dimension.
102 Linear Algebra Chapter 5
Theorem Suppose each of U and V is an n-dimensional IPS. Then ∃ an isometry
f : U →V. In particular, U is isometric to R
n
with its standard inner product.
Proof There exist orthonormal bases ¦u
1
, .., u
n
¦ for U and ¦v
1
, .., v
n
¦ for V .
By the first theorem on page 79, there exists a homomorphism f : U → V with
f(u
i
) = v
i
, and by the previous theorem, f is an isometry.
Exercise Let f : R
3
→ R be the homomorphism defined by the matrix (2,1,3).
Find a linear transformation h : R
2
→R
3
which gives an isometry from R
2
to ker(f).
Orthogonal Matrices
As noted earlier, linear algebra is not so much the study of vector spaces as it is
the study of endomorphisms. We now wish to study isometries from R
n
to R
n
.
We know from a theorem on page 90 that an endomorphism preserves volume iff
its determinant is ±1. Isometries preserve inner product, and thus preserve angle and
distance, and so certainly preserve volume.
Theorem Suppose A ∈ R
n
and f : R
n
→ R
n
is the homomorphism defined by
f(B) = AB. Then the following are equivalent.
1) The columns of A form an orthonormal basis for R
n
, i.e., A
t
A = I.
2) The rows of A form an orthonormal basis for R
n
, i.e., AA
t
= I.
3) f is an isometry.
Proof A left inverse of a matrix is also a right inverse (see the exercise on
page 64). Thus 1) and 2) are equivalent because each of them says A is invert-
ible with A
−1
= A
t
. Now ¦e
1
, .., e
n
¦ is the canonical orthonormal basis for R
n
, and
f(e
i
) is column i of A. Thus by the previous section, 1) and 3) are equivalent.
Definition If A ∈ R
n
satisfies these three conditions, A is said to be orthogonal.
The set of all such A is denoted by O(n), and is called the orthogonal group.
Theorem
1) If A is orthogonal, [ A [= ±1.
2) If A is orthogonal, A
−1
is orthogonal. If A and C are orthogonal, AC is
orthogonal. Thus O(n) is a multiplicative subgroup of GL
n
(R).
Chapter 5 Linear Algebra 103
3) Suppose A is orthogonal and f is defined by f(B) = AB. Then f preserves
distances and angles. This means that if u, v ∈ R
n
, |u −v| =
|f(u)−f(v)| and the angle between u and v is equal to the angle between
f(u) and f(v).
Proof Part 1) follows from [A[
2
= [A[ [A
t
[ = [I[ = 1. Part 2) is imme-
diate, because isometries clearly form a subgroup of the multiplicative group of
all automorphisms. For part 3) assume f : R
n
→ R
n
is an isometry. Then
|u − v|
2
= (u − v) (u − v) = f(u − v) f(u − v) = |f(u − v)|
2
= |f(u) − f(v)|
2
.
The proof that f preserves angles follows from u v = |u||v|cosΘ.
Exercise Show that if A ∈ O(2) has [A[ = 1, then A =
cosΘ −sinΘ
sinΘ cosΘ
for
some number Θ. (See the exercise on page 56.)
Exercise (topology) Let R
n
≈ R
n
2
have its usual metric topology. This means
a sequence of matrices ¦A
i
¦ converges to A iff it converges coordinatewise. Show
GL
n
(R) is an open subset and O(n) is closed and compact. Let h : GL
n
(R) →
O(n) be defined by Gram-Schmidt. Show H : GL
n
(R) [0, 1] → GL
n
(R) defined
by H(A, t) = (1 −t)A+th(A) is a deformation retract of GL
n
(R) to O(n).
Diagonalization of Symmetric Matrices
We continue with the case F = R. Our goals are to prove that, if A is a symmetric
matrix, all of its eigenvalues are real and that ∃ an orthogonal matrix C such that
C
−1
AC is diagonal. As background, we first note that symmetric is the same as
self-adjoint.
Theorem Suppose A ∈ R
n
and u, v ∈ R
n
. Then (A
t
u) v = u (Av).
Proof If y, z ∈ R
n
, then the dot product y z, is the matrix product y
t
z, and
matrix multiplication is associative. Thus (A
t
u) v = (u
t
A)v = u
t
(Av) = u (Av).
Definition Suppose A ∈ R
n
. A is said to be symmetric provided A
t
= A. Note
that any diagonal matrix is symmetric. A is said to be self-adjoint if (Au)v = u(Av)
for all u, v ∈ R
n
. The next theorem is just an exercise using the previous theorem.
Theorem A is symmetric iff A is self-adjoint.
104 Linear Algebra Chapter 5
Theorem Suppose A ∈ R
n
is symmetric. Then ∃ real numbers λ
1
, .., λ
n
(not
necessarily distinct) such that CP
A
(x) = (x − λ
1
)(x − λ
2
) (x − λ
n
). That is, all
the eigenvalues of A are real.
Proof We know CP
A
(x) factors into linears over C. If µ = a + bi is a complex
number, its conjugate is defined by ¯ µ = a −bi. If h : C →C is defined by h(µ) = ¯ µ,
then h is a ring isomorphism which is the identity on R. If w = (a
i,j
) is a complex
matrix or vector, its conjugate is defined by ¯ w = (¯ a
i,j
). Since A ∈ R
n
is a real
symmetric matrix, A = A
t
=
¯
A
t
. Now suppose λ is a complex eigenvalue of A
and v ∈ C
n
is an eigenvector with Av = λv. Then λ(v
t
¯ v) = (λv)
t
¯ v = (Av)
t
¯ v =
(v
t
A)¯ v = v
t
(A¯ v) = v
t
(Av) = v
t
(λv) =
¯
λ(v
t
¯ v). Thus λ =
¯
λ and λ ∈ R. Or
you can define a complex inner product on C
n
by (w v) = w
t
¯ v. The proof then
reads as λ(v v) = (λv v) = (Av v) = (v Av) = (v λv) =
¯
λ(v v). Either way,
λ is a real number.
We know that eigenvectors belonging to distinct eigenvalues are linearly indepen-
dent. For symmetric matrices, we show more, namely that they are perpendicular.
Theorem Suppose A is symmetric, λ
1
, λ
2
∈ R are distinct eigenvalues of A, and
Au = λ
1
u and Av = λ
2
v. Then u v = 0.
Proof λ
1
(u v) = (Au) v = u (Av) = λ
2
(u v).
Review Suppose A ∈ R
n
and f : R
n
→ R
n
is defined by f(B) = AB. Then A
represents f w.r.t. the canonical orthonormal basis. Let S = ¦v
1
, .., v
n
¦ be another
basis and C ∈ R
n
be the matrix with v
i
as column i. Then C
−1
AC is the matrix
representing f w.r.t. S. Now S is an orthonormal basis iff C is an orthogonal matrix.
Summary Representing f w.r.t. an orthonormal basis is the same as conjugating
A by an orthogonal matrix.
Theorem Suppose A ∈ R
n
and C ∈ O(n). Then A is symmetric iff C
−1
AC
is symmetric.
Proof Suppose A is symmetric. Then (C
−1
AC)
t
= C
t
A(C
−1
)
t
= C
−1
AC.
The next theorem has geometric and physical implications, but for us, just the
incredibility of it all will suffice.
Chapter 5 Linear Algebra 105
Theorem If A ∈ R
n
, the following are equivalent.
1) A is symmetric.
2) ∃ C ∈ O(n) such that C
−1
AC is diagonal.
Proof By the previous theorem, 2) ⇒ 1). Show 1) ⇒ 2). Suppose A is a
symmetric 22 matrix. Let λ be an eigenvalue for A and ¦v
1
, v
2
¦ be an orthonormal
basis for R
2
with Av
1
= λv
1
. Then w.r.t this basis, the transformation determined
by A is represented by
λ b
0 d
. Since this matrix is symmetric, b = 0.
Now suppose by induction that the theorem is true for symmetric matrices in
R
t
for t < n, and suppose A is a symmetric n n matrix. Denote by λ
1
, .., λ
k
the
distinct eigenvalues of A, k ≤ n. If k = n, the proof is immediate, because then there
is a basis of eigenvectors of length 1, and they must form an orthonormal basis. So
suppose k < n. Let v
1
, .., v
k
be eigenvectors for λ
1
, .., λ
k
with each | v
i
|= 1. They
may be extended to an orthonormal basis v
1
, .., v
n
. With respect to this basis, the
transformation determined by A is represented by
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
λ
1
λ
k
¸
(B)
(0) (D)
¸
.
Since this is a symmetric matrix, B = 0 and D is a symmetric matrix of smaller
size. By induction, ∃ an orthogonal C such that C
−1
DC is diagonal. Thus conjugating
by
I 0
0 C
makes the entire matrix diagonal.
This theorem is so basic we state it again in different terminology. If V is an IPS, a
linear transformation f : V →V is said to be self-adjoint provided (uf(v)) = (f(u)v)
for all u, v ∈ V .
Theorem If V is an n-dimensional IPS and f : V →V is a linear transformation,
then the following are equivalent.
1) f is self-adjoint.
2) ∃ an orthonormal basis ¦v
1
, ..., v
n
¦ for V with each
v
i
an eigenvector of f.
106 Linear Algebra Chapter 5
Exercise Let A =
2 2
2 2
. Find an orthogonal C such that C
−1
AC is diagonal.
Do the same for A =
2 1
1 2
.
Exercise Suppose A, D ∈ R
n
are symmetric. Under what conditions are A and D
similar? Show that, if A and D are similar, ∃ an orthogonal C such that D = C
−1
AC.
Exercise Suppose V is an n-dimensional real vector space. We know that V is
isomorphic to R
n
. Suppose f and g are isomorphisms from V to R
n
and A is a subset
of V . Show that f(A) is an open subset of R
n
iff g(A) is an open subset of R
n
. This
shows that V , an algebraic object, has a god-given topology. Of course, if V has
an inner product, it automatically has a metric, and this metric will determine that
same topology. Finally, suppose V and W are finite-dimensional real vector spaces
and h : V →W is a linear transformation. Show that h is continuous.
Exercise Define E : C
n
→C
n
by E(A) = e
A
= I +A+(1/2!)A
2
+. This series
converges and thus E is a well defined function. If AB = BA, then E(A + B) =
E(A)E(B). Since A and −A commute, I = E(0
¯
) = E(A − A) = E(A)E(−A), and
thus E(A) is invertible with E(A)
−1
= E(−A). Furthermore E(A
t
) = E(A)
t
, and
if C is invertible, E(C
−1
AC) = C
−1
E(A)C. Now use the results of this section to
prove the statements below. (For part 1, assume the Jordan form, i.e., assume any
A ∈ C
n
is similar to a lower triangular matrix.)
1) If A ∈ C
n
, then [ e
A
[= e
trace(A)
. Thus if A ∈ R
n
, [ e
A
[= 1
iff trace(A) = 0.
2) ∃ a non-zero matrix N ∈ R
2
with e
N
= I.
3) If N ∈ R
n
is symmetric, then e
N
= I iff N = 0
¯
.
4) If A ∈ R
n
and A
t
= −A, then e
A
∈ O(n).
Chapter 6
Appendix
The five previous chapters were designed for a year undergraduate course in algebra.
In this appendix, enough material is added to form a basic first year graduate course.
Two of the main goals are to characterize finitely generated abelian groups and to
prove the Jordan canonical form. The style is the same as before, i.e., everything is
right down to the nub. The organization is mostly a linearly ordered sequence except
for the last two sections on determinants and dual spaces. These are independent
sections added on at the end.
Suppose R is a commutative ring. An R-module M is said to be cyclic if it can
be generated by one element, i.e., M ≈ R/I where I is an ideal of R. The basic
theorem of this chapter is that if R is a Euclidean domain and M is a finitely generated
R-module, then M is the sum of cyclic modules. Thus if M is torsion free, it is a
free R-module. Since Z is a Euclidean domain, finitely generated abelian groups
are the sums of cyclic groups – one of the jewels of abstract algebra.
Now suppose F is a field and V is a finitely generated F-module. If T : V →V is
a linear transformation, then V becomes an F[x]-module by defining vx = T(v). Now
F[x] is a Euclidean domain and so V
F[x]
is the sum of cyclic modules. This classical
and very powerful technique allows an easy proof of the canonical forms. There is a
basis for V so that the matrix representing T is in Rational canonical form. If the
characteristic polynomial of T factors into the product of linear polynomials, then
there is a basis for V so that the matrix representing T is in Jordan canonical form.
This always holds if F = C. A matrix in Jordan form is a lower triangular matrix
with the eigenvalues of T displayed on the diagonal, so this is a powerful concept.
In the chapter on matrices, it is stated without proof that the determinant of the
product is the product of the determinants. A proof of this, which depends upon the
classification of certain types of alternating multilinear forms, is given in this chapter.
The final section gives the fundamentals of dual spaces.
107
108 Appendix Chapter 6
The Chinese Remainder Theorem
On page 50 in the chapter on rings, the Chinese Remainder Theorem was proved
for the ring of integers. In this section this classical topic is presented in full generality.
Surprisingly, the theorem holds even for non-commutative rings.
Definition Suppose R is a ring and A
1
, A
2
, ..., A
m
are ideals of R. Then the sum
A
1
+ A
2
+ + A
m
is the set of all a
1
+ a
2
+ + a
m
with a
i
∈ A
i
. The product
A
1
A
2
A
m
is the set of all finite sums of elements a
1
a
2
a
m
with a
i
∈ A
i
. Note
that the sum and product of ideals are ideals and A
1
A
2
A
m
⊂ (A
1
∩A
2
∩ ∩A
m
).
Definition Ideals A and B of R are said to be comaximal if A+B = R.
Theorem If A and B are ideals of a ring R, then the following are equivalent.
1) A and B are comaximal.
2) ∃ a ∈ A and b ∈ B with a +b = 1
¯
.
3) π(A) = R/B where π : R →R/B is the projection.
Theorem If A
1
, A
2
, ..., A
m
and B are ideals of R with A
i
and B comaximal for
each i, then A
1
A
2
A
m
and B are comaximal. Thus A
1
∩ A
2
∩ ∩ A
m
and B
are comaximal.
Proof Consider π : R →R/B. Then π(A
1
A
2
A
m
) = π(A
1
)π(A
2
) π(A
m
) =
(R/B)(R/B) (R/B) = R/B.
Chinese Remainder Theorem Suppose A
1
, A
2
, ..., A
n
are pairwise comaximal
ideals of R, with each A
i
= R. Then the natural map π : R →R/A
1
R/A
2
R/A
n
is a surjective ring homomorphism with kernel A
1
∩ A
2
∩ ∩ A
n
.
Proof There exists a
i
∈ A
i
and b
i
∈ A
1
A
2
A
i−1
A
i+1
A
n
with a
i
+b
i
= 1
¯
. Note
that π(b
i
) = (0, .., 0, 1
¯
i
, 0, .., 0). If (r
1
+ A
1
, r
2
+ A
2
, ..., r
n
+ A
n
) is an element of the
range, it is the image of r
1
b
1
+r
2
b
2
++r
n
b
n
= r
1
(1
¯
−a
1
)+r
2
(1
¯
−a
2
)++r
n
(1
¯
−a
n
).
Theorem If R is commutative and A
1
, A
2
, ..., A
n
are pairwise comaximal ideals
of R, then A
1
A
2
A
n
= A
1
∩ A
2
∩ ∩ A
n
.
Proof for n = 2. Show A
1
∩A
2
⊂ A
1
A
2
. ∃ a
1
∈ A
1
and a
2
∈ A
2
with a
1
+a
2
= 1
¯
.
If c ∈ A
1
∩ A
2
, then c = c(a
1
+a
2
) ∈ A
1
A
2
.
Chapter 6 Appendix 109
Prime and Maximal Ideals and UFD
s
In the first chapter on background material, it was shown that Z is a unique
factorization domain. Here it will be shown that this property holds for any principle
ideal domain. Later on it will be shown that every Euclidean domain is a principle
ideal domain. Thus every Euclidean domain is a unique factorization domain.
Definition Suppose R is a commutative ring and I ⊂ R is an ideal.
I is prime means I = R and if a, b ∈ R have ab ∈ I, then a or b ∈ I.
I is maximal means I = R and there are no ideals properly between I and R.
Theorem 0
¯
is a prime ideal of R iff R is
0
¯
is a maximal ideal of R iff R is
Theorem Suppose J ⊂ R is an ideal, J = R.
J is a prime ideal iff R/J is
J is a maximal ideal iff R/J is
Corollary Maximal ideals are prime.
Proof Every field is a domain.
Theorem If a ∈ R is not a unit, then ∃ a maximal ideal I of R with a ∈ I.
Proof This is a classical application of the Hausdorff Maximality Principle. Con-
sider ¦J : J is an ideal of R containing a with J = R¦. This collection contains a
maximal monotonic collection ¦V
t
¦
t∈T
. The ideal V =
¸
t∈T
V
t
does not contain 1
¯
and
thus is not equal to R. Therefore V is equal to some V
t
and is a maximal ideal
containing a.
Note To properly appreciate this proof, the student should work the exercise in
group theory at the end of this section (see page 114).
Definition Suppose R is a domain and a, b ∈ R. Then we say a ∼ b iff there
exists a unit u with au = b. Note that ∼ is an equivalence relation. If a ∼ b, then a
110 Appendix Chapter 6
and b are said to be associates.
Examples If R is a domain, the associates of 1
¯
are the units of R, while the only
associate of 0
¯
is 0
¯
itself. If n ∈ Z is not zero, then its associates are n and −n.
If F is a field and g ∈ F[x] is a non-zero polynomial, then the associates of g are
all cg where c is a non-zero constant.
The following theorem is elementary, but it shows how associates fit into the
scheme of things. An element a divides b (a[b) if ∃! c ∈ R with ac = b.
Theorem Suppose R is a domain and a, b ∈ (R − 0
¯
). Then the following are
equivalent.
1) a ∼ b.
2) a[b and b[a.
3) aR = bR.
Parts 1) and 3) above show there is a bijection from the associate classes of R to
the principal ideals of R. Thus if R is a PID, there is a bijection from the associate
classes of R to the ideals of R. If an element of a domain generates a non-zero prime
ideal, it is called a prime element.
Definition Suppose R is a domain and a ∈ R is a non-zero non-unit.
1) a is irreducible if it does not factor, i.e., a = bc ⇒ b or c is a unit.
2) a is prime if it generates a prime ideal, i.e., a[bc ⇒ a[b or a[c.
Note If a is a prime and a[c
1
c
2
c
n
, then a[c
i
for some i. This follows from the
definition and induction on n. If each c
j
is irreducible, then a ∼ c
i
for some i.
Note If a ∼ b, then a is irreducible (prime) iff b is irreducible (prime). In other
words, if a is irreducible (prime) and u is a unit, then au is irreducible (prime).
Note a is prime ⇒a is irreducible. This is immediate from the definitions.
Theorem Factorization into primes is unique up to order and associates, i.e., if
d = b
1
b
2
b
n
= c
1
c
2
c
m
with each b
i
and each c
i
prime, then n = m and for some
permutation σ of the indices, b
i
and c
σ(i)
are associates for every i. Note also ∃ a unit
u and primes p
1
, p
2
, . . . , p
t
where no two are associates and du = p
s
1
1
p
s
2
2
p
st
t
.
Chapter 6 Appendix 111
Proof This follows from the notes above.
Definition R is a factorization domain (FD) means that R is a domain and if a is
a non-zero non-unit element of R, then a factors into a finite product of irreducibles.
Definition R is a unique factorization domain (UFD) means R is a FD in which
factorization is unique (up to order and associates).
Theorem If R is a UFD and a is a non-zero non-unit of R, then a is irreducible
⇔ a is prime. Thus in a UFD, elements factor as the product of primes.
Proof Suppose R is a UFD, a is an irreducible element of R, and a[bc. If either
b or c is a unit or is zero, then a divides one of them, so suppose each of b and c is
a non-zero non-unit element of R. There exists an element d with ad = bc. Each of
b and c factors as the product of irreducibles and the product of these products is
the factorization of bc. It follows from the uniqueness of the factorization of ad = bc,
that one of these irreducibles is an associate of a, and thus a[b or a[c. Therefore
the element a is a prime.
Theorem Suppose R is a FD. Then the following are equivalent.
1) R is a UFD.
2) Every irreducible element of R is prime, i.e., a irreducible ⇔ a is prime.
Proof We already know 1) ⇒ 2). Part 2) ⇒ 1) because factorization into primes
is always unique.
This is a revealing and useful theorem. If R is a FD, then R is a UFD iff each
irreducible element generates a prime ideal. Fortunately, principal ideal domains
have this property, as seen in the next theorem.
Theorem Suppose R is a PID and a ∈ R is non-zero non-unit. Then the following
are equivalent.
1) aR is a maximal ideal.
2) aR is a prime ideal, i.e., a is a prime element.
3) a is irreducible.
Proof Every maximal ideal is a prime ideal, so 1) ⇒ 2). Every prime element is
an irreducible element, so 2) ⇒ 3). Now suppose a is irreducible and show aR is a
maximal ideal. If I is an ideal containing aR, ∃ b ∈ R with I = bR. Since b divides
a, the element b is a unit or an associate of a. This means I = R or I = aR.
112 Appendix Chapter 6
Our goal is to prove that a PID is a UFD. Using the two theorems above, it
only remains to show that a PID is a FD. The proof will not require that ideals be
principally generated, but only that they be finitely generated. This turns out to
be equivalent to the property that any collection of ideals has a "maximal" element.
We shall see below that this is a useful concept which fits naturally into the study of
unique factorization domains.
Theorem Suppose R is a commutative ring. Then the following are equivalent.
1) If I ⊂ R is an ideal, ∃ a finite set ¦a
1
, a
2
, ..., a
n
¦ ⊂ R such that I =
a
1
R +a
2
R + +a
n
R, i.e., each ideal of R is finitely generated.
2) Any non-void collection of ideals of R contains an ideal I which is maximal in
the collection. This means if J is an ideal in the collection with J ⊃ I, then
J = I. (The ideal I is maximal only in the sense described. It need not contain
all the ideals of the collection, nor need it be a maximal ideal of the ring R.)
3) If I
1
⊂ I
2
⊂ I
3
⊂ ... is a monotonic sequence of ideals, ∃ t
0
≥ 1 such that I
t
= I
t
0
for all t ≥ t
0
.
Proof Suppose 1) is true and show 3). The ideal I = I
1
∪ I
2
∪ . . . is finitely
generated and ∃ t
0
≥ 1 such that I
t
0
contains those generators. Thus 3) is true. Now
suppose 2) is true and show 1). Let I be an ideal of R, and consider the collection
of all finitely generated ideals contained in I. By 2) there is a maximal one, and it
must be I itself, and thus 1) is true. We now have 2)⇒1)⇒3), so suppose 2) is false
and show 3) is false. So there is a collection of ideals of R such that any ideal in the
collection is properly contained in another ideal of the collection. Thus it is possible
to construct a sequence of ideals I
1
⊂ I
2
⊂ I
3
. . . with each properly contained in
the next, and therefore 3) is false. (Actually this construction requires the Hausdorff
Maximality Principle or some form of the Axiom of Choice, but we slide over that.)
Definition If R satisfies these properties, R is said to be Noetherian, or it is said
to satisfy the ascending chain condition. This property is satisfied by many of the
classical rings in mathematics. Having three definitions makes this property useful
and easy to use. For example, see the next theorem.
Theorem A Noetherian domain is a FD. In particular, a PID is a FD.
Proof Suppose there is a non-zero non-unit element that does not factor as the
finite product of irreducibles. Consider all ideals dR where d does not factor. Since
R is Noetherian, ∃ a maximal one cR. The element c must be reducible, i.e., c = ab
where neither a nor b is a unit. Each of aR and bR properly contains cR, and so each
Chapter 6 Appendix 113
of a and b factors as a finite product of irreducibles. This gives a finite factorization
of c into irreducibles, which is a contradiction.
Corollary A PID is a UFD. So Z is a UFD and if F is a field, F[x] is a UFD.
You see the basic structure of UFD
s
is quite easy. It takes more work to prove
the following theorems, which are stated here only for reference.
Theorem If R is a UFD then R[x
1
, ..., x
n
] is a UFD. Thus if F is a field,
F[x
1
, ..., x
n
] is a UFD. (This theorem goes all the way back to Gauss.)
If R is a PID, then the formal power series R[[x
1
, ..., x
n
]] is a UFD. Thus if F
is a field, F[[x
1
, ..., x
n
]] is a UFD. (There is a UFD R where R[[x]] is not a UFD.
See page 566 of Commutative Algebra by N. Bourbaki.)
Theorem Germs of analytic functions on C
n
form a UFD.
Proof See Theorem 6.6.2 of An Introduction to Complex Analysis in Several Vari-
ables by L. H¨ ormander.
Theorem Suppose R is a commutative ring. Then R is Noetherian ⇒R[x
1
, ..., x
n
]
and R[[x
1
, ..., x
n
]] are Noetherian. (This is the famous Hilbert Basis Theorem.)
Theorem If R is Noetherian and I ⊂ R is a proper ideal, then R/I is Noetherian.
(This follows immediately from the definition. This and the previous theorem show
that Noetherian is a ubiquitous property in ring theory.)
Domains With Non-unique Factorizations Next are presented two of the
standard examples of Noetherian domains that are not unique factorization domains.
Exercise Let R = Z(
√
5) = ¦n +m
√
5 : n, m ∈ Z¦. Show that R is a subring of
R which is not a UFD. In particular 2 2 = (1 −
√
5) (−1 −
√
5) are two distinct
irreducible factorizations of 4. Show R is isomorphic to Z[x]/(x
2
−5), where (x
2
−5)
represents the ideal (x
2
− 5)Z[x], and R/(2) is isomorphic to Z
2
[x]/(x
2
− [5]) =
Z
2
[x]/(x
2
+ [1]), which is not a domain.
114 Appendix Chapter 6
Exercise Let R = R[x, y, z]/(x
2
− yz). Show x
2
− yz is irreducible and thus
prime in R[x, y, z]. If u ∈ R[x, y, z], let ¯ u ∈ R be the coset containing u. Show R
is not a UFD. In particular ¯ x ¯ x = ¯ y ¯ z are two distinct irreducible factorizations
of ¯ x
2
. Show R/(¯ x) is isomorphic to R[y, z]/(yz), which is not a domain. An easier
approach is to let f : R[x, y, z] → R[x, y] be the ring homomorphism defined by
f(x) = xy, f(y) = x
2
, and f(z) = y
2
. Then S = R[xy, x
2
, y
2
] is the image of
f and S is isomorphic to R. Note that xy, x
2
, and y
2
are irreducible in S and
(xy)(xy) = (x
2
)(y
2
) are two distinct irreducible factorizations of (xy)
2
in S.
Exercise In Group Theory If G is an additive abelian group, a subgroup H
of G is said to be maximal if H = G and there are no subgroups properly between
H and G. Show that H is maximal iff G/H ≈ Z
p
for some prime p. For simplicity,
consider the case G = Q. Which one of the following is true?
1) If a ∈ Q, then there is a maximal subgroup H of Q which contains a.
2) Q contains no maximal subgroups.
Splitting Short Exact Sequences
Suppose B is an R-module and K is a submodule of B. As defined in the chapter
on linear algebra, K is a summand of B provided ∃ a submodule L of B with
K+L = B and K∩L = 0
¯
. In this case we write K⊕L = B. When is K a summand
of B? It turns out that K is a summand of B iff there is a splitting map from
B/K to B. In particular, if B/K is free, K must be a summand of B. This is used
below to show that if R is a PID, then every submodule of R
n
is free.
Theorem 1 Suppose R is a ring, B and C are R-modules, and g : B → C is a
surjective homomorphism with kernel K. Then the following are equivalent.
1) K is a summand of B.
2) g has a right inverse, i.e., ∃ a homomorphism h : C →B with g ◦h = I : C →C.
(h is called a splitting map.)
Proof Suppose 1) is true, i.e., suppose ∃ a submodule L of B with K ⊕ L = B.
Then (g[L) : L → C is an isomorphism. If i : L → B is inclusion, then h defined
by h = i ◦ (g[L)
−1
is a right inverse of g. Now suppose 2) is true and h : C → B
is a right inverse of g. Then h is injective, K + h(C) = B and K ∩ h(C) = 0
¯
.
Thus K ⊕h(C) = B.
Chapter 6 Appendix 115
Definition Suppose f : A → B and g : B → C are R-module homomorphisms.
The statement that 0 →A
f
→B
g
→C →0 is a short exact sequence (s.e.s) means
f is injective, g is surjective and f(A) = ker(g). The canonical split s.e.s. is A →
A ⊕ C → C where f = i
1
and g = π
2
. A short exact sequence is said to split if ∃
an isomorphism B
≈
→A⊕C such that the following diagram commutes.
0 → A B C →0
A⊕C
≈
f
g
i
1
π
2
`
`
`
`
`
`
·
We now restate the previous theorem in this terminology.
Theorem 1.1 A short exact sequence 0 → A → B → C → 0 splits iff f(A) is
a summand of B, iff B → C has a splitting map. If C is a free R-module, there is
a splitting map and thus the sequence splits.
Proof We know from the previous theorem f(A) is a summand of B iff B → C
has a splitting map. Showing these properties are equivalent to the splitting of the
sequence is a good exercise in the art of diagram chasing. Now suppose C has a free
basis T ⊂ C, and g : B → C is surjective. There exists a function h : T → B such
that g ◦ h(c) = c for each c ∈ T. The function h extends to a homomorphism from
C to B which is a right inverse of g.
Theorem 2 If R is a domain, then the following are equivalent.
1) R is a PID.
2) Every submodule of R
R
is a free R-module of dimension ≤ 1.
This theorem restates the ring property of PID as a module property. Although
this theorem is transparent, 1)⇒2) is a precursor to the following classical result.
Theorem 3 If R is a PID and A ⊂ R
n
is a submodule, then A is a free R-module
of dimension ≤ n. Thus subgroups of Z
n
are free Z-modules of dimension ≤ n.
Proof From the previous theorem we know this is true for n = 1. Suppose n > 1
and the theorem is true for submodules of R
n−1
. Suppose A ⊂ R
n
is a submodule.
116 Appendix Chapter 6
Consider the following short exact sequences, where f : R
n−1
→R
n−1
⊕R is inclusion
and g = π : R
n−1
⊕R →R is the projection.
0 −→R
n−1
f
−→R
n−1
⊕R
π
−→R −→0
0 −→A∩ R
n−1
−→A −→π(A) −→0
By induction, A∩ R
n−1
is free of dimension ≤ n −1. If π(A) = 0
¯
, then A ⊂ R
n−1
.
If π(A) = 0
¯
, it is free of dimension 1 and thus the sequence splits by Theorem 1.1.
In either case, A is a free submodule of dimension ≤ n.
Exercise Let A ⊂ Z
2
be the subgroup generated by ¦(6, 24), (16, 64)¦. Show A
is a free Z-module of dimension 1. Also show the s.e.s. Z
4
×3
−→ Z
12
−→ Z
3
splits
but Z
×2
−→Z −→Z
2
and Z
2
×2
−→Z
4
−→Z
2
do not (see top of page 78).
Euclidean Domains
The ring Z possesses the Euclidean algorithm and the polynomial ring F[x] has
the division algorithm (pages 14 and 45). The concept of Euclidean domain is an
abstraction of these properties, and the efficiency of this abstraction is displayed in
this section. Furthermore the first axiom, φ(a) ≤ φ(ab), is used only in Theorem
2, and is sometimes omitted from the definition. Anyway it is possible to just play
around with matrices and get some deep results. If R is a Euclidean domain and M
is a finitely generated R-module, then M is the sum of cyclic modules. This is one of
the great classical theorems of abstract algebra, and you don't have to worry about
it becoming obsolete. Here N will denote the set of all non-negative integers, not
just the set of positive integers.
Definition A domain R is a Euclidean domain provided ∃ φ : (R−0
¯
) −→N such
that if a, b ∈ (R −0
¯
), then
1) φ(a) ≤ φ(ab).
2) ∃ q, r ∈ R such that a = bq +r with r = 0
¯
or φ(r) < φ(b).
Examples of Euclidean Domains
Z with φ(n) = [n[.
A field F with φ(a) = 1 ∀ a = 0
¯
or with φ(a) = 0 ∀ a = 0
¯
.
F[x] where F is a field with φ(f = a
0
+a
1
x + +a
n
x
n
) = deg(f).
Z[i] = ¦a +bi : a, b ∈ Z¦ = Gaussian integers with φ(a +bi) = a
2
+b
2
.
Chapter 6 Appendix 117
Theorem 1 If R is a Euclidean domain, then R is a PID and thus a UFD.
Proof If I is a non-zero ideal, then ∃ b ∈ I −0
¯
satisfying φ(b) ≤ φ(a) ∀ a ∈ I −0
¯
.
Then b generates I because if a ∈ I − 0
¯
, ∃ q, r with a = bq + r. Now r ∈ I and
r = 0
¯
⇒φ(r) < φ(b) which is impossible. Thus r = 0
¯
and a ∈ bR so I = bR.
Theorem 2 If R is a Euclidean domain and a, b ∈ R −0
¯
, then
φ(1
¯
) is the smallest integer in the image of φ.
a is a unit in R iff φ(a) = φ(1
¯
).
a and b are associates ⇒ φ(a) = φ(b).
Proof This is a good exercise. However it is unnecessary for Theorem 3 below.
The following remarkable theorem is the foundation for the results of this section.
Theorem 3 If R is a Euclidean domain and (a
i,j
) ∈ R
n,t
is a non-zero matrix,
then by elementary row and column operations (a
i,j
) can be transformed to
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
d
1
0 0
0 d
2
.
.
.
.
.
.
d
m
0
0 0
¸
where each d
i
= 0
¯
, and d
i
[d
i+1
for 1 ≤ i < m. Also d
1
generates the ideal of R
generated by the entries of (a
i,j
).
Proof Let I ⊂ R be the ideal generated by the elements of the matrix A = (a
i,j
).
If E ∈ R
n
, then the ideal J generated by the elements of EA has J ⊂ I. If E is
invertible, then J = I. In the same manner, if E ∈ R
t
is invertible and J is the ideal
generated by the elements of AE, then J = I. This means that row and column
operations on A do not change the ideal I. Since R is a PID, there is an element
d
1
with I = d
1
R, and this will turn out to be the d
1
displayed in the theorem.
The matrix (a
i,j
) has at least one non-zero element d with φ(d) a miminum.
However, row and column operations on (a
i,j
) may produce elements with smaller
118 Appendix Chapter 6
φ values. To consolidate this approach, consider matrices obtained from (a
i,j
) by a
finite number of row and column operations. Among these, let (b
i,j
) be one which
has an entry d
1
= 0 with φ(d
1
) a minimum. By elementary operations of type 2, the
entry d
1
may be moved to the (1, 1) place in the matrix. Then d
1
will divide the other
entries in the first row, else we could obtain an entry with a smaller φ value. Thus
by column operations of type 3, the other entries of the first row may be made zero.
In a similar manner, by row operations of type 3, the matrix may be changed to the
following form.
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
d
1
0 0
0
.
.
. c
ij
0
¸
Note that d
1
divides each c
i,j
, and thus I = d
1
R. The proof now follows by induction
on the size of the matrix.
This is an example of a theorem that is easy to prove playing around at the
blackboard. Yet it must be a deep theorem because the next two theorems are easy
consequences.
Theorem 4 Suppose R is a Euclidean domain, B is a finitely generated free R-
module and A ⊂ B is a non-zero submodule. Then ∃ free bases ¦a
1
, a
2
, ..., a
t
¦ for A
and ¦b
1
, b
2
, ..., b
n
¦ for B, with t ≤ n, and such that each a
i
= d
i
b
i
, where each d
i
= 0
¯
,
and d
i
[d
i+1
for 1 ≤ i < t. Thus B/A ≈ R/d
1
⊕R/d
2
⊕ ⊕R/d
t
⊕R
n−t
.
Proof By Theorem 3 in the section Splitting Short Exact Sequences, A has a
free basis ¦v
1
, v
2
, ..., v
t
¦. Let ¦w
1
, w
2
, ..., w
n
¦ be a free basis for B, where n ≥ t. The
composition
R
t
≈
−→A
⊂
−→B
≈
−→R
n
e
i
−→v
i
w
i
−→e
i
is represented by a matrix (a
i,j
) ∈ R
n,t
where v
i
= a
1,i
w
1
+a
2,i
w
2
+ +a
n,i
w
n
. By
the previous theorem, ∃ invertible matrixes U ∈ R
n
and V ∈ R
t
such that
Chapter 6 Appendix 119
U(a
i,j
)V =
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
d
1
0 0
0 d
2
0
.
.
. 0
.
.
.
d
t
0 0
¸
with d
i
[d
i+1
. Since changing the isomorphisms R
t
≈
−→A and B
≈
−→R
n
corresponds
to changing the bases ¦v
1
, v
2
, ..., v
t
¦ and ¦w
1
, w
2
, ..., w
n
¦, the theorem follows.
Theorem 5 If R is a Euclidean domain and M is a finitely generated R-module,
then M ≈ R/d
1
⊕R/d
2
⊕ ⊕R/d
t
⊕R
m
where each d
i
= 0
¯
, and d
i
[d
i+1
for 1 ≤ i < t.
Proof By hypothesis ∃ a finitely generated free module B and a surjective homo-
morphism B −→M −→0. Let A be the kernel, so 0 −→A
⊂
−→B −→M −→0 is
a s.e.s. and B/A ≈ M. The result now follows from the previous theorem.
The way Theorem 5 is stated, some or all of the elements d
i
may be units, and for
such d
i
, R/d
i
= 0
¯
. If we assume that no d
i
is a unit, then the elements d
1
, d
2
, ..., d
t
are
called invariant factors. They are unique up to associates, but we do not bother with
that here. If R = Z and we select the d
i
to be positive, they are unique. If R = F[x]
and we select the d
i
to be monic, then they are unique. The splitting in Theorem 5
is not the ultimate because the modules R/d
i
may split into the sum of other cyclic
modules. To prove this we need the following Lemma.
Lemma Suppose R is a PID and b and c are non-zero non-unit elements of R.
Suppose b and c are relatively prime, i.e., there is no prime common to their prime
factorizations. Then bR and cR are comaximal ideals. (See p 108 for comaximal.)
Proof There exists an a ∈ R with aR = bR + cR. Since a[b and a[c, a is a
unit, so R = bR +cR.
Theorem 6 Suppose R is a PID and d is a non-zero non-unit element of R.
Assume d = p
s
1
1
p
s
2
2
p
st
t
is the prime factorization of d (see bottom of p 110). Then
the natural map R/d
≈
−→R/p
s
1
1
⊕ ⊕ R/p
st
t
is an isomorphism of R-modules.
(The elements p
s
i
i
are called elementary divisors of R/d.)
Proof If i = j, p
s
i
i
and p
s
j
j
are relatively prime. By the Lemma above, they are
120 Appendix Chapter 6
comaximal and thus by the Chinese Remainder Theorem, the natural map is a ring
isomorphism (page 108). Since the natural map is also an R-module homomorphism,
it is an R-module isomorphism.
This theorem carries the splitting as far as it can go, as seen by the next exercise.
Exercise Suppose R is a PID, p ∈ R is a prime element, and s ≥ 1. Then the
R-module R/p
s
has no proper submodule which is a summand.
Torsion Submodules This will give a little more perspective to this section.
Definition Suppose M is a module over a domain R. An element m ∈ M is said
to be a torsion element if ∃ r ∈ R with r = 0
¯
and mr = 0
¯
. This is the same as
saying m is dependent. If R = Z, it is the same as saying m has finite order. Denote
by T(M) the set of all torsion elements of M. If T(M) = 0
¯
, we say that M is torsion
free.
Theorem 7 Suppose M is a module over a domain R. Then T(M) is a submodule
of M and M/T(M) is torsion free.
Proof This is a simple exercise.
Theorem 8 Suppose R is a Euclidean domain and M is a finitely generated
R-module which is torsion free. Then M is a free R-module, i.e., M ≈ R
m
.
Proof This follows immediately from Theorem 5.
Theorem 9 Suppose R is a Euclidean domain and M is a finitely generated
R-module. Then the following s.e.s. splits.
0 −→T(M) −→M −→M/T(M) −→0
Proof By Theorem 7, M/T(M) is torsion free. By Theorem 8, M/T(M) is a free
R-module, and thus there is a splitting map. Of course this theorem is transparent
anyway, because Theorem 5 gives a splitting of M into a torsion part and a free part.
Chapter 6 Appendix 121
Note It follows from Theorem 9 that ∃ a free submodule V of M such that T(M)⊕
V = M. The first summand T(M) is unique, but the complementary summand V is
not unique. V depends upon the splitting map and is unique only up to isomorphism.
To complete this section, here are two more theorems that follow from the work
we have done.
Theorem 10 Suppose T is a domain and T
∗
is the multiplicative group of units
of T. If G is a finite subgroup of T
∗
, then G is a cyclic group. Thus if F is a finite
field, the multiplicative group F
∗
is cyclic. Thus if p is a prime, (Z
p
)
∗
is cyclic.
Proof This is a corollary to Theorem 5 with R = Z. The multiplicative group G
is isomorphic to an additive group Z/d
1
⊕Z/d
2
⊕ ⊕ Z/d
t
where each d
i
> 1 and
d
i
[d
i+1
for 1 ≤ i < t. Every u in the additive group has the property that ud
t
= 0
¯
.
So every g ∈ G is a solution to x
dt
− 1
¯
= 0
¯
. If t > 1, the equation will have degree
less than the number of roots, which is impossible. Thus t = 1 and so G is cyclic.
Exercise For which primes p and q is the group of units (Z
p
Z
q
)
∗
a cyclic group?
We know from Exercise 2) on page 59 that an invertible matrix over a field is the
product of elementary matrices. This result also holds for any invertible matrix over
a Euclidean domain.
Theorem 11 Suppose R is a Euclidean domain and A ∈ R
n
is a matrix with
non-zero determinant. Then by elementary row and column operations, A may be
transformed to a diagonal matrix
¸
¸
¸
¸
¸
d
1
0
d
2
.
.
.
0 d
n
¸
where each d
i
= 0
¯
and d
i
[d
i+1
for 1 ≤ i < n. Also d
1
generates the ideal generated
by the entries of A. Furthermore A is invertible iff each d
i
is a unit. Thus if A is
invertible, A is the product of elementary matrices.
122 Appendix Chapter 6
Proof It follows from Theorem 3 that A may be transformed to a diagonal matrix
with d
i
[d
i+1
. Since the determinant of A is not zero, it follows that each d
i
= 0
¯
.
Furthermore, the matrix A is invertible iff the diagonal matrix is invertible, which is
true iff each d
i
is a unit. If each d
i
is a unit, then the diagonal matrix is the product
of elementary matrices of type 1. Therefore if A is invertible, it is the product of
elementary matrices.
Exercise Let R = Z, A =
3 11
0 4
and D =
3 11
1 4
. Perform elementary
operations on A and D to obtain diagonal matrices where the first diagonal element
divides the second diagonal element. Write D as the product of elementary matri-
ces. Find the characteristic polynomials of A and D. Find an elementary matrix B
over Z such that B
−1
AB is diagonal. Find an invertible matrix C in R
2
such that
C
−1
DC is diagonal. Show C cannot be selected in Q
2
.
Jordan Blocks
In this section, we define the two special types of square matrices used in the
Rational and Jordan canonical forms. Note that the Jordan block B(q) is the sum
of a scalar matrix and a nilpotent matrix. A Jordan block displays its eigenvalue
on the diagonal, and is more interesting than the companion matrix C(q). But as
we shall see later, the Rational canonical form will always exist, while the Jordan
canonical form will exist iff the characteristic polynomial factors as the product of
linear polynomials.
Suppose R is a commutative ring, q = a
0
+ a
1
x + + a
n−1
x
n−1
+ x
n
∈ R[x]
is a monic polynomial of degree n ≥ 1, and V is the R[x]-module V = R[x]/q.
V is a torsion module over the ring R[x], but as an R-module, V has a free basis
¦1, x, x
2
, . . . , x
n−1
¦. (See the last part of the last theorem on page 46.) Multipli-
cation by x defines an R-module endomorphism on V , and C(q) will be the ma-
trix of this endomorphism with respect to this basis. Let T : V → V be defined
by T(v) = vx. If h(x) ∈ R[x], h(T) is the R-module homomorphism given by
multiplication by h(x). The homomorphism from R[x]/q to R[x]/q given by
multiplication by h(x), is zero iff h(x) ∈ qR[x]. That is to say q(T) = a
0
I + a
1
T+
+ T
n
is the zero homomorphism, and h(T) is the zero homomorphism iff
h(x) ∈ qR[x]. All of this is supposed to make the next theorem transparent.
Theorem Let V have the free basis ¦1, x, x
2
, ..., x
n−1
¦. The companion matrix
Chapter 6 Appendix 123
representing T is
C(q) =
The characteristic polynomial of B(q) is q, and [B(q)[ = λ
n
= (−1)
n
a
0
. Finally, if
h(x) ∈ R[x], h(B(q)) is zero iff h(x) ∈ qR[x].
Note For n = 1, C(a
0
+ x) = B(a
0
+x) = (−a
0
). This is the only case where a
block matrix may be the zero matrix.
Note In B(q), if you wish to have the 1
s
above the diagonal, reverse the order of
the basis for V .
Jordan Canonical Form
We are finally ready to prove the Rational and Jordan forms. Using the previous
sections, all that's left to do is to put the pieces together. (For an overview of Jordan
form, read first the section in Chapter 5, page 96.)
124 Appendix Chapter 6
Suppose R is a commutative ring, V is an R-module, and T : V → V is an
R-module homomorphism. Define a scalar multiplication V R[x] → V by
v(a
0
+a
1
x + +a
r
x
r
) = va
0
+T(v)a
1
+ +T
r
(v)a
r
.
Theorem 1 Under this scalar multiplication, V is an R[x]-module.
This is just an observation, but it is one of the great tricks in mathematics.
Questions about the transformation T are transferred to questions about the module
V over the ring R[x]. And in the case R is a field, R[x] is a Euclidean domain and so
we know almost everything about V as an R[x]-module.
Now in this section, we suppose R is a field F, V is a finitely generated F-module,
T : V →V is a linear transformation and V is an F[x]-module with vx = T(v). Our
goal is to select a basis for V such that the matrix representing T is in some simple
form. A submodule of V
F[x]
is a submodule of V
F
which is invariant under T. We
know V
F[x]
is the sum of cyclic modules from Theorems 5 and 6 in the section on
Euclidean Domains. Since V is finitely generated as an F-module, the free part of
this decomposition will be zero. In the section on Jordan Blocks, a basis is selected
for these cyclic modules and the matrix representing T is described. This gives the
Rational Canonical Form and that is all there is to it. If all the eigenvalues for T are
in F, we pick another basis for each of the cyclic modules (see the second theorem in
the section on Jordan Blocks). Then the matrix representing T is called the Jordan
Canonical Form. Now we say all this again with a little more detail.
From Theorem 5 in the section on Euclidean Domains, it follows that
V
F[x]
≈ F[x]/d
1
⊕F[x]/d
2
⊕ ⊕F[x]/d
t
where each d
i
is a monic polynomial of degree ≥ 1, and d
i
[d
i+1
. Pick ¦1, x, x
2
, . . . , x
m−1
¦
as the F-basis for F[x]/d
i
where m is the degree of the polynomial d
i
.
Theorem 2 With respect to this basis, the matrix representing T is
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
C(d
1
)
C(d
2
)
.
.
.
C(d
t
)
¸
Chapter 6 Appendix 125
The characteristic polynomial of T is p = d
1
d
2
d
t
and p(T) = 0
¯
. This is a type
of canonical form but it does not seem to have a name.
Now we apply Theorem 6 to each F[x]/d
i
. This gives V
F[x]
≈ F[x]/p
s
1
1
⊕ ⊕
F[x]/p
sr
r
where the p
i
are irreducible monic polynomials of degree at least 1. The p
i
need not be distinct. Pick an F-basis for each F[x]/p
s
i
i
as before.
Theorem 3 With respect to this basis, the matrix representing T is
¸
¸
¸
¸
¸
¸
¸
¸
C(p
s
1
1
)
C(p
s
2
2
) 0
0
.
.
.
C(p
sr
r
)
¸
The characteristic polynomial of T is p = p
s
1
1
p
sr
r
and p(T) = 0
¯
. This is called
the Rational canonical form for T.
Now suppose the characteristic polynomial of T factors in F[x] as the product of
linear polynomials. Thus in the Theorem above, p
i
= x −λ
i
and
V
F[x]
≈ F[x]/(x −λ
1
)
s
1
⊕ ⊕F[x]/(x −λ
r
)
sr
is an isomorphism of F[x]-modules. Pick ¦1, (x − λ
i
), (x − λ
i
)
2
, . . . , (x − λ
i
)
m−1
¦ as
the F-basis for F[x]/(x −λ
i
)
s
i
where m is s
i
.
Theorem 4 With respect to this basis, the matrix representing T is
126 Appendix Chapter 6
The characteristic polynomial of T is p = (x−λ
1
)
s
1
(x−λ
r
)
sr
and p(T) = 0
¯
. This
is called the Jordan canonical form for T. Note that the λ
i
need not be distinct.
Note A diagonal matrix is in Rational canonical form and in Jordan canonical
form. This is the case where each block is one by one. Of course a diagonal matrix
is about as canonical as you can get. Note also that if a matrix is in Jordan form,
its trace is the sum of the eigenvalues and its determinant is the product of the
eigenvalues. Finally, this section is loosely written, so it is important to use the
transpose principle to write three other versions of the last two theorems.
Exercise Suppose F is a field of characteristic 0 and T ∈ F
n
has trace(T
i
) = 0
¯
for 0 < i ≤ n. Show T is nilpotent. Let p ∈ F[x] be the characteristic polynomial of
T. The polynomial p may not factor into linears in F[x], and thus T may have no
conjugate in F
n
which is in Jordan form. However this exercise can still be worked
using Jordan form. This is based on the fact that there exists a field
¯
F containing F
as a subfield, such that p factors into linears in
¯
F[x]. This fact is not proved in this
book, but it is assumed for this exercise. So ∃ an invertible matrix U ∈
¯
F
n
so that
U
−1
TU is in Jordan form, and of course, T is nilpotent iff U
−1
TU is nilpotent. The
point is that it sufficies to consider the case where T is in Jordan form, and to show
the diagonal elements are all zero.
So suppose T is in Jordan form and trace (T
i
) = 0
¯
for 1 ≤ i ≤ n. Thus trace
(p(T)) = a
0
n where a
0
is the constant term of p(x). We know p(T) = 0
¯
and thus
trace (p(T)) = 0
¯
, and thus a
0
n = 0
¯
. Since the field has characteristic 0, a
0
= 0
¯
and so 0
¯
is an eigenvalue of T. This means that one block of T is a strictly lower
triangular matrix. Removing this block leaves a smaller matrix which still satisfies
the hypothesis, and the result follows by induction on the size of T. This exercise
illustrates the power and facility of Jordan form. It also has a cute corollary.
Corollary Suppose F is a field of characteristic 0, n ≥ 1, and (λ
1
, λ
2
, .., λ
n
) ∈ F
n
satisfies λ
i
1
+λ
i
2
+ +λ
i
n
= 0
¯
for each 1 ≤ i ≤ n. Then λ
i
= 0
¯
for 1 ≤ i ≤ n.
Minimal polynomials To conclude this section here are a few comments on the
minimal polynomial of a linear transformation. This part should be studied only if
you need it. Suppose V is an n-dimensional vector space over a field F and T : V →V
is a linear transformation. As before we make V a module over F[x] with T(v) = vx.
Chapter 6 Appendix 127
Definition Ann(V
F[x]
) is the set of all h ∈ F[x] which annihilate V , i.e., which
satisfy V h = 0
¯
. This is a non-zero ideal of F[x] and is thus generated by a unique
monic polynomial u(x) ∈ F(x), Ann(V
F[x]
) = uF[x]. The polynomial u is called the
minimal polynomial of T. Note that u(T) = 0
¯
and if h(x) ∈ F[x], h(T) = 0
¯
iff
h is a multiple of u in F[x]. If p(x) ∈ F[x] is the characteristic polynomial of T,
p(T) = 0
¯
and thus p is a multiple of u.
Now we state this again in terms of matrices. Suppose A ∈ F
n
is a matrix
representing T. Then u(A) = 0
¯
and if h(x) ∈ F[x], h(A) = 0
¯
iff h is a multiple of
u in F[x]. If p(x) ∈ F[x] is the characteristic polynomial of A, then p(A) = 0
¯
and
thus p is a multiple of u. The polynomial u is also called the minimal polynomial of
A. Note that these properties hold for any matrix representing T, and thus similar
matrices have the same minimal polynomial. If A is given to start with, use the linear
transformation T : F
n
→F
n
determined by A to define the polynomial u.
Now suppose q ∈ F[x] is a monic polynomial and C(q) ∈ F
n
is the compan-
ion matrix defined in the section Jordan Blocks. Whenever q(x) = (x − λ)
n
, let
B(q) ∈ F
n
be the Jordan block matrix also defined in that section. Recall that q is
the characteristic polynomial and the minimal polynomial of each of these matrices.
This together with the rational form and the Jordan form will allow us to understand
the relation of the minimal polynomial to the characteristic polynomial.
Exercise Suppose A
i
∈ F
n
i
has q
i
as its characteristic polynomial and its minimal
polynomial, and A =
¸
¸
¸
¸
¸
A
1
0
A
2
.
.
.
0 A
r
¸
. Find the characteristic polynomial
and the minimal polynomial of A.
Exercise Suppose A ∈ F
n
.
1) Suppose A is the matrix displayed in Theorem 2 above. Find the characteristic
and minimal polynomials of A.
2) Suppose A is the matrix displayed in Theorem 3 above. Find the characteristic
and minimal polynomials of A.
3) Suppose A is the matrix displayed in Theorem 4 above. Find the characteristic
and minimal polynomials of A.
128 Appendix Chapter 6
4) Suppose λ ∈ F. Show λ is a root of the characteristic polynomial of A iff λ
is a root of the minimal polynomial of A. Show that if λ is a root, its order
in the characteristic polynomial is at least as large as its order in the minimal
polynomial.
5) Suppose
¯
F is a field containing F as a subfield. Show that the minimal poly-
nomial of A ∈ F
n
is the same as the minimal polynomial of A considered as a
matrix in
¯
F
n
. (This funny looking exercise is a little delicate.)
6) Let F = R and A = | 677.169 | 1 |
Translating Expressions Bundle
Be sure that you have an application to open
this file type before downloading and/or purchasing.
1 MB|45 pages
Product Description
This is a bundle that includes everything you need to teach translating expressions in Algebra 1. It includes Cornell notes, practices, homework, activities and SOL practice. Included in the set are
- Cornell notes on translating
- practice to go with the lesson
- 2 homework assignments
- SOL multiple choice set
- Activity: BINGO
The lessons are made to last over a 90 minutes block and there is enough material to fill 2 block periods. I use this material with my Algebra 1 double block class.
This material is also available in a bundle with evaluating expressions and simplifying radicals. The activities are available as just a bundle as well. All can be found by visiting my store. | 677.169 | 1 |
11+ Essentials Numerical Reasoning
Maths Worded Problems. Book 2. 11 + Essentials (First Past the Post)
Publisher:
The University of Buckingham Press, Buckingham, United Kingdom
Published:
30th Nov 2013
Dimensions:
w 210mm h 297mm d 4mm
Weight:
222g
ISBN-10:
1908684364
ISBN-13:
9781908684363
Barcode No:
9781908684363
Synopsis
Numerical Reasoning tests are increasingly common in 11 plus and common entrance exams. Packaged as mini-tests, this Book 2 edition in the excellent new 11+ Essentials First Past the Post series, provides further graduated and multi-part worded problems, typically requiring comprehension and mathematical skillsets, relevant to the new CEM exam styles. Questions are typically worded problems requiring comprehension and mathematical manipulation of facts and figures presented. They are designed to measure the ability to interpret, analyse and sometimes draw conclusions from tables and graphs. Problems are often multi-part, requiring several conclusions to be drawn or evaluated from the same initial information, with answers occasionally being contingent on a preceding part answered accurately. Although some mathematical ability is required, the primary element being measured is the ability to reason with the facts. This series of mini-tests is representative of the numerical reasoning section of contemporary multi-discipline 11 plus exams that typically have two papers covering several subjects such as verbal and non-verbal reasoning, comprehensions and numerical reasoning.
Numerical reasoning papers usually have a long multi-part worded numerical reasoning problems section and another containing more conventional short, quick-fire maths questions. This publication complements the Book 1 edition, with questions packaged as mini-tests. They are suitable as practice for independent schools and new, CEM-style 11 plus exams. As with all the First Past the Post editions these are the only test practice materials that provide your child with individual on-line evaluation and access to peer group comparison. These offer ideal preparation for new, CEM-style exams in 11 plus; further graduated Numerical Reasoning questions relevant to new styles; multi-part worded questions, now increasingly common; fully explained answers; a peer-compare on-line access code enclosed; complete interactive evaluation to enhance individual skills; and are suitable also for independent schools testing. Extensively road-tested by ElevenPlusExams, these tests are best used as real exam practice to benchmark your child s performance. The on-line service helps identify areas for improvement and, crucially, gauge peer comparison anonymously.
They are perfect for use both in the classroom and at home | 677.169 | 1 |
Mathematics has spread its influence far beyond the realm of numbers. The concepts and methods of mathematics are crucially important to all of culture and affect the way countless people in all spheres of life look at the world. Consider these cases:
The Trigonometry And Pre-Calculus Tutor is a 5 hour DVD course geared to fully prepare a student to enter university level Calculus. Most students that have trouble with Calculus discover quickly that the root cause of their difficulty is actually that they have never mastered essential material in Trigonometry.
Calculus II is the payoff for mastering Calculus I. This second course in the calculus sequence introduces you to exciting new techniques and applications of one of the most powerful mathematical tools ever invented. Equipped with the skills of Calculus II, you can solve a wide array of problems in the physical, biological, and social sciences, engineering, economics, and other areas. Success at Calculus II also gives you a solid foundation for the further study of mathematics, and it meets the math requirement for many undergraduate majors.
What's the sure road to success in calculus? The answer is simple: Precalculus. Traditionally studied after Algebra II, this mathematical field covers advanced algebra, trigonometry, exponents, logarithms, and much more. These interrelated topics are essential for solving calculus problems, and by themselves are powerful methods for describing the real world, permeating all areas of science and every branch of mathematics. Little wonder, then, that precalculus is a core course in high schools throughout the country and an important review subject in college | 677.169 | 1 |
This text, first published in 1964, is a classic work in numerical linear algebra. Its purpose is to develop the tools and analytical methods needed to evaluate computational techniques for solving systems of linear equations and finding eigenvalues. It is in no sense an introductory book. It assumes a decent background in linear algebra, and then it digs deeply into the analysis of linear systems right from the beginning.
Householder wrote this book in an era when people wrote much of their own software for numerical linear algebra. Householder's work provided guidance and theoretical foundations, but he generally stayed away from any kind of implementation details. Because of the work of Householder and his colleagues, some very good software packages for numerical linear algebra have been developed over the years — so good that software engineers are now strongly encouraged not to write their own code. Householder's name remains prominent, not least because of the "Householder transformation", a reduction of an \(n \times n\) symmetric matrix to tridiagonal form using \(n – 2\) orthogonal transformations. It is also an essential step in the QR algorithm — decomposition of a real matrix into the product of an orthogonal matrix Q and an upper triangular matrix R.
The book begins with elementary matrices, projections, determinantal identities, orthogonal polynomials and the Lanczos algorithm for tridiagonalization. Chapter two develops the idea of matrix norms, a concept that is particularly useful in error analysis. The third chapter, the last of the preliminary work, takes up localization. Localization results pertain to eigenvalues and their presence or absence in regions of the complex plane. Associated with these are separation theorems for the eigenvalues of normal matrices.
The remainder of the book surveys the main methods for solving a linear system of equations, calculating a matrix inverse, and finding eigenvalues and eigenvectors. The emphasis throughout is on exploring the mathematical foundations that underlie these methods and the mathematical relationships between them.
The book has descriptions of methods, proofs, and plenty of exercises. But there is not a single numerical example, and very little of anything else that might be called an example. This makes the subject — one that it is intrinsically numerical — look very rarified.
This would not be a book of choice for learning numerical linear algebra. It gets too quickly into the details and would soon overwhelm someone new to the field. A good alternative might be G. W. Stewart's Introduction to Matrix Computations.
Bill Satzer (wjsatzer@mmm.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics. | 677.169 | 1 |
Compressed Zip File
Be sure that you have an application to open this file type before downloading and/or purchasing. How to unzip files.
1.53 MB | 11 pages
PRODUCT DESCRIPTION
In this playlist, students explore standard HSG.GPE.A.2. Students will learn a new way to define parabolas. They will write the equation of a parabola from its directrix and focus mishandling the p value
• Links to resources for students needing extra practice
• An answer guide with correct answers and DOK levels | 677.169 | 1 |
This ebook is available for the following devices:
iPad
Windows
Mac
Sony Reader
Cool-er Reader
Nook
Kobo Reader
iRiver Story
more
Starting from first principles, this book covers all of the foundational material needed to develop a clear understanding of the Mathematica language, with a practical emphasis on solving problems. Concrete examples throughout the text demonstrate how Mathematica can be used to solve problems in science, engineering, economics/finance, computational linguistics, geoscience, bioinformatics, and a range of other fields. The book will appeal to students, researchers and programmers wishing to further their understanding of Mathematica. Designed to suit users of any ability, it assumes no formal knowledge of programming so it is ideal for self-study. Over 290 exercises are provided to challenge the reader's understanding of the material covered and these provide ample opportunity to practice using the language. Mathematica notebooks containing examples, programs and solutions to exercises are available from less
In the press
' … has been revised from cover to cover, with many organizational changes and a substantial amount of new material. It includes coverage up to and including Mathematica 5.1. While the second edition was very well received, the third edition strikes me as an essential document … I was struck immediately by the lucidity of the prose …This is not the first book to be written in Mathematica, but it is far and away the most beautiful. From page layout to production quality, the work is on par with any technical text produced by any publisher.' Bruce Torrence, The Mathematica Journal | 677.169 | 1 |
Overview. Formal mathematical
writing is the primary means by which mathematical knowledge is
organized, shared, and preserved. New discoveries are often
initially presented orally at conferences and in small groups, but
without a formal written record, this knowledge will quickly be
forgotten. A major goal of this course is to give students
practice
with mathematical writing, which involves some special notational and
formatting conventions. For each quiz problem you are asked
to prepare a careful formal solution following the
guidelines on this page. This sort of writing will be referred to
as Polished Work. The
notation
and formatting conventions should already be familiar: they are almost
always followed in mathematics textbooks. Organization. Each
polished
problem should begin on its own page. The left margin should be
at least 3 inches wide for me to write comments. Also leave white
space between paragraphs for the same purpose. Each polished
problem should begin with a
complete statement of the problem (which will often be a proposition to
prove), as well as a problem and section number (or something similar
for extra problems that do not come from the text). You
may wish
to
organize your work by proving lemmas that you can cite in the main
proof.
In this case, each lemma should have a clear statement and proof
separate
from the main question.
Solutions may
either be handwritten or prepared with word processing software.
Word processing has some advantages: revisions and corrections are
easier to produce, and the finished product is easier to read.
But if you are inexperienced using word processing software for
mathematical writing, a hand written approach may be faster, at least
initially. Each student is asked to use software for at least
some of the work in his or her portfolio, so that one outcome of the
course will be a familiarity with this approach to mathematical
writing. More information about word processing software appears
below.
Format.
Mathematical writing follows the usual rules of grammar, including the
use of complete sentences, organization into paragraphs, correct
punctuation and capitalization, etc. In addition, there are a few
conventions that are specific to mathematics:
(Using word processing software) Variables should
be italicized, or for vectors,
set in bold typeface;
Mathematical equations and inequalities may be
included in symbolic form (although, when read aloud, they should make
sense in the context of the surrounding material);
Equations, inequalities, and expressions may
either appear in-line within the surrounding text, or may be displayed
on separate lines. Displayed lines should be centered on the page, and
may be numbered for reference. Follow the format of the textbook
for this.
All writing should appear in either normal
paragraph formatting or centered displayed lines of mathematical
symbols, but not a combination. Do not introduce unusual
indentation schemes.
Feel free to include tables or figures if
appropriate. These can be formatted as on pages 6 and 100 of the
text, with a label and/or caption. This is usefule when one
problem solution includes more
than one table or figure, and you want a way to refer to a specific
figure. But sometimes no label or caption is needed, and you can
refer to the the figure below or above without any confusion.
Do not use mathematical symbols as
shorthand. For example, do not insert a ∃ in a sentence
to mean there exists and do
not use arrows as a substitute for words. Logical symbols
are
generally only permitted as part of symbolic portrayals of formal
logical propositions. The logical symbols
for element
of and subset of
are permitted within
running text though as a general rule, such
mathematical symbols
should appear only as part of a larger complete mathematical
statement.
Thus, it is permitted to write "Suppose n ∊ ℤ" but not "Suppose n ∊ the integers." The 'word' iff is a permitted contraction of if and only if.
Word Processing
Software.The industry standarrd for writing in
mathematics and several other technical fields is LaTeX, and its
variants. Two share-ware packages for LaTeX are Lyx and Miktex,
and there may be others.
Although I am pretty familiar with LaTeX
, I know very little about these particular packages. Learning to
use some version of LaTeX is probably worthwhile for math majors, and
you may wish to experiment with this type of system. But it is
not required and should not distract you from the primary objectives of
the course.
Another option is to use MS Word, which includes an equation
editor for formatting complicated mathematical expressions.
In my 2010 version of Word, I start the equation editor by selecting an
option from the insert menu, as shown below.
Something similar should work for newer
versions of word. Once you have the equation editor open, with a
little
experimentation you will see how to create mathematical
expressions. Feel free to ask me for assistance with
this. If you use word, you can manually italicize variables
and use font properties to create subscripts and exponents that appear in the running text.
For anything more complicated than that, you should use the equation
editor, either in-line or centered on a separate line. | 677.169 | 1 |
This is a complete guide to statistics and SPSS for social science students. Statistics with SPSS for Social Science provides a step-by-step explanation of all the important statistical concepts, tests and procedures.
By adding pay-per-click (PPC) campaigns to your marketing mix, you can more effectively connect your products and services with potential customers who are browsing the web in search of what you're offering.
This introduction has been designed to teach Mathematica as a programming language to scientists, engineers, mathematicians and computer scientists. The text may be used in a first or second course on programming at the undergraduate level or in a Mathematica-related course in engineering, mathematics or the sciences. | 677.169 | 1 |
Writing logs in exponential form
Cheat Sheets & Tables Algebra, Trigonometry and Calculus cheat sheets and a variety of tables. Class Notes Each class has notes available. Most of the classes have. 8 0. Complex numbers, power series, and exponentials A complex number has the form (0.1) z = x + iy; where x and y are real numbers. These numbers make up the. Make any video your lesson.
Free Algebra 2 worksheets created with Infinite Algebra 2. Printable in convenient PDF format. Many applications involve using an exponential expression with a base of e. Applications of exponential growth and decay as well as interest that is. In mathematics, the logarithm is the inverse operation to exponentiation. That means the logarithm of a number is the exponent to which another fixed number, the base. Using and Deriving Algebraic Properties of Logarithms miscellaneous on-line topics for Calculus Applied to the Real World : Return to Main Page. Research-based recommendations for effective instruction in 21st-century literacies For teachers. Research shows that effective instruction in 21st-century literacies.
Writing logs in exponential form
- Elementary Arithmetic - High School Math - College Algebra - Trigonometry - Geometry - Calculus. But let's start at the beginning and work our way up through the. Welcome to Babylon Floral Design, Denver's most unique flower boutique, specializing in cutting edge floral design and unique gift items. We strive to provide the. Majorgolflesson.com is the official site of Torrey Pines PGA teaching pro Michael Major. : Social norms have traditionally been difficult to quantify. In any particular society, their sheer number and complex interdependencies often limit a system-level.
A lesson on what logs are and how they can be changed from logarithmic form to exponential form and back from Getting Started. USATestprep is very user-friendly! Students and teachers can use the site effectively from the first day of purchase. We also offer a live online. Be careful with negative exponents. The temptation is to negate the base, which would not be a correct thing to do.
"We don't understand how a single star forms, yet we want to understand how 10 billion stars form." –Carlos Frenck When we look out into the. Check out SVP member Evelia Coyotzi profiled in this great video about why we need to #liftthecaps and get the City to issue more vending permits Thanks to Nowhere. Flume is a distributed, reliable, and available service for efficiently moving large amounts of data soon after the data is produced. This release provides a scalable. The Singularity is an era in which our intelligence will become increasingly nonbiological and trillions of times more why the Common Core is important for your child. What parents should know; Myths vs. facts.
Interpreting the rate of change of exponential models (Algebra 2 level) Constructing exponential models according to rate of change (Algebra 2 level) Advanced. Paris in the Fall: Chef Q Inspires with New Cookbook. New cookbook by Chef Q for your recipe collection offers inspiration and healthy, tasty meals. The history of hypnosis is full of contradictions. On the one hand, a history of hypnosis is a bit like a history of breathing. Like breathing, hypnosis is an. Logarithmic functions and exponential functions are connected to one another in that they are inverses of each other. You may recall that when two functions are. Pearson Prentice Hall and our other respected imprints provide educational materials, technologies, assessments and related services across the secondary curriculum. | 677.169 | 1 |
Synopses & Reviews
Publisher Comments
Differential Equations: An Introduction to Modern Methods and Applications is a textbook designed for a first course in differential equations commonly taken by undergraduates majoring in engineering or science. It emphasizes a systems approach to the subject and integrates the use of modern computing technology in the context of contemporary applications from engineering and science. Section exercises throughout the text are designed to give students hands-on experience in modeling, analysis, and computer experimentation. Optional projects at the end of each chapter provide additional opportunitites for students to explore the role played by differential equations in scientific and engineering problems of a more serious nature.
Synopsis
Written by one of the most well known names in mathematics, this book provides readers with a more modern approach to differential equations. It is streamlined for easier readability while incorporating the latest topics and technologies. The modeling- and technology-intensive format allows readers who may normally struggle with learning the subject to feel confident. It also incorporates numerous exercises that have been developed and tested over decades. | 677.169 | 1 |
The concepts included are limits, derivatives, antiderivatives and definite integrals. These concepts will be applied to solve problems of rates of change, maximum and minimum, curve sketching and areas. The classes of functions used to develop these concepts and applications are: polynomial, rational, trigonometric, exponential and logarithmic.
An introduction to applications of algebra to business, the behavioural sciences, and the social sciences. Topics will be chosen from set theory, permutations and combinations, binomial theorem, probability theory, systems of linear equations, vectors and matrices, mathematical induction. [Offered: F,W,S]
Prereq: Open only to students in the following Faculties: ARTS, AHS, ES.
An introduction to applications of calculus in business, the behavioural sciences, and the social sciences. The models studied will involve polynomial, rational, exponential and logarithmic functions. The major concepts introduced to solve problems are rate of change, optimization, growth and decay, and integration. [Offered: F]
Prereq: Open only to students in the following Faculties: ARTS, AHS, ES.
Functions of a real variable: powers, rational functions, trigonometric, exponential and logarithmic functions, their properties and inverses. Intuitive discussion of limits and continuity. Definition and interpretation of the derivative, derivatives of elementary functions, derivative rules and applications. Riemann sums and other approximations to the definite integral. Fundamental Theorems and antiderivatives; change of variable. Applications to area, rates, average value. [Offered: F,W,S]
Prereq: OAC Calculus or 4U Advanced Functions and Introductory Calculus; Not open to Honours Mathematics students.
Rational, trigonometric, exponential, and power functions of a real variable; composites and inverses. Absolute values and inequalities. Limits and continuity. Derivatives and the linear approximation. Applications of the derivative, including curve sketching, optimization, related rates, and Newton's method. The Mean Value Theorem and error bounds. Introduction to the Riemann Integral and approximations. Antiderivatives and the Fundamental Theorem. Change of variable, areas and rate integrals. Suitable topics are illustrated using computer software.
[Note: Offered at St. Jerome's University in the Fall term. Offered: F,W,S]
First order equations, second order linear equations with constant coefficients, series solutions, the Laplace transform method, systems of linear differential equations. Applications in engineering are emphasized. [Offered: F,W,S]
The Undergraduate Calendar is published by the Office of the Registrar, University of Waterloo,
Waterloo, ON N2L 3G1 Canada
Contact Information: Need academic advisement help? If so, please direct your inquiry to the appropriate
Undergraduate Faculty Advisor by visiting the
Undergraduate Faculty Advisors page on the Registrar's Office website for contact information.
If you are reporting technical problems and broken links in the calendar, send an email to roucal@uwaterloo.ca.
All other inquiries may be directed to: registrar@uwaterloo.ca. | 677.169 | 1 |
Chapter 6 More on Matrices
Similar presentations
Presentation on theme: "Chapter 6 More on Matrices"— Presentation transcript:
1Chapter 6 More on Matrices Fletcher DunnValve SoftwareIan ParberryUniversity of North Texas3D Math Primer for Graphics & Game Development
2What You'll See in This Chapter This chapter completes our coverage of matrices by discussing a few more interesting and useful matrix operations. It is divided into five sections.Section 6.1 covers the determinant of a matrix.Section 6.2 covers the inverse of a matrix.Section 6.3 discusses orthogonal matrices.Section 6.4 introduces homogeneous vectors and 4×4 matrices, and shows how they can be used to perform affine transformations in 3D.Section 6.5 discusses perspective projection and shows how to do it with a 4×4 matrix.Chapter 6 Notes3D Math Primer for Graphics & Game Dev
93D Math Primer for Graphics & Game Dev Triple ProductIf we interpret the rows of a 3x3 matrix as three vectors, then the determinant of the matrix is equivalent to the so-called triple product of the three vectors:Chapter 6 Notes3D Math Primer for Graphics & Game Dev
103D Math Primer for Graphics & Game Dev MinorsLet M be an r x c matrix.Consider the matrix obtained by deleting row i and column j from M.This matrix will obviously be r-1 x c-1.The determinant of this submatrix, denoted M{ij} is known as a minor of M.For example, the minor M{12} is the determinant of the 2 x 2 matrix that is the result of deleting the 1st row and 2nd column from M:Chapter 6 Notes3D Math Primer for Graphics & Game Dev
113D Math Primer for Graphics & Game Dev CofactorsThe cofactor of a square matrix M at a given row and column is the same as the corresponding minor, only every alternating minor is negated.We will use the notation C{12} to denote the cofactor of M in row i, column j.Use (-1)(i+j) term to negate alternating minors.Chapter 6 Notes3D Math Primer for Graphics & Game Dev
133D Math Primer for Graphics & Game Dev n x n DeterminantThe definition we will now consider expresses a determinant in terms of its cofactors.This definition is recursive, since cofactors are themselves signed determinants.First, we arbitrarily select a row or column from the matrix.Now, for each element in the row or column, we multiply this element by the corresponding cofactor.Summing these products yields the determinant of the matrix.Chapter 6 Notes3D Math Primer for Graphics & Game Dev
18Important Determinant Facts The identity matrix has determinant 1:|I| = 1.The determinant of a matrix product is equal to the product of the determinants:|AB| = |A||B|.This extends to multiple matrices:|M1M2…Mn-1 Mn| = |M1||M2|… |Mn-1||Mn|.The determinant of the transpose of a matrix is equal to the original.|MT| = |M|.Chapter 6 Notes3D Math Primer for Graphics & Game Dev
193D Math Primer for Graphics & Game Dev Zero Row or ColumnIf any row of column in a matrix contains all zeros, then the determinant of that matrix is zero.Chapter 6 Notes3D Math Primer for Graphics & Game Dev
20Exchanging Rows or Columns Exchanging any pair of rows or columns negates the determinant.Chapter 6 Notes3D Math Primer for Graphics & Game Dev
21Adding a Multiple of a Row or Column Adding any multiple of a row (or column) to another row (or column) does not change the value of the determinant.This explains why shear matrices from Chapter 5 have a determinant of 1.Chapter 6 Notes3D Math Primer for Graphics & Game Dev
23Geometric Interpretation In 2D, the determinant is equal to the signed area of the parallelogram or skew box that has the basis vectors as two sides.By signed area, we mean that the area is negative if the skew box is flipped relative to its original orientation.Chapter 6 Notes3D Math Primer for Graphics & Game Dev
253 x 3 Determinant as Volume In 3D, the determinant is the volume of the parallelepiped that has the transformed basis vectors as three edges.It will be negative if the object is reflected (turned inside out) as a result of the transformation.Chapter 6 Notes3D Math Primer for Graphics & Game Dev
26Uses of the Determinant The determinant is related to the change in size that results from transforming by the matrix.The absolute value of the determinant is related to the change in area (in 2D) or volume (in 3D) that will occur as a result of transforming an object by the matrix.The determinant of the matrix can also be used to help classify the type of transformation represented by a matrix.If the determinant of a matrix is zero, then the matrix contains a projection.If the determinant of a matrix is negative, then the matrix contains a reflection.Chapter 6 Notes3D Math Primer for Graphics & Game Dev
283D Math Primer for Graphics & Game Dev Inverse of a MatrixThe inverse of a square matrix M, denoted M-1 is the matrix such that when we multiply by M-1, the result is the identity matrix.M M-1 = M-1M = I.Not all matrices have an inverse.An obvious example is a matrix with a row or column of zeros: no matter what you multiply this matrix by, you will end up with the zero matrix.If a matrix has an inverse, it is said to be invertible or non-singular. A matrix that does not have an inverse is said to be non-invertible or singular.Chapter 6 Notes3D Math Primer for Graphics & Game Dev
29Invertibility and Linear Independence For any invertible matrix M, the vector equality vM = 0 is true only when v = 0.Furthermore, the rows of an invertible matrix are linearly independent, as are the columns.The rows and columns of a singular matrix are linearly dependent.Chapter 6 Notes3D Math Primer for Graphics & Game Dev
30Determinant and Invertibility The determinant of a singular matrix is zero and the determinant of a non-singular matrix is non-zero.Checking the magnitude of the determinant is the most commonly used test for invertibility, because it's the easiest and quickest.Chapter 6 Notes3D Math Primer for Graphics & Game Dev
313D Math Primer for Graphics & Game Dev The Classical AdjointOur method for computing the inverse of a matrix is based on the classical adjoint.The classical adjoint of a matrix M, denoted adj M, is defined to be the transpose of the matrix of cofactors of M.For example, let:Chapter 6 Notes3D Math Primer for Graphics & Game Dev
333D Math Primer for Graphics & Game Dev Classical Adjoint of MThe classical adjoint of M is the transpose of the matrix of cofactors:Chapter 6 Notes3D Math Primer for Graphics & Game Dev
343D Math Primer for Graphics & Game Dev Back to the InverseThe inverse of a matrix is its classical adjoint divided by its determinant:M-1 = adj M / |M|.If the determinant is zero, the division is undefined, which jives with our earlier statement that matrices with a zero determinant are non-invertible.Chapter 6 Notes3D Math Primer for Graphics & Game Dev
363D Math Primer for Graphics & Game Dev Gaussian EliminationThere are other techniques that can be used to compute the inverse of a matrix, such as Gaussian elimination.Many linear algebra textbooks incorrectly assert that such techniques are better suited for implementation on a computer because they require fewer arithmetic operations.This is true for large matrices, or for matrices with a structure that can be exploited.However, for arbitrary matrices of smaller order like the 2 x 2, 3 x 3, and 4 x 4 used most often in geometric applications, the classical adjoint method is faster.The reason is that the classical adjoint method provides for a branchless implementation, meaning there are no if statements or loops that cannot be unrolled statically.This is a big win on today's superscalar architectures and vector processors.Chapter 6 Notes3D Math Primer for Graphics & Game Dev
37Facts About Matrix Inverse The inverse of the inverse of a matrix is the original matrix. If M is nonsingular, (M-1)-1 = M.The identity matrix is its own inverse: I-1 = I.Note that there are other matrices that are their own inverse, such as any reflection matrix, or a matrix that rotates 180° about any axis.The inverse of the transpose of a matrix is the transpose of the inverse: (MT)-1 = (M-1)TChapter 6 Notes3D Math Primer for Graphics & Game Dev
38More Facts About Matrix Inverse The inverse of a product is equal to the product of the inverses in reverse order.(AB)-1 = B-1A-1This extends to more than two matrices:(M1M2…Mn-1 Mn)-1 = Mn-1Mn-1-1…. M2-1M1-1The determinant of the inverse is the inverse of the determinant: |M-1| = 1/|M|.Chapter 6 Notes3D Math Primer for Graphics & Game Dev
39Geometric Interpretation of Inverse The inverse of a matrix is useful geometrically because it allows us to compute the reverse or opposite of a transformation – a transformation that undoes another transformation if they are performed in sequence.So, if we take a vector v, transform it by a matrix M, and then transform it by the inverse M-1 of M, then we will get v back.We can easily verify this algebraically:(vM)M-1 = v(MM-1) = vI = vChapter 6 Notes3D Math Primer for Graphics & Game Dev
413D Math Primer for Graphics & Game Dev Orthogonal MatricesA square matrix M is orthogonal if and only if the product of the matrix and its transpose is the identity matrix: MMT = I.If a matrix is orthogonal, its transpose and the inverse are equal: MT = M-1.If we know that our matrix is orthogonal, we can essentially avoid computing the inverse, which is a relatively costly computation.For example, rotation and reflection matrices are orthogonal.Chapter 6 Notes3D Math Primer for Graphics & Game Dev
459 Equations Using Dot Product Now we can re-write the 9 equations more compactly:Chapter 6 Notes3D Math Primer for Graphics & Game Dev
463D Math Primer for Graphics & Game Dev Two ObservationsFirst, the dot product of a vector with itself is 1 if and only if the vector is a unit vector.Therefore, the equations with a 1 on the right hand side of the equals sign will only be true when r1, r2, and r3 are unit vectors.Second, the dot product of two vectors is 0 if and only if they are perpendicular.Therefore, the other six equations (with 0 on the right hand side of the equals sign) are true when r1, r2, and r3 are mutually perpendicular.Chapter 6 Notes3D Math Primer for Graphics & Game Dev
473D Math Primer for Graphics & Game Dev ConclusionSo, for a matrix to be orthogonal, the following must be true:Each row of the matrix must be a unit vector.The rows of the matrix must be mutually perpendicular.Similar statements can be made regarding the columns of the matrix, since if M is orthogonal, then MT must be orthogonal.Chapter 6 Notes3D Math Primer for Graphics & Game Dev
48Orthonormal Bases Revisited Notice that these criteria are precisely those that we said in Chapter 3 were satisfied by an orthonormal set of basis vectors.There we also noted that an orthonormal basis was particularly useful because we could perform the "opposite" coordinate transform from the one that is always available, using the dot product.When we say that the transpose of an orthogonal matrix is its inverse, we are just restating this fact in the formal language of linear algebra.Chapter 6 Notes3D Math Primer for Graphics & Game Dev
493D Math Primer for Graphics & Game Dev 9 is Actually 6Also notice that 3 of the orthogonality equations are duplicates (since dot product is commutative), and between these 9 equations, we actually have 6 constraints, leaving 3 degrees of freedom.This is interesting, since 3 is the number of degrees of freedom inherent in a rotation matrix.But again note that rotation matrices cannot compute a reflection, so there is slightly more freedom in the set of orthogonal matrices than in the set of orientations in 3D.Chapter 6 Notes3D Math Primer for Graphics & Game Dev
503D Math Primer for Graphics & Game Dev CaveatsWhen computing a matrix inverse we will usually only take advantage of orthogonality if we know a priori that a matrix is orthogonal.If we don't know in advance, it's probably a waste of time checking.Finally, even matrices which are orthogonal in the abstract may not be exactly orthogonal when represented in floating point, and so we must use tolerances, which have to be tuned.Chapter 6 Notes3D Math Primer for Graphics & Game Dev
513D Math Primer for Graphics & Game Dev A Note on TerminologyIn linear algebra, we described a set of basis vectors as orthogonal if they are mutually perpendicular.It is not required that they have unit length. If they do have unit length, they are an orthonormal basis.Thus the rows and columns of an orthogonal matrix are orthonormal basis vectors.However, constructing a matrix from a set of orthogonal basis vectors does not necessarily result in an orthogonal matrix (unless the basis vectors are also orthonormal).Chapter 6 Notes3D Math Primer for Graphics & Game Dev
52Scary Monsters (Matrix Creep) Recall that that rotation matrices (and products of them) are orthogonal.Recall that the rows of an orthogonal matrix form an orthonormal basis.Or at least, that's the way we'd like them to be.But the world is not perfect. Floating point numbers are subject to numerical instability.Aka "matrix creep" (apologies to David Bowie)We need to orthogonalize the matrix, resulting in a matrix that has mutually perpendicular unit vector axes and is (hopefully) as close to the original matrix as possible.Chapter 6 Notes3D Math Primer for Graphics & Game Dev
53Gramm-Schmidt Orthogonalization Here's how to control matrix creep.Go through the rows of the matrix in order.For each, subtract off the component that is parallel to the other rows.More details: let r1, r2, r3 be the rows of a 3 x 3 matrix M.Remember, you can also think of these as the x-, y-, and z-axes of a coordinate space.Then an orthogonal set of row vectors, r1, r2, r3 can be computed as follows:Chapter 6 Notes3D Math Primer for Graphics & Game Dev
583D Math Primer for Graphics & Game Dev BiasThis is biased towards r1, meaning that r1 doesn't change but the other basis vectors do change.Option: instead of subtracting off the whole amount, subtract off a fraction of the original axis.Let k be a fraction – say 1/4Chapter 6 Notes3D Math Primer for Graphics & Game Dev
61Homogenous Coordinates Extend 3D into 4D.The 4th dimension is not "time".The 4th dimension is really just a kluge to help the math work out (later in this lecture).The 4th dimension is called w.Chapter 6 Notes3D Math Primer for Graphics & Game Dev
62Extending 1D into Homogenous Space Start with 1D, its easier to visualize than 3D.Homogenous 1D coords are of the form (x, w).Imagine the vanilla 1D line lying at w = 1.So the 1D point x has homogenous coords (x, 1).Given a homogenous point (x, w), the corresponding 1D point is its projection onto the line w = 1 along a line to the origin, which turns out to be (x/w, 1).Chapter 6 Notes3D Math Primer for Graphics & Game Dev
63Projecting Onto 1D Space Each point x in 1D space corresponds to an infinite number of points in homogenous space, those on the line from the origin through the point (x, 1).The homogenous points on this line project onto its intersection with the line w = 1.Chapter 6 Notes3D Math Primer for Graphics & Game Dev
64What are the 2D Coords of Homogenous Point (p,q)? Chapter 6 Notes3D Math Primer for Graphics & Game Dev
68Projecting Onto 2D Space Each point (x, y) in 2D space corresponds to an infinite number of points in homogenous space.Those on the line from the origin thru (x, y, 1).The homogenous points on this line project onto its intersection with the plane w = 1.Chapter 6 Notes3D Math Primer for Graphics & Game Dev
702D Homogenous Coordinates Just like before (argument omitted), the homogenous point (x, y, w) corresponds to the 2D point (x/w, y/w, 1).That is, the 2D equivalent of the homogenous point (p, q, r) is (p/r, q/r).Chapter 6 Notes3D Math Primer for Graphics & Game Dev
723D Math Primer for Graphics & Game Dev Point at Infinityw can be any value except 0 (divide by zero error).The point (x,y,z,0) can be viewed as a "point at infinity"Chapter 6 Notes3D Math Primer for Graphics & Game Dev
73Why Use Homogenous Space? It will let us handle translation with a matrix transformation.Embed 3D space into homogenous space by basically ignoring the w component.Vector (x, y, z) gets replaced by (x, y, z, 1).Does that "1" at the end sound familiar?Chapter 6 Notes3D Math Primer for Graphics & Game Dev
743D Math Primer for Graphics & Game Dev Homogenous MatricesEmbed 3D transformation matrix into 4D matrix by using the identity in the w row and column.Chapter 6 Notes3D Math Primer for Graphics & Game Dev
78Translation vs. Orientation Just like in 3D, compose 4D operations by multiplying the corresponding matrices.The translation and orientation parts of a composite matrix are independent.For example, let R be a rotation matrix and T be a translation matrix.What does M = RT look like?Chapter 6 Notes3D Math Primer for Graphics & Game Dev
803D Math Primer for Graphics & Game Dev Rotate then TranslateThen we could rotate and then translate a point v to a new point v using v = vRT.We are rotating first and then translating.The order of transformations is important, and since we use row vectors, the order of transformations coincides with the order that the matrices are multiplied, from left to right.Chapter 6 Notes3D Math Primer for Graphics & Game Dev
833D Math Primer for Graphics & Game Dev In ReverseApplying this information in reverse, we can take a 4 x 4 matrix M and separate it into a linear transformation portion, and a translation portion.We can express this succinctly by letting the translation vector t = [Δx, Δy, Δz].Chapter 6 Notes3D Math Primer for Graphics & Game Dev
84Points at Infinity Again Points at infinity are actually useful.They orient just like points with w 0: multiply by the orientation matrix.But they don't translate: translation matrices have no effect on them.Chapter 6 Notes3D Math Primer for Graphics & Game Dev
873D Math Primer for Graphics & Game Dev So What?The translation part of 4D homogenous transformation matrices has no effect on points at infinity.Use points at infinity for things that don't need translating (eg. Surface normals).Use regular points (with w = 1) for things that do need translating (eg. Points that make up game objects).Chapter 6 Notes3D Math Primer for Graphics & Game Dev
883D Math Primer for Graphics & Game Dev 4x3 MatricesThe last column of 4D homogenous transformation matrices is always [0, 0, 0, 1]T.Technically it always needs to be there for the algebra to work out.But we know what it's going to do, so there's no reason to implement it in code.Chapter 6 Notes3D Math Primer for Graphics & Game Dev
89General Affine Transformations Armed with 4 x 4 transform matrices, we can now create more general affine transformations that contain translation. For example:Rotation about an axis that does not pass through the originScale about a plane that does not pass through the originReflection about a plane that does not pass through the originOrthographic projection onto a plane that does not pass through the originChapter 6 Notes3D Math Primer for Graphics & Game Dev
90General Affine Transformations The basic idea is to translate the center of the transformation to the origin, perform the linear transformation using the techniques developed in Chapter 5, and then transform the center back to its original location.We will start with a translation matrix T that translates the point p to the origin, and a linear transform matrix R from Chapter 5 that performs the linear transformation.The final transformation matrix A will be the equal to the matrix product TRT-1.T-1 is of course the translation matrix with the opposite translation amount as T.Chapter 6 Notes3D Math Primer for Graphics & Game Dev
923D Math Primer for Graphics & Game Dev ObservationThus, the extra translation in an affine transformation only changes the last row of the 4 x 4 matrix.The upper 3 x 3 portion, which contains the linear transformation, is not affected.Our use of homogenous coordinates so far has really been nothing more than a mathematical kludge to allow us to include translation in our transformations.Chapter 6 Notes3D Math Primer for Graphics & Game Dev
943D Math Primer for Graphics & Game Dev ProjectionsWe've only used w = 1 and w = 0 so far.There's a use for the other values of w too.We've seen how to do orthographic projection before.Now we'll see how to do perspective projection too.Chapter 6 Notes3D Math Primer for Graphics & Game Dev
95Orthographic Projection Orthographic projection has parallel projectors.The projected image is the same size no matter how far the object is from the projection plane.We want objects to get smaller with distance.This is known as perspective foreshortening.Chapter 6 Notes3D Math Primer for Graphics & Game Dev
993D Math Primer for Graphics & Game Dev The Pinhole CameraThe math is based on a pinhole camera.Take a closed box that's very dark inside.Make a pinhole.If you point the pinhole at something bright, an image of the object will be projected onto the back of the box.That's kind of how the human eye works too.Chapter 6 Notes3D Math Primer for Graphics & Game Dev
1013D Math Primer for Graphics & Game Dev Projection GeometryLet's project on a plane parallel to the x-y plane.Choose a distance d from the pinhole to the projection plane, called the focal distance.The pinhole is called the focal point.Put the focal point at the origin and the projection plane at z = -d.(Remember the concept of camera space?)Chapter 6 Notes3D Math Primer for Graphics & Game Dev
1033D Math Primer for Graphics & Game Dev Do the MathView it from the side.Consider where a point p gets projected onto the plane – at a point pStart with the y coordinate for now.Chapter 6 Notes3D Math Primer for Graphics & Game Dev
1123D Math Primer for Graphics & Game Dev NotesMultiplication by this matrix doesn't actually perform the perspective transform, it just computes the proper denominator into w. The perspective division actually occurs when we convert from 4D to 3D by dividing by w.There are many variations. For example, we can place the plane of projection at z = 0, and the center of projection at [0, 0, -d]. This results in a slightly different equation.Chapter 6 Notes3D Math Primer for Graphics & Game Dev
113This Seems Overly Complicated. It seems like it would be simpler to just divide by z, rather than bothering with matrices.So why is homogenous space interesting?4 x 4 matrices provide a way to express projection as a transformation that can be concatenated with other transformations.Projection onto non-axially aligned planes is possible.Basically, we don't need homogenous coordinates , but 4 x 4 matrices provide a compact way to represent and manipulate projection transformations.Chapter 6 Notes3D Math Primer for Graphics & Game Dev
114Real Projection Matrices The projection matrix in a real graphics geometry pipeline (perhaps more accurately known as the clip matrix) does more than just copy z into w. It will differ from the one we derived in two important respects:Most graphics systems apply a normalizing scale factor such that w = 1 at the far clip plane. This ensures that the values used for depth buffering are distributed appropriately for the scene being rendered, in order to maximize precision of depth buffering.The projection matrix in most graphics systems also scales the x and y values according to the field of view of the camera.We'll get into these details in Chapter 10, when we show what a projection matrix looks like in practice in DirectX and OpenGL.Chapter 6 Notes3D Math Primer for Graphics & Game Dev | 677.169 | 1 |
7 Best Algebra Textbooks | December 2016Prentice Hall Mathematics' Algebra 1 First Edition textbook promises to provide comprehensive content coverage, introduce concepts and skills, and provide opportunities to practice and reinforce what the student has learned. With the support of a teacher, it succeeds.
The cover image of the textbook McDougal Littell Algebra 1, which features a kayaker hurtling down a waterfall, makes an apt metaphor for the book, which dives down into the core concepts of algebra and reinforces the learning with scores of great practice problems.
It will come as no surprise that the textbook Algebra for College Students, Eighth Edition, takes its user on a path through the more advanced levels of algebraic studies, though it also covers the basic building blocks of the discipline, such as variables and functions.
The Algebra 1 Common Core Student Edition textbook is rapidly catching on as a must-have teaching tool for countless classrooms around America as this new set of standards becomes more widely accepted. It covers crucial topics, like linear function, probability, and more.
Former middle school and high school teacher Josh Rappaport's engaging and enjoyable Algebra Survival Guide Workbook comes packed with "Thousands of Problems To Sharpen Skills and Enhance Understanding" and will help you internalize the concepts underlying algebra.
Algebra: Structure and Method is part of a series of textbooks that build upon each other. Students will start with the basics of algebraic math and work their way through concepts, including factoring and variables, slowly gaining the info needed for advanced studies.
The author of Null Pearson Education's textbook Algebra and Trigonometry, 4th Edition, Bob Blitzer, helps teach math "with vivid applications that use math to solve real-life problems," so a student can truly understand both what they are learning and why.
Buyer's Guide
A Lesson In Mathematics
Few subjects in school send students into a state of cross-eyes confusion more readily than mathematics. Even among students who naturally excel at it, who are somehow more attuned to the principals of the mathematical world, there can be a resistance to the subject.
I've had some good conversations with mathematician friends of mine, as well as a couple current math teachers, about why so many students balk at math more than other classes. I expected most of them to tell me that math is just harder to learn, or that the majority of human brains simply aren't built to wrap themselves around concepts beyond a certain level.
To my surprise, the bulk of the conversations I had shared a common thread. It's not that higher levels of math are necessarily more difficult to learn; most of the people I talked to said that certain higher levels of math are actually harder to teach.
That may seem like a matter of semantics, but take a look at how we physically interact with our world using math. Learning addition, subtraction, multiplication, and division through elementary school, we find ourselves applying what we learn on a near daily basis. Going to the store for snacks, saving up our allowance for a new video game–everything about this mathematical experience is tangible.
It's when we get into the intangibles that things get hairy. Most school textbooks covering algebra or geometry spend 90% of their pages explaining concepts and formulas as dryly as possible, and finish off each chapter with a petty attempt at exemplifying a scenario in which you might actually apply a given lesson.
That's all well and good for the one student in a thousand who goes on to become an architect, but there's no hook in it for the rest of us. When I got to college, after sloughing through years of unintelligible, inapplicable math textbooks, I took a practical physics class, and I realized what all math text books needed, which is something these algebra books have in varying degrees: a sense of wonder.
The seven books on our list all combine centuries worth of algebraic discovery into a few hundred pages of lessons, quizzes, and examples, but in recent years–likely in response to this sense among math teachers that there was something missing from their books–math texts have been imbued with a greater sense of magic, with the idea that numbers have meanings and implications we can only begin to grasp.
A Method To Your Mathness
As important as I think a sense of mystery is to the education of young mathematical minds, you may completely disagree. And that's fine. Like I said, there are varying degrees of wonder spread through the books on our list, so you can go with the driest among them if that suits your style.
After all, each of us learns in slightly different ways. I know that I need my sense of suspicion and curiosity stimulated for my brain to open up to new information. I suspect that this is true of most brains, though I'm sure some shut down as soon as things get philosophical.
As you read up on the descriptions offered for each of the algebra textbooks on our list, it'd be worth keeping in mind your personal learning style. If you're investigating these books as teaching tools, then a close look at your teaching style will be just as useful.
When I taught English for a year at my old high school before heading to graduate school, I actually preferred the drier texts to the more evocative ones. I found that I flew so far off the handle imaginatively that if I had a textbook that did the same, we'd never get anything done. I used a more conservative packet of materials to ground my teaching insanity.
You might be just the opposite, preferring a text that can elevate your lessons to new heights. Whatever your approach, there's a book on this list that'll fit the mold. Not all of the algebra texts on this list approach the same educational levels, either, so make sure that fit is suited to the height you wish to teach or to attain.
Ages Of Algebra
It's a comfortable thing for westerners to attribute all of the great historical achievements of mankind to Greek and Roman thinkers, but the roots of algebra and other mathematical works reache back much farther than that.
Archeologists and math historians have dated the oldest texts known to man that elucidate algebraic concepts to sometime between 2000 and 1500 BCE. These texts–the Babylonian Plimpton 322 tablet and the Egyptian Rhind papyrus–both put forth models of linear equations.
The work of these ancient mathematicians spread to the critical thinkers of Greek antiquity, whose work then spread to the Persian empire. By the time the Persian mathematicians made their own great strides in the art, they'd already distributed the thought eastward and into India and China. At the height of the Persian empire, western Europe received a new taste of evolved algebra up through Spain.
All this rich history covering centuries of exploration, experimentation, and expression, all so sleepy students could have the luxury of checking out in the middle of math class and thinking, instead, about their crush sitting across the classroom authoritative | 677.169 | 1 |
ISBN 13: 9780026880527
Math Lab 2B
There are 300 Activity Cards divided into 24 sections with 8 to 19 cards in each section. Some of the Activity Cards are Skill Cards, and others are Mixed Practice Cards. Skill Cards These cards contain examples that address a single math concept with three sets of problems. Mixed Practice Cards Most sections have at least two Mixed Practice Cards. The problems on these cards provide mixed practice of skills in that section, as well as practice in a multiple-choice format. The Activity Cards-Skill Cards and Mixed Practice Cards-are correlated to several standardized tests in the Scope and Sequence.
Top Search Results from the AbeBooks Marketplace
Book Description Sra, 1997. Paperback. Book Condition: Used: Like New. Complete set in original box. The books in very good shape and clean, No writing and No highlight marks in it. !!!FAST SHIPPING IN PADDED ENVELOPE!!! FREE TRACKING!!! (Case 9, shelve 1). Bookseller Inventory # 1704190015 | 677.169 | 1 |
This text is designed for courses in technical mathematics with calculus. The book provides students in technology and pre-engineering with the necessary comprehensive mathematics skills required, including practical calculus. Features of this text include: clear explanations supported by detailed and well-illustrated examples; calculator examples that are integrated throughout the text, including calculator screen images to illustrate the step-by-step calculator operations; more than 8400 exercises; and two appendices of instructions for using graphing calculators.
"synopsis" may belong to another edition of this title.
Product Description:
For courses in Technical Mathematics with Calculus. This text provides students in technology and pre-engineering with the necessary comprehensive mathematics skills required including practical calculus. With basic mathematics concepts presented through algebra, trigonometry, analytic geometry and calculus, the text is written in an intuitive manner, with technical applications integrated whenever possible.
From the Inside Flap:
Preface
Technical Mathematics with Calculus provides the necessary comprehensive mathematics skills for students in an engineering technology program that requires a development of practical calculus.
The text presents the following major areas: fundamental concepts and measurement; fundamental algebraic concepts; exponential and logarithmic functions; right-triangle trigonometry, the trigonometric functions, and trigonometric formulas and identities; complex numbers; matrices; polynomial and rational functions; statistics for process control; analytic geometry; differential and integral calculus with applications; partial derivatives and double integrals; series; and differential equations. KEY FEATURES Numerous detailed, illustrated examples Chapter review summaries Chapter review exercises Important formulas and principles are highlighted Abundant two-color illustrations Two-color format that effectively highlights and illustrates important principles Comprehensive development and consistent use of measurement and significant digits throughout the text Instruction using a basic graphing calculator (Appendix C) and an advanced graphing calculator (Appendix D) is developed in the appendices. Calculator examples are integrated throughout the text; graphing calculator may be used as a faculty option. Chapter introductions and chapter objectives More than 8400 exercises Essential geometry is reviewed in Appendix A The metric system is developed in Appendix B StudyWizard CD-ROM that contains additional exercises keyed to each section Companion Website that contains different additional exercises keyed to each section Instructor's Manual with solutions for selected odd-numbered exercises, answers for even-numbered exercises, and sample chapter tests and answers Illustration of Some Key Features
Examples: Since many students learn by example, a large number of detailed and well-illustrated examples are used throughout the text.
Exercises: To reinforce key concepts for students, we have provided a large variety of well-illustrated exercises.
Chapter End Matter: A chapter summary and a chapter review are provided at the end of each chapter to review concept understanding and to help students review for quizzes and examinations.
Calculator Story Boards: Calculator story boards, including screens, are used to show students the sequence of the step-by-step operations.
Illustrations and Boxes are abundantly and effectively used to highlight important principles. TO THE FACULTY
The topics have been arranged with the assistance of faculty who teach in a variety of technical programs. However, we have also allowed for many other compatible arrangements. The topics are presented in an intuitive manner, with technical applications integrated throughout whenever possible. The large number of detailed examples and exercises is a feature that students and faculty alike find essential.
The text is written at a language level and a mathematics level that are cognizant of and beneficial to most students in technical programs. The students are assumed to have a mathematics background that includes one year of high school algebra or its equivalent and some geometry. The introductory chapters are written so that students who are deficient in some topics may also be successful. The material in this book should be completed in three or four semesters or equivalent and serves as a foundation for more advanced work in mathematics. This text is intended for use in Associate Degree programs as well as ABET (Accrediting Board for Engineering Technology) programs and BIT (Bachelor of Industrial Technology) programs.
Chapters 1 and 2 provide the basic skills that are needed early in almost any technical program. Chapters 3 through 8 complete the basic algebraic foundation, and Chapters 9 through 13 include the trigonometry necessary for the technologies. Chapters 14 through 17 include some advanced topics needed for some programs. Chapter 18 addresses the basics of statistics for process control. Chapter 19 (analytic geometry) completes a comprehensive mathematics background needed in many programs; some programs include this chapter at the end of the first year while other programs include this chapter at the beginning of the introductory calculus. Chapters 20 through 22 present intuitive discussions about the limit and develop basic techniques and applications of differentiation. Chapters 23 through 25 develop basic integration concepts, some appropriate applications, and more complicated methods of integration. Chapter 26 presents partial derivatives and double integrals. Chapters 27 through 29 provide an introduction to series and differential equations with technical applications.
We have included Appendix C on the basic graphing calculator and Appendix D on the advanced calculator so that faculty have the option of which, if any, graphing calculator is used in their course. Some graphing calculator uses are integrated into some of the examples in the text.
A companion Instructor's Manual with solutions for selected odd-numbered exercises, answers for even-numbered exercises, and sample chapter tests and answers is available. TO THE STUDENT
Mathematics provides the essential framework for and is the basic language of all the technologies. With this basic understanding of mathematics, you will be able to quickly understand your chosen field of study and then be able to independently pursue your own lifelong education. Without this basic understanding, you will likely struggle and often feel frustrated not only in your mathematics and support sciences courses but also in your technical courses.
Technology and the world of work will continue to change rapidly. Your own working career will likely change several times during your working lifetime. Mathematical, problem-solving, and critical-thinking skills will be crucial as opportunities develop in your own career path in a rapidly changing world. ACKNOWLEDGMENTS
We extend our sincere and special thanks to our reviewers: Joe Jordan, John Tyler Community College (VA); Maureen Kelly, North Essex Community College (MA); Carol A. McVey, Florence-Darlington Technical College (SC); John D. Meese, DeVry Institute of Technology (OH); Kenneth G. Merkel, Ph.D., PE, University of Nebraska-Lincoln; Susan L. Miertschin, University of Houston; and Pat Velicky, Florence-Darlington Technical College (SC). We would also like to express thanks to our Prentice Hall editor, Stephen Helba; to our Prentice Hall associate editor, Michelle Churma; to our production editor, Louise Sette; to Kirsten Kauffman (York Production Services); and to Joyce Ewen for her superb proofing assistance. | 677.169 | 1 |
Details
West Virginia Next Generation Open Educational Resources
Welcome to Math Grade 8
Welcome to Math 8!
ForWelcome to WV Math 8. This course covers the WV Next Generation Standards and the Common Core Standards for grade 8 and will prepare you for your adventure into high school mathematics, specifically Math I or Algebra 1. There are three BIG IDEAS focused on in eighth grade. So what are these big ideas? The first idea involves linear equations. You will create, model, and reason about linear equations and systems of linear equations so that you will be able to use them to solve a variety of real world problems. The next big idea deals with functions. You will learn about functions and their properties and use functions to describe and compare functional situations in real world situations. The final major focus is geometry. You will transform 2-dimensional figures and relate them using congruence and similarity. You will develop an understanding about relationships between angles, and about distance, and its relationship to the Pythagorean Theorem. You will also be able to find volume of cylinders, cones, and spheres. | 677.169 | 1 |
This volume is a republication and expansion of the much-loved Wohascum County Problem Book, published in 1993. The original 130 problems have been retained and supplemented by an additional 78 problems. The puzzles contained within, which are accessible but never routine, have been specially selected for their mathematical appeal, and detailed solutions are provided. The reader will encounter puzzles involving calculus, algebra, discrete mathematics, geometry and number theory, and the volume includes an appendix identifying the prerequisite knowledge for each problem. A second appendix organises the problems by subject matter so that readers can focus their attention on particular types of problems if they wish. This collection will provide enjoyment for seasoned problem solvers and for those who wish to hone their skills.
This book is a celebration of mathematical problem solving at the level of the high school American Invitational Mathematics Examination. There is no other book on the market focused on the AIME. It is intended, in part, as a resource for comprehensive study and practice for the AIME competition for students, teachers, and mentors. After all, serious AIME contenders and competitors should seek a lot of practice in order to succeed. However, this book is also intended for anyone who enjoys solving problems as a recreational pursuit. The AIME contains many problems that have the power to foster enthusiasm for mathematics – the problems are fun, engaging, and addictive. The problems found within these pages can be used by teachers who wish to challenge their students, and they can be used to foster a community of lovers of mathematical problem solving! There are more than 250 fully-solved problems in the book, containing examples from AIME competitions of the 1980's, 1990's, 2000's, and 2010's. In some cases, multiple solutions are presented to highlight variable approaches. To help problem-solvers with the exercises, the author provides two levels of hints to each exercise in the book, one to help stuck starters get an idea how to begin, and another to provide more guidance in navigating an approach to the solution.
This guide covers the story of trigonometry. It is a swift overview, but it is complete in the context of the content discussed in beginning and advanced high-school courses. The purpose of these notes is to supplement and put into perspective the material of any course on the subject you may have taken or are currently taking. (These notes will be tough going for those encountering trigonometry for the very first time!)
This is a challenging problem-solving book in Euclidean geometry, assuming nothing of the reader other than a good deal of courage. Topics covered included cyclic quadrilaterals, power of a point, homothety, triangle centers; along the way the reader will meet such classical gems as the nine-point circle, the Simson line, the symmedian and the mixtilinear incircle, as well as the theorems of Euler, Ceva, Menelaus, and Pascal. Another part is dedicated to the use of complex numbers and barycentric coordinates, granting the reader both a traditional and computational viewpoint of the material. The final part consists of some more advanced topics, such as inversion in the plane, the cross ratio and projective transformations, and the theory of the complete quadrilateral. The exposition is friendly and relaxed, and accompanied by over 300 beautifully drawn figures. The emphasis of this book is placed squarely on the problems. Each chapter contains carefully chosen worked examples, which explain not only the solutions to the problems but also describe in close detail how one would invent the solution to begin with. The text contains as selection of 300 practice problems of varying difficulty from contests around the world, with extensive hints and selected solutions. This book is especially suitable for students preparing for national or international mathematical olympiads, or for teachers looking for a text for an honor class.
Mathematics Galore! Showcases some of the best activities and student outcomes of the St. Mark's Institute of Mathematics and invites you to engage the mathematics yourself! Revel in the delight of deep intellectual play and marvel at the heights to which young scholars can rise. See some great mathematics explained and proved via natural and accessible means.Based on 26 essays ("newsletters") and eight additional pieces, Mathematics Galore! offers a large sample of mathematical tidbits and treasures, each immediately enticing, and each a gateway to layers of surprising depth and conundrum. Pick and read essays in no particular order and enjoy the mathematical stories that unfold. Be inspired for your courses, your math clubs and your math circles, or simply enjoy for yourself the bounty of research questions and intriguing puzzlers that lie within. | 677.169 | 1 |
You are here
Approximation
Perhaps more than any other field of applied mathematics, continuum and fluid mechanics is intertwined with both the concept and practice of approximation. Of course any student of first year calculus learns about Taylor series as approximations to a function, and indeed local approximations by piecewise linear functions are an important part of numerical analysis. However, the idea of approximation from the point of view of continuum mechanics runs much deeper.
In some sense the very idea of simple mathematics-based laws to describe the deformation and flow of matter are an approximation, since matter is made up of countless numbers of molecules. It is hopeless to try to describe the behaviour of all the molecules in a piece of material, and so a simplified description is sought. The two disciplines of physics that bridge the gap between the microscopic and the length scales of the every day world are thermodynamics and statistical mechanics. It is the task of thermodynamics to describe the macroscopic, or en masse, effects of molecular motion, while statistical mechanics seeks to justify thermodynamic laws (such as heat flowing from hot to cold) from a more basic point of view. Notice that I am being careful not to say "more fundamental", since statistical mechanics requires drastic simplifying assumptions to carry out its work.
If we accept this first level of approximation, then we are left with the task of deriving the general laws of flow and deformation. This is the task of continuum mechanics and is taken up in the course AMATH 361. At this point we can take a sentence or two to describe the difficulties one would expect in the process of deriving the general laws. First of all, a sample of material occupies a region of three dimensional space. This means that the functions (often called field variables) we are trying to solve for will have to be functions of up to three space variables, and possibly time as well. Typical examples of variables of interest are: density, pressure, temperature, chemical concentration (such as the salinity of the ocean). All these are scalars, or functions that accept several inputs but produce only a single output. Hence they could be well described by what a student learns in the calculus courses MATH 137, 138 and 237. However, if one thinks about what makes a fluid a fluid, they probably conclude that it is the ability to flow when perturbed (pouring juice out of a jug). This means that the fluid velocity is the variable of interest, and velocity, as we know is a vector quantity. Thus we are talking about a function which takes several inputs and produces several outputs. The calculus of vector-valued functions is more complicated than that for scalar valued functions and is generally not taught until later, in the course AMATH 231.
This means a student is faced with the somewhat difficult task of building up two years of calculus knowledge before tackling even the simplest of tasks, such as defining what the conservation of mass means for a flowing gas.
In the past this has attracted many great minds to study continuum mechanics, and the general theory does contain a great deal of mathematical beauty. However there is no avoiding the fact that even if the project is carried out successfully, one is left with a highly nonlinear set of partial differential equations. This means that no general solution in terms of a mathematical formula is possible (I think this fact is intuitively obvious from the vast array of fluid phenomena we observe on a daily basis). We are thus left to wonder how any practical problems can be solved?
The answer lies in the process sometimes called mathematical modeling. The whole idea is to take the complete mathematical description and leave out as many pieces as possible. The choice of what must be kept is to be driven by the application of interest. Thus for example, if we do not wish to consider sound waves, we treat air, a gas, as if it were incompressible (the way water is).
The idea of a mathematical method dependent on the question we are trying to answer is a deeply disturbing notion and is largely at odds with the bulk of our mathematical training. Still there is no doubt that it has yielded impressive results. Long before computers, or even slide rules, estimates of phenomena as diverse as flow over air plane wings and the propagation of energy by waves on the surface of a lake were available.
Let's consider a simple example. Consider a river bend such as the one pictured below.
The first approximation would take the rather complex geometry of an actual river bend and replace it with an idealized one, say a quarter circle such as that pictured below.
Next we choose coordinates. It seems natural to adopt polar coordinates. But should they be two-dimensional polar coordinates or a full 3D set of cylindrical polar coordinates? This will depend on the level of detail in the desired description. For a first guess you might argue that most rivers are much wider than they are deep and hence we expect the horizontal components of velocity to be much larger than the vertical component. This reduces the number of variables from three to two. You might also argue that the horizontal velocities won't vary much with depth (except at the very bottom) and so we can assume that all functions are independent of the vertical coordinate (often labeled z).
This is a considerable simplification, but we can do better. Ask yourself, what is the most important thing water does as it rounds a river bend? It goes around the bend, of course. So the most important component of velocity surely must the one in the angular direction. Thus, to a first approximation we simply drop all others. Now, we have already argued that this single component of velocity does not depend on depth, however if we consider regions away from the inflow and outflow into the bend we could argue that how far along the bend we are should not influence the structure our flow has. Furthermore, it is consistent with the other approximations we have made, to say that the flow should not vary much in time. This would leave only a single component of the velocity that is a function of a single spatial coordinate (the radial distance r).
It may be a good idea to go through the assumptions for yourself. Once you have done so you will find that the problem has undergone a drastic simplification. Indeed, mathematics-wise we have gone from the troubling realm of partial differential equations to that of ordinary differential equations. Even though the actual equations of interest are beyond a discussion at the first and second year level, I think anyone can appreciate the strength of the approach.
A final point should, of course, be how one tests whether the approximations made were actually valid. Traditionally the only choice was to conduct a well controlled experiment, collect data, and confront that data against the predictions of the approximate theory. In the digital world it is more often the case that the differential equations governing the system BEFORE approximation are compared with the predictions of the approximate theory. Even if the approximate theory is not quantitatively accurate, it may well turn out that it makes the correct qualitative predictions and as such has considerable utility. | 677.169 | 1 |
Mathematical Society
About
The purpose of the Dartmouth College Mathematical Society is to create a community where students interested in mathematics can interact and discuss mathematics in a setting outside of the classroom environment. Our goal is to encourage, foster interest in and enrich the understanding of mathematics. We intend to promote the pursuit of mathematics at all levels.
The Society is a community of enthusiastic students where people can discuss news, share experiences in courses, study abroad programs, and research internships, and explore interesting topics in mathematics.In mathematics, when graduate students perform research, they regularly create seminars in their specialized areas in order to discuss new ideas with their peers.
Meetings
The primary goal of the Society is to create an undergraduate seminar for members to meet regularly and learn about interesting topics in mathematics. The topics are presented by undergraduate guest lecturers, professors and graduate students at Dartmouth College.
Competition
The other goal of the Society is to prepare students for the William Lowell Putnam Mathematical Competition and for the Thayer Examination (administered to First-Years only) in order to promote the benefits of a rigorous problem solving education. The Society encourages even those who are not interested in our math contests to consider attending some of our problem solving seminars anyway. | 677.169 | 1 |
The Official SAT Question of the Day
Sunday, May 15, 2011
Week 36
Algebra II embarks on a new and what will be our final function, the radical function. It is another example of a function with a restricted domain as well as a different pattern of growth than our family of polynomials. We will work this week on our exponent rules and try to solidify your ability to accurately perform operations with expressions that contain rational exponents. Please find guided notes below as well as relevant videos and online resources. For online practice or general resources please check the tabs at the top of the screen entitled, "online practice" and "online resources". Every week until June 17th, I will post the Scratch project guidelines and a link to a scratch reference guide. This project is available to all juniors for bonus Also, don't forget about Khan Academy--a good place to review those basic skills. | 677.169 | 1 |
Matrices and Society: Matrix Algebra and Its Applications in the Social Sciences
Hardcover
Item is available through our marketplace sellers.
Overview structures in tribal societies. All these invaluable techniques and many more are explained clearly and simply in this wide-ranging bookPRINCETON UNIVERSITY PRESS
You are at home in the evening; there is nothing good on television; and you are at a loose end. There are three possibilities open to you: to go out to the pub, to go out to the theatre, or to stay at home and invite some friends round for a game of cards. In order to weigh up the comparative advantages and disadvantages of these three alternatives, you decide to put certain basic facts about each of them down on paper. And, being an orderly, methodical type you put them down in the form of a table, like this:
If you went to the pub, you would have to take your car out and drive three miles. It would cost you £1 to get in, since there is a special entertainment on there tonight, and you would also have to pay for the four pints of liquor and three packets of crisps which you calculate that you would consume on the premises. If you went to the theatre, it would cost you £3 to get in, but as compared with the pub you would save a little on motoring costs and quite a lot on liquor and crisps. If you stayed at home and asked some friends round for cards, you would have to provide a comparatively large quantity of liquor and crisps, but to compensate for this you would not have to take the car out and there would not of course be any admission charge.
Suppose now that just for fun you extracted the array of numbers from this table and put a pair of large square brackets around them, like this:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
No doubt much to your surprise, you would then have succeeded in constructing a matrix, which is simply a rectangular array of numbers. The individual numbers in a matrix are called its components or elements. This particular matrix, since it has three rows and four columns, is said to be a 3 × 4 matrix. If you had considered only two alternative courses of action – omitting, say, the 'Go to theatre' possibility – the matrix you constructed would have looked like this:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
This matrix has two rows and four columns, and is therefore said to be a 2 × 4 matrix. If on the other hand you had retained all three rows but omitted one of the columns, your matrix would have been a square one with three rows and three columns – that is, a 3 × 3 matrix.
Suppose now that you had seriously considered only one possible course of action – going to the pub, say. Your table would then have consisted of a 1 × 4 matrix – that is, a single row of four numbers:
[3 1 4 3].
An ordered collection of numbers written in a single row like this is a special (and very important) kind of matrix which is called a row vector.
Suppose finally that you had been interested only in the different quantities of liquor consumption associated with the three alternative courses of action. Your table would then have consisted of a 3 × 1 matrix – a single column of three numbers:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
An ordered collection of numbers written in a single column like this is another special (and equally important) kind of matrix which is called a column vector.
Matrices in the Social Sciences
The next question, of course, is why matrices are important in the social sciences, and why so many textbooks spend so much time instructing you how to play around with them. What is it all about?
A short answer is that the social sciences are often concerned with unravelling complex interrelationships of various kinds, and that it is often extremely convenient and illuminating to put these interrelationships down on paper in matrix form.
In economics, for example, we may be interested in the implications of the fact that some of the things which an industry produces (that is, its 'output') may be used as ingredients (that is, as 'inputs') in the production of other things – or even in their own production. A large part of the electricity produced in this country, for example, is not consumed directly by you and me, but is used as an input in the production of things like corn, machines, clothes and so on – and of electricity itself. Imagine, then, a very simple economy where there are only three industries, which we shall imaginatively call A, B and C. Industry A produces 300 units of its particular product – tons of steel, kilowatt hours of electricity, or whatever we suppose it to be – every year. It sells 50 of these 300 units to itself (as it were) for use as an input in its own production process; it sells 100 units to industry B and 50 to industry C for use as inputs in their production processes; and the remaining 100 units are sold to final consumers like you and me. Industry B produces 150 units, 70 of which go to itself, 25 to A, 5 to C, and the remaining 50 to final consumers. Industry C produces 180 units, 60 of which go to itself, 30 to A, 10 to B, and 80 to final consumers.
The way in which the total outputs of the three industries are disposed of can be very conveniently set out in the form of a simple matrix like this:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
One of the advantages of setting out the facts in this way is that two interrelated aspects of the overall situation are presented to us at one and the same time: the three rows tell us where each industry's output goes to, and the first three columns tell us where each industry's physical inputs come from.
Another type of matrix which often crops up in the social sciences is one which sets out the gains and losses accruing from some kind of 'game' which two (or more) participants are supposed to be playing. Suppose, for example, that two persons, Tom and Jerry, find themselves in some sort of conflict situation in which they are obliged to choose (independently of one another) between several alternative courses of action, and in which the final outcome – the gain or loss for each 'player' – depends upon the particular combination of choices which they make. Tom and Jerry, let us say, are two rival candidates for political office. At a certain stage in the election campaign a crisis arises, in which Tom has to choose between two possible strategies and Jerry between three. If we could calculate the numbers of votes which would be gained (or lost) by Tom – and therefore we assume, lost (or gained) by Jerry – in the event of each of the six possible outcomes, our calculations would be usefully presented in matrix form. The matrix might appear like this:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
From this we can immediately see, for example, that if Tom adopts his first strategy and Jerry adopts his third strategy, the outcome will be that Tom will lose (and Jerry will therefore gain) 1000 votes. If Tom adopts his second strategy and Jerry adopts his second, then Tom will gain (and Jerry will therefore lose) 2000 votes. And so on. The advantage of this way of presenting the facts is once again that it puts them before our eyes simultaneously from two points of view – the rows, as it were, from the point of view of Tom, and the columns from the point of view of Jerry.
In sociology, again, we might be interested in what is called a dominance situation, in which the pattern of dominance between three individuals (Pip, Squeak and Wilfred, say) can be represented in a matrix, where the entry 1 indicates that the person whose row the entry is in dominates the person whose column it is in; at the same time an entry 0 in a row means that the person whose row it is in does not dominate the person whose column the 0 entry is in.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Thus the rows of this matrix show us that Pip dominates Wilfred, Squeak dominates Pip, and Wilfred dominates Squeak; and if we look at the columns we can immediately see whom Pip, Squeak and Wilfred respectively are dominated by.
Or, to take a final example, we might be interested in some kind of transition matrix, setting out the probabilities of a person's proceeding from (for instance) one social class to another in some given time period – a generation, say. Take the following matrix, which might represent the probability of the sons of upper-, middle- and lower-class fathers moving into the upper, middle and lower classes respectively:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Here the rows show us the probabilities of the son's class when that of the father is known. The son of a middle-class father, for example, has a one in ten chance of moving to the upper class, a seven in ten chance of staying in the middle class, and a one in five chance of moving to the lower class. In this case the interpretation of the columns is not quite so straightforward. Obviously the first column, for instance, does not tell us what the probabilities are of an upper-class son having an upper-, middle- or lower-class father. Such probabilities must depend on the number of fathers there are in each class.
The Manipulation of Matrices
So far, all we have done is to explain what a matrix is, and to establish that the matrix form may sometimes be a neat and convenient way in which to set out some of the interrelationships in which social scientists are interested. But if that was all there was to it, there would hardly be any need for a book like the present one. The point is that it is often useful not merely to set out the interrelationships in matrix form, but also to be able to manipulate the matrices themselves in various ways. What is meant by this?
In ordinary arithmetic, where we deal with individual numbers, we use certain simple techniques, which we all take in with our mother's milk, in order to add, subtract, multiply and divide them. Suppose, however, that you want to deal not with individual numbers but with arrays of numbers in matrix form. Suppose you think that it might be useful to treat each of these matrices as a unit, and to perform operations upon them analogous to those of addition, subtraction, multiplication and division in ordinary arithmetic. How would you go about it?
Essentially, what you would require is a set of conventions establishing what you are going to mean by addition, subtraction, multiplication and division, when you are dealing with matrices rather than with individual numbers. And the conventions you adopted would depend largely upon their convenience in relation to the particular problems which you were hoping to be able to solve with the aid of these operations.
So far as addition is concerned, the convention usually adopted is a fairly simple and commonsense one. To add two matrices of the same dimensions, or order as it is often called (that is, having the same number of rows and columns), you simply add the corresponding components. For example:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Here are two more examples, in the first of which we add two 1 × 3 row vectors, and in the second 2 × 2 square matrices:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
But, you may be asking, what about matrices which do not have the same number of rows and the same number of columns? How do you add them? The simple answer is that you don't, and can't. The operation 'addition', when applied to matrices, is defined in terms of the addition of the corresponding components, and is therefore applicable only to matrices of the same dimensions.
The operation subtraction is defined analogously to that of addition, and it is also therefore applicable only to matrices of the same shape. Examples:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
That is all easy enough, I suppose. But multiplication is defined in a different, less obvious way, and will take a little longer to explain.
First let us get what is called scalar multiplication out of the way. Sometimes we may want to multiply each component of a matrix by a single number, say 2, in which case we just do precisely that, setting out the operation as in the following example:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
More often, however, as we shall see later in this book, we will want to multiply the matrix not by a single number but by another matrix, or by itself. What meaning is it most useful for us to give to such an operation?
Let us approach this problem indirectly by having another look at the matrix which we constructed at the beginning of this chapter:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
In order to identify this matrix, let us call it A. Suppose now that you want to take a further step in weighing up the advantages and disadvantages of the three alternatives by calculating the respective total money costs involved in each of them. Suppose also that for some reason best known to yourself you want to separate out the tax element in these money costs, so that you finish up with two separate figures relating to each of the three options – one showing the total money costs of the option, and the other showing the total amount of tax included in these costs.
The motoring cost per mile, reckoned in pence, is, let us say, 20p which includes a tax element of 10p; the admission charge per £1 is (not unnaturally) 100p, which includes a tax element of 25p; the price of liquor per pint is 80p, which includes a tax element of 20p; and the price of crisps per packet is 10p, the tax element here being zero. It will be convenient to put this information in the form of a second matrix – a 4 × 2 one this time – which we will identify as matrix B:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Given these two matrices A and B, it is of course simplicity itself to make the calculations you have in mind. To work out the total money costs involved in going to the pub, you multiply each of the components in the first row of A by the corresponding components in the first column of B, and then add up the four products, thus:
And to work out the tax element included in this total of 510p, you multiply each of the components in the first row of A by the corresponding components in the second column of B, and then add up the four products, thus:
Similarly to work out the total money costs involved in going to the theatre, you multiply each of the components in the second row of A by the corresponding components in the first column of B, and then add up the four products, thus:
And to work out the tax element included in this total of 430p, you multiply each of the components in the second row of A by the corresponding components in the second column of B, and then add up the four products, thus:
You should now have no difficulty in making the third and last pair of calculations yourself, or in seeing that the most convenient way of presenting the final results is in the form of a third matrix – a 3 × 2 one which we shall identify as matrix C: | 677.169 | 1 |
Math Course Takes 'Real Life' Approach to Algebra
Educational courseware publisher American Education Corp. is taking a new approach to answering the age-old question, "What does algebra have to do with real life?" The company has announced the release of a new course for its A+nyWhere Learning System program. Algebra I: A Function Approach Part 1 is the first semester segment of a full-year algebra course geared to grades 9 and 10, and, in addition to the fundamental concepts and tools of algebra, the course aims to relate the material to "real life."
Taking the fundamentals and applying them to real-world situations using exercises in relevant scenarios allows students to realize the practical uses of linear and quadratic equations, graphs and coordinates, functions, and other algebraic concepts.
The A+nyWhere program is computer based, so students taking courses like Algebra I can use a number of tools incorporated into the software to aid in their assignments and overall comprehension of the material. These tools include onscreen standard and scientific calculators, pictures and diagrams, video tutorials, exercises, practice exams, and, for upper-level courses, interactive feedback | 677.169 | 1 |
Saxon
Filter by Grade
Filter by Subject
Filter by Publisher
Saxon Math has a proven record of success among schools and homeschools. As a result, Saxon is a top math choice from kindergarten through high school.
Saxon's approach to math is to focus on individual skills and essential concepts rather than classroom interaction. Therefore, with Saxon, each concept is introduced and then reviewed and expanded on consistently throughout the school year. Students gain new concepts, practice them and build on them incrementally as they learn new concepts that apply to the old.
When used in its entirety, Saxon's innovative approach ensures that students gain and retain critical math skills, ensuring success in more complex skills.
Topics and concepts are never taught and then dropped to be reviewed next year. Instead, more complex concepts build on the old as students practice them every day.
Saxon does not provide colorful textbooks; instead it provides a step-by-step proven approach to math that leads to math success.
Buying your Saxon Curriculum, home school supplies and other homeschooling curriculum at Curriculumexpress.com can be done with confidence because our home education materials, science kits, home schooling curriculum, books and more are backed by our 30 day money back guarantee. | 677.169 | 1 |
Search This Blog
Math curriculum support workshop number three
On Friday September 21 the third in a series of work shops intended to provide support to the instructors at Madolehnihmw high school was held from 4:00 to 6:00 P.M. I left the college at about 3:10 and arrived at 3:56 at MHS.
Whereas in earlier sessions I had some foreknowledge of what was to be covered, for this session I had no foreknowledge and responded to questions. The first question was the effect of the coefficient a in the vertex form of a quadratic, which I write as (y - k) = a(x - h)^2. This led into coverage of the vertex form, its relationship to y = ax^2 + bx + c, and on into imaginary roots (x-intercepts) as the x-axis crossing of the reflection of the parabola. I wanted a mirror tile, but I had none with me.
There were also questions on ways to graph from the vertex form without plugging in a series of x-values. The text presumes the class is using graphing calculators. The students are not and are hand graphing the functions. I showed how one can use the vertex, y-intercept, and x-intercepts to often get a good start on a sketch. I also covered obtaining the x-intercepts from the vertex form.
Along the way I mentioned the focus on slope and y-intercept for linear equations, a focus that effectively disappears as soon as the book turns to the topic of quadratic equations. I noted that c in y = ax^2 + bx + c is the y-intercept and the slope is 2ax + b. I then explain that this requires calculus to generate the slope function. The students, however, can see that a slope does exist and that the slope changes as x changes - hence the curved line.
Although MHS dismissed at noon, the instructors in the program waited until 4:00 when I arrived. This is true commitment to the profession that is education.
The instructors then had questions on rational functions. Rational functions is section 4.1 of the MS 100 College Algebra text. When I was division chair I was aware that reaching 4.1 was nigh on impossible if one covered all sections from chapter one to three. When I taught MS 100 I used to jump out of chapter three just after polynomial division to pick up 4.1 and, if possible, 4.2. While I am not necessarily enamored of rational functions, I do think asymptotic behavior is a worthy concept in and of itself.
The session wrapped up with questions on direct, inverse, and joint variations.
The team at MHS kindly provided refreshments for everyone.
The workshop wrapped up at 6:00 in the evening.
The drive home included a hint of a rainbow in a sunset lit cloud.
The group will next meet on 26 October at MHS at 4:00 in the afternoon | 677.169 | 1 |
Computers and Communications : Toward a Computer Utility
Description
Algebra success for allBasic concepts and properties of algebra are introduced early to prepare students for equation solving. Abundant exercises graded by difficulty level address a wide range of student abilities. The Basic Algebra Planning Guide assures that even the at-risk student can acquire course content.Multiple representations of conceptsConcepts and skills are introduced algebraically, graphically, numerically, and verbally-often in the same lesson to help students make the connection and to address diverse learning styles.Focused on developing algebra concepts and skillsKey algebraic concepts are introduced early and opportunities to develop conceptual understanding appear throughout the text, including in Activity Labs. Frequent and varied skill practice ensures student proficiency and success.show more | 677.169 | 1 |
Ensure
• Bring awe and wonder with a chapter opener that puts the maths in context • Access the right level of content with the progress indicators on the page • Provide rigorous maths practice with hundreds of high quality questions • Focus on literacy skills with key words per topic and a glossary at the back • Achieve fluency through 100s of practice questions • Develop mathematical reasoning with flagged practice questions and longer activities at the end of exercises • Practise multistep and problem solving skills with flagged practice questions and longer activities at the end of exercises • Measure progress with 'Ready to progress?' learning outcomes at end of chapters • Make connections across different areas of mathematics with synoptic extended questions at the end of each chapter that use maths from previous chapters • Break up lessons and add variety and engagement with longer, colourful real-life tasks and contexts which could be: investigations, challenges, activities, problem solving, using financial skills, or mathematical reasoning • Access answers in the accompanying Teacher Pack 1.3 ISBN 978-0-00-753783-9
"synopsis" may belong to another edition of this title.
Product Description:
Ensure progress at the right pace with Pupil Book 1.3. Track your progress with indicators that show....
Book Description HarperCollins Publishers, 2014. PAP. Book Condition: New. New Book. Shipped from US within 10 to 14 business days. Established seller since 2000. Bookseller Inventory # IB-9780007537730
Book Description Collins Educational 2014-03777036
Book Description HarperCollins Publishers. Paperback. Book Condition: new. BRAND NEW, Maths Frameworking: Pupil Book 1.3, Kevin Evans, Keith Gordon, Trevor Senior, Brian Speed, Chris Pearce, Ensure 'Ready to progress?' ISBN 978-0-00-753783-9. Bookseller Inventory # B HARPER COLLINS, 2014. Paperback. Book Condition: NEW. 9780007537730 This listing is a new book, a title currently in-print which we order directly and immediately from the publisher. Bookseller Inventory # HTANDREE0994272 | 677.169 | 1 |
Exponent Properties for Algebra [guided notes, warm-up, practice]
Be sure that you have an application to open
this file type before downloading and/or purchasing.
3 MB|14 pages
Product Description
2 pages of guided notes to introduce your students to the properties of exponents, 1 critical thinking warm-up, 4 pages of practice
properties covered:
product of powers, power of a product, power of a quotient, quotient of powers, zero exponent, negative exponents
answer key provided
thanks for checking out this product at the miss jude math! shop
Happy Solving! | 677.169 | 1 |
Beginning Algebra A Text/Workbook, 9th Edition
For the modern student like you--Pat McKeague's BEGINNING ALGEBRA, 9E--offers concise writing, continuous review, and contemporary applications to show you how mathematics connects to your modern world. The new edition continues to reflect the author's passion for teaching mathematics by offering guided practice, review, and reinforcement to help you build skills through hundreds of new examples and applications. Use the examples, practice exercises, tutorials, videos, and e-Book sections in Enhanced WebAssign to practice your skills and demonstrate your knowledge. | 677.169 | 1 |
It's like if you want to be a good pianist,
you have to do a lot of scales and a lot of practice,
and a lot of that is kind of boring, it's work.
But you need to do that before you can really be very expressive and really play beautiful music.
You have to go through that phase of practice and drill.
- Terry Tao
About this article:
What for? OK, there is a way to become a good theoretical physicist. Here is a guide to study pure mathematics, or even more. This list is written for those who want to learn mathematics but have no idea how to start. Yup, a list for beginners. I don't claim that this list makes you a good pure mathematician, since I belong to the complement of good pure mathematician. I make no attempt to define what pure mathematics is, but hopefully it will be clear as you proceed. I also highlighted several books that you would really like to keep in your own library. You probably like to read those books again and again in your life. Free material excluded. Note the highlighted list does NOT indicate those books are good for beginners. I shall try to keep this list up to date whenever I exist.
Assumed knowledge. I assumed you have high school mathematics background (i.e. basic trigonometry, Euclidean geometry, etc). The aim of this page is to introduce what different branches of mathematics are; and recommended a few notes or texts. Scientists in other fields and engineers may skip first or second stages and begin at later stages, according to their prior knowledge.
Time. It takes approximately one year for each stage (except for stage 4, I list more material in each field for more advanced studies), for a full time student. Part time students may double the time. But its better for anyone to understand most parts of stage n before proceeding to n+1, for some integer n in {1, 2, 3, 4}. If you decided to attend a class, don't expect the professor can teach, it always happen, especially in higher level courses. What's the order of courses to study within a stage doesn't really matter, usually. One doesn't need to read every listed book within a subject to master the subject. I listed more than enough so that you can scout around to find one that you feel comfortable with. Some people like to consult a few books, beware of the symbols from different books in such cases. Moreover, it often happens that you couldn't solve a problem within an hour. It's not surprise to spend a week or more to tackle one problem. Things may come to your head suddenly. Shouting eureka is the high point of a mathematician.
"Axiom of choice". My selection will not be bounded by any publication press, author's nationality or religion. It relies on two factors: well written or cheap. These two factors are not mutually exclusive. I treat "free" as an element in "cheap". Note the price factor may by irrelevant, sometimes I get a HK $2xx book and Amazon says its US $1xx (~HK $7xx)...... with the only difference is, perhaps, I got the international edition. Moreover, some Chinese press in mainland China published photocopied of English text with a relatively cheap price. Bear in mind that, just because one is a good mathematician doesn't imply he's a good author or educator. Perhaps Terry Tao is an exceptional case. To study science, reading the classics (the Elements, Dialogo sopra i due massimi sistemi del mondo, the Principia, Disquisitiones Arithmeticae, Principia Mathematica, etc) is optional. While for literature or philosophy, I wonder if any well educated student has never study Shakespeare or Plato. To view [.pdf] get Adobe Reader, to view [.ps] download this and this, or visit this page, to view [.djvu] get this. Get WinRAR for [.rar] files.
Comments. Links to Amazon for most of the listed book are included, so that you have an easy access to other users' comments. Note that the comments are sometimes quite extreme: for the same book, one rated it with 5 star (with dozens of people supporting) and at the same time another rated 1 star (with dozens of people supporting again), especially for introductory discrete mathematics, probability and statistics books. It seemed to me that lots of people study these subjects because they need to, they want to apply mathematics. Large proportion of these readers are lack of mathematical maturity. If they can't pass the exam, you know... In contrast, most pure mathematics students study because they like the subject and enjoy it. So, ask yourself, why do you study?
Other resources. Although I'm not into reading books online, I should remind you that MIT's open recourse, the Archimedeans and Wikibooks provide another great sources of materials. These are excluded in the following list. The list below aimed to recommend books or (usually) printable notes. Google books allow you to preview sections from a book. Schaum's Outlines series are cheap, but I seldom include them, you may search the relevant if you like.
Some nice stuff! I'm a physics student, (In September 2nd year) and I want to study theoretical physics, so this is my time to get a strong math background.
Thinking in doing Stage 1 in four-five months. As long as I "know" some of that stuff, I just want to keep it formal, and learn how to prove things, just take the mathematical point of view, not just the physical one.
even though I am an agnostic the only thing I have to say is "May god bless you!". I loved mathematics till my class 10 (India) after that I could not understand why the fuck was I studying what I was studying...school text books did not provide me any help...after that I studied electrical engineering where I sucked in mathematics up till the point that I thought that I cannot do it anymore. This was around 5 years ago. About a few days back I have decided to study maths again. I don't think I will die peacefully unless I understand things on my own. and again I thank you and may you live a happy life and thanks for creating this list.
What more could I say, YOU ROCK!!! This is really great, keep on adding more good textbooks, notes, solutions etc. I was quite weak in maths, but I re-discover my interest in the subject, and your blog will definitely be a great source of information and inspiration, Thank you, Buddy!
Actually, I have visited your blog for several times, and every time I found it useful. I just want to leave a message to encourage you, and hope that you will keep updating the blog from time to time.
I'm a Mathematics graduate (2014) at University College London. But right after I finished my graduation exam, I fell down in my room. And I was diagnosed with brain AVM. Although I received a upper-second class honour degree (bachelor), I almost forgot all the Maths. | 677.169 | 1 |
Provide students with a college-prep Algebra II course that will allow them to easily progress onto even more difficult mathematical challenges. Saxon Algebra 2, 4th Edition prepares students for calculus through explicit embedded geometry instruction. Trigonometry concepts, statistics, and applications for other subjects such as physics and chemistry are also included. Incremental lessons include a Warm Up activity; New Concepts section that introduces new concepts through examples with sidebar hints and notes; and Lesson Practice questions with lesson reference numbers underneath the question number. Real-world applications and continual practice & review provide the time needed to master each concept, helping students to build confidence in their mathematical abilities. Distributed,...
Less
NEW, in plastic wrap / Includes MathXL access code ISBN-10: 0-13-313211- 0 / 0133132110 ISBN-13: 978-0-13-313211-3 / 9780133132113 Ships securely SAME or NEXT day with tracking number Select expedited shipping for delivery in 2-4 business days Buyer will receive email notification once order is sent out ***Offers welcome if found cheaper elsewhere. Along with a superior service we strive to provide the most competitive pricing***
Advanced Algebra: Expanded Edition is the second book in the Life of Fred High School Mathematics Series, and is designed for students in 10th grade who have already finished the preceding Beginning Algebra, Expanded Edition. This new edition of Algebra replaces the both the earlier Life of Fred Advanced Algebra and Fred's Home Companion Advanced Algebra books; it also contains all problems completely worked out. The sold-separately Zillions of Practice Problems for Advanced Algebra book is an optional resource for students who want more practice opportunities. Ten chapters with multiple sub-lessons (105 total) are included. Each lesson ends with a Your Turn to Play segment with a small number of thought- provoking questions. Answers are provided on the next page for students to go over...
Less
Algebra 1/2 Home Study Kit includes the hardcover student text, softcover answer key and softcover test booklet. Containing 123 lessons, this text is the culmination of prealgebra mathematics, a full pre-algebra course and an introduction to geometry and discrete mathematics. Some topics covered include Prime and Composite numbers; fractions & decimals; order of operations, coordinates, exponents, square roots, ratios, algebraic phrases, probability, the Pythagorean Theorem and more. Utilizing an incremental approach to math, your students will learn in small doses at their own pace, increasing retention of knowledge and satisfaction!
Product Description Reduce your student loan debt, save time, and get rewarded for what you already know! CLEP tests are a quick way to earn college credits...you'll just have to invest the time to prepare for the test! Using this convenient guide, review the main elements of the subject and take the test yourself to see how well you do. Test tips, a study plan, and an overview of the exam are also included. CLEP College Algebra covers algebraic operations and reviews math essentials, functions and their properties, equations and inequalities, and number systems and operations. Two practice tests are also included (in the book and online) with detailed explanations of why correct answers are correct, and why the incorrect answers are incorrect. A free-response essay section is also...
Less Math 7/6 covers 120 Lessons, 12 Investigations, and 1 Supplemental Topic. 4th Edition.
This teacher's guide accompanies BJU Press' sold-separately Algebra 2 Student Text, 3rd Edition. Higher-level math can be intimidating for homeschooling parents to teach. BJU Press' comprehensive teacher's guides, however, can take away some of that anxiety with thorough, pre-prepared lesson plans! This teacher's edition provides lesson guidance, notes on common student errors, interactive activities, motivational ideas, additional problems, and more. Reduced-size student pages are provided, and are surrounded in the margins by teacher notes, objectives, vocabulary, step by step lesson instruction, additional problems, and more. Units open with an overview as well as presentation instructions that include biblical links. The included teacher's toolkit CD includes PDF files of visuals,...
Less
Guide your students to mastering advanced algebraic concepts with BJU Algebra 2, 3rd Edition! Students will learn concepts such as linear, quadratic, polynomial, radical, and rational functions, exponential and logarithmic functions, and probability and statistics. Complex numbers are also covered throughout the text. Each lesson develops key concepts and detailed examples with practical applications integrated throughout. Exercise sets are graded by difficulty level and each lesson's cumulative review exercises will help students retain previously learned information. This third edition features two new chapters that cover matrix features and sequencing and series. New textbook features include selected Internet keyword searches for helping students locate online tools and enrichment,...
Less
Beginning Algebra: Expanded Edition is the first book in the Life of Fred High School Mathematics Series, and is designed for students in 9th grade. This expanded edition of Algebra replaces the both the earlier Life of Fred Beginning Algebra and Fred's Home Companion Algebra books; it also contains all problems completely worked out. The sold-separately Zillions of Practice Problems for Beginning Algebra book is an optional add-on for students who want more practice opportunities. Twelve chapters with multiple sub-lessons are included. Each lesson ends with a Your Turn to Play segment with a small number of thought- provoking questions. Answers are provided on the next page for students to go over themselves after attempting to solve the problems. Chapters conclude with three problem...
Less
Provide students with a college-prep math course that will give them the foundation they need to successfully move into higher levels of math. Saxon Algebra 1, 4th Edition covers all of the traditional first-year algebra topics while helping students build higher-order thinking skills, real-world application skills, reasoning, and an understanding of interconnecting math strands. Saxon Algebra 1 focuses on algebraic thinking through multiple representations, including verbal, numeric, symbolic, and graphical, while graphing calculator labs model mathematical situations. Incremental lessons include a Warm Up activity; New Concepts section that introduces new concepts through examples with sidebar hints and notes; and Lesson Practice questions with lesson reference numbers underneath the...
Less
This Saxon Algebra 2 Home Study Kit includes the Student Textbook, Testing Book and Answer Key. Traditional second-year algebra topics, as well as a full semester of informal geometry, are included with both real-world, abstract and interdisciplinary applications. Topics include geometric functions like angles, perimeters, and proportional segments; negative exponents; quadratic equations; metric conversions; logarithms; and advanced factoring. Student Text is 558 pages, short answers for problem/practice sets, an index and glossary are included; hardcover. The Test book contains both student tests and solutions with work shown along with the final answer. 32 tests are included. The Answer key shows only the final solution for the practice and problem sets found in the student text. 44...
Less
This teacher's guide accompanies BJU Press' Pre-Algebra Grade 8 Student Text, 2nd Edition. Reduced student pages are included, and have teacher lesson notes in the margins. Algebraic expressions and linear equations are applied throughout a thorough review of operations on integers, fractions, decimals, percents, and radicals. Students explore relations and functions using equations, tables, and graphs, while chapters on statistics and geometry extend foundational concepts in preparation for high school courses. This resource is also known as Bob Jones Pre-Algebra Grade 8 Teacher's Edition, 2nd Edition.
Clearly written and comprehensive, the tenth edition of Gustafson/Frisk/H ughes; popular book provides in-depth and precise coverage, incorporated into a framework of tested teaching strategy. The authors combine carefully selected pedagogical features and patient explanation to give students a book that preserves the integrity of mathematics, yet does not discourage them with material that is confusing or too rigorous. Long respected for its ability to help students quickly master difficult problems, this book also helps them develop the skills they;ll need in future courses and in everyday life. This new edition has the mathematical precision instructors have come to expect, and by bringing in new co-author, Jeff Hughes, the authors have focused on making the text more modern to better...
Less
This Saxon Algebra 2 Home Study Kit includes the Student Textbook, Test Book/Answer Key Packet, and Solutions Manual. Traditional second-year algebra topics, as well as a full semester of informal geometry, are included with both real-world, abstract and interdisciplinary applications. 129 lessons cover topics such as geometric functions, angles, perimeters, and proportional segments; negative exponents; quadratic equations; metric conversions; logarithms; and advanced factoring. Student Text is 577 pages, short answers for problem/practice sets, an index and glossary are included; hardcover. The Homeschool Packet includes contains 32 student tests and test solutions with work shown along with the final answer. The Answer key portion shows only the final solution for the practice and...
Less
This solutions manual accompanies Saxon Math's Algebra 1/2 Student Text. Make grading easy with solutions to all textbook practices, problem sets, and additional topic practices. Early solutions that contain every step, and later solutions omit obvious steps; final answers are given in bold type for accurate, efficient grading. Paperback.
Key To Algebra offers a unique, proven way to introduce algebra to your students. New concepts are explained in simple language and examples are easy to follow. Word problems relate algebra to familiar situations, helping students understand abstract concepts. Students develop understanding by solving equations and inequalities intuitively before formal solutions are introduced. Students begin their study of algebra in Books 1-4 using only integers. Books 5-7 introduce rational numbers and expressions. Books 8-10 extend coverage to the real number system. This kit contains only Books 1-10. Answers Notes for Books 1-4 Books 5-7 and Books 8-10 are available separately, as well as the Key to Algebra Reproducible Tests.
Algebra Readiness is designed for the middle school learner and provides a smooth transition from Course 1 (Grade 6) and Course 2 (Grade 7) Math. The This Algebra Readiness transition course covers algebraic expressions and integers, solving one-step equations and inequalities, area and volume, and linear functions. In the Getting Ready to Learn portion of the textbook lesson, Check your readiness exercises help students see where they might need to review before the lesson. Check skills you'll need...
Less
Students will develop the understanding they need to resolve more complex problems and functions with step-by-step instruction in Algebra 1. 120 lessons cover signed numbers, exponents; roots; absolute value; equations and inequalities; scientific notations; unit conversions; polynomials; graphs; factoring; quadratic equations; direct and inverse variations; exponential growth; statistics; and probability. This kit includes: Student Text; 564 pages, hardcover. Short answers for problem/practice sets, an index and glossary are included. The Homeschool Packet includes contains 30 student tests & test answers. The Answer key portion shows only the final solution for the practice and problem sets found in the student text. The Solutions Manual features solutions to all textbook practices and...
Less
Bob Blitzer has inspired thousands of students with his engaging approach to mathematics, making this beloved series the #1 in the market. Blitzer draws on his unique background in mathematics and behavioral science to present the full scope of mathematics with vivid applications in real-life situations. Students stay engaged because Blitzer often uses pop-culture and up-to-date references to connect math to students' lives, showing that their world is profoundly mathematical.
Administered be the College Board and accepted by over 2900 colleges and universities nationwide,CLEP and AP exams allow students to earn college credits by passing an exam instead of taking the college course. However, many students struggle with learning college level material without a teacher. CLEP Professor provides that teacher. Unlike other CLEP preparation programs, CLEP Professor actually teaches every concept on the CLEP and AP exam. The digital whiteboard lectures are like being in a real classroom, with the added advantage of being able to pause, rewind, and fast forward the teacher. Each lesson is taught from a Christian worldview. Diagnostic test saves time Interactive video lectures teach every concept on the exam Practice problems with video solutions promote mastery...
Less
College Algebra: Concepts and Models provides a solid understanding of algebra, using modeling techniques and real-world data applications. The text is effective for students who will continue on in mathematics, as well as for those who will end their mathematics education with college algebra. Instructors may also take advantage of optional discovery and exploration activities that use technology and are integrated throughout the text. A brief version of this text, College Algebra: A Concise Course, provides a shorter version of the text without the introductory review/3, 3rd Edition includes 123 lessons plus 10 supplemental topics, 3rd Edition includes 120 Lessons. DVDs come in plastic clamshell case
College Algebra, First Edition will appeal to those who want to give important topics more in-depth, higher-level coverage. This text offers streamlined approach accompanied with accessible definitions across all chapters to allow for an easy-to-understand read. College Algebra contains prose that is precise, accurate, and easy to read, with straightforward definitions of even the topics that are typically most difficult for students.
Free Delivery Worldwide : The Learning Guide for Intermediate Algebra for College Students : Paperback : Pearson Education (US) : 9780321760425 : 0321760425 : 24 Mar 2012 : The Learning Guide helps students learn how to make the most of their textbook and its companion learning tools, including MyMathLab. Organized by the textbookâ s learning objectives, this workbook provides additional practice for each section and guidance for test preparation. Published in an unbound, binder-ready format, the Learning Guide can serve as the foundation for studentsâ course notebooks...
Brush up on your first year of college-level math with our new College Algebra guide! Pinpointed essentials of college algebra are covered in our easy-to-access format that includes succinct explanations of step-by-step problem solving, as well as the related mathematical rules. Whether you are in high school or college, taking the course for your first time or tackling higher-level math, this guide is an essential resource for reviewing this fundamental area of mathematics | 677.169 | 1 |
.
Fit2Go is suited for building a conceptual understanding of mathematical facts that are usually known only as "rules of thumb." Everyone knows that two points define a line. Fewer would know that three points define a parabola. High school students can prove it either in their algebra course by solving a system of equations or in their analytic geometry studies by implementing the geometric properties of the parabola. Fit2Go provides a wide repertoire of choices that fit given sets or subsets of data, and elicits questions and conjectures that can lead to formal solutions and proofs.
Features
Fit2Go is a linear and quadratic function graphing tool and curve fitter. Students can view a phenomenon, identify variables, conduct experiments, and take measurements in order to construct models of the phenomena. Fit2Go offers linear or quadratic models by presenting graphs and expressions of functions that can fit the data. Fit2Go provides an easy visual way of enter the data by dynamically viewing the point and reading its values. After choosing the type of model (an important decision that should be made by the user rather than automatically interpreted by the tool), Fit2Go presents a specific line (if only two data points are marked) or a specific quadratic function (if three points are marked). Interesting cases occur when too many or too few points are marked. Fit2Go does not attempt to fit a model to all points by interpolation. Rather, it randomly plots optional curves that fit a subset of the marked points and allows the user to alternate between the random options. If there are too few constrains, Fit2Go graphs a family of graphs, which it alternates according to the user requests.
Suggested Activities
Construct a function with a given values-table
a value table is given:
Start by graphing the points on paper. Chart and write down the expression of the function you conjecture would best fit the points.
Enter the order pairs as points into Fit2Go
Click on the points to which you wish to fit your model and choose a family that you think is a good candidate for fitting the points you have selected.
Find relationships between the height of a ramp and the time it takes a skateboard to travel the length of the ramp
If possible, let the students set up an experiment and take measurements. Assume a specific constant length of the board. If it cannot be done, use some data sets that could fit such an experiment
Consider what happens to the run time when the board becomes flatter. Or higher. Could the runtime be zero? Could it be a linear model?Math Training Math Training is a fun way to practice your basic arithmeticTrigonometry Tr | 677.169 | 1 |
Complete Idiot's Guide to Algebra
ISBN-10: 1592576486
ISBN-13: 9781592576487 the facts (and figures) to understanding algebra. The Complete Idiots Guide to Algebrahas been updated to include easier-to-read graphs and additional practice problems. It covers variations of standard problems that will assist students with their algebra courses, along with all the basic concepts, including linear equations and inequalities, polynomials, exponents and logarithms, conic sections, discrete math, word problems and more. Written in an easy-to-comprehend style to make math concepts approachable Award-winning math teacher and author of The Complete Idiots Guide to Calculusand the bestselling advanced placement book in ARCOs Master series | 677.169 | 1 |
ISBN-10: 0321989929
ISBN-13: 9780321989925.Note: You are purchasing a standalone product; MyMathLab does not come packaged with this content. MyMathLab is not a self-paced technology and should only be purchased when required by an instructor. If you would like to purchase "both "the physical text and MyMathLab, search for: 9780134022697 / 0134022696 Linear Algebra and Its Applications plus New MyMathLab with Pearson eText -- Access Card Package, 5/eWith traditional linear algebra texts, the course is relatively easy for students during the early stages as material is presented in a familiar, concrete setting. However, when abstract concepts are introduced, students often hit a wall. Instructors seem to agree that certain concepts (such as linear independence, spanning, subspace, vector space, and linear transformations) are not easily understood and require time to assimilate. These concepts are fundamental to the study of linear algebra, so students' understanding of them is vital to mastering the subject. This text makes these concepts more accessible by introducing them early in a familiar, concrete "Rn" setting, developing them gradually, and returning to them throughout the text so that when they are discussed in the abstract, students are readily able to understand | 677.169 | 1 |
Introduction to Differential Equations Using Sage by David Joyner
Since its release in 2005, Sage has acquired a substantial following among mathematicians, but its first user was Joyner, who is credited with helping famed mathematician William Stein turn the program into a usable and popular choice.
Introduction to Differential Equations Using Sage extends Stein's work by creating a classroom tool that allows both differential equations and Sage to be taught concurrently. It's a creative and forward-thinking approach to math instruction.
David Joyner is a professor in the Mathematics Department at the U.S. Naval Academy. He is the author of Adventures in Group Theory: Rubik's Cube, Merlin's Machine, and Other Mathematical Toys, also published by Johns Hopkins. Marshall Hampton is a professor in the Department of Mathematics and Statistics at the University of Minnesota, Duluth. | 677.169 | 1 |
Mathemetics-Statistics
The Edexcel A level (9371)
Mathematics qualification is designed as a two year course. All pupils study a
set of Core modules, covering Pure Mathematics.
At the start of Year 12, pupils
begin their units of application. Here at Mesaieed International School, we
offer one route which includes Core 1 and Core 2 plus Statistics 1 in Year 12 followed by Core 3 and Core 4 plus Mechanics 1 in Year 13. No other combination
of application units is offered or allowed.
Mathematics
AS and A level are extremely demanding courses and require complete dedication
towards the subject. Pupils are expected to develop and demonstrate an in-depth
understanding of the concepts and processes being covered. Many pupils who
consider themselves to be numerate initially find Mathematics AS level to be
difficult. It is only by complete immersion that a pupil may begin to grasp the
level of understanding required, if one wishes to be successful in this
subject. | 677.169 | 1 |
Relating the Domain of a Function to Its Graph - Playlist and Teaching Notes
Compressed Zip File
Be sure that you have an application to open this file type before downloading and/or purchasing. How to unzip files.
1.26 MB | 7 pages
PRODUCT DESCRIPTION
In this playlist, students explore standard HSF.IF.B.5. Students will practice relating the domain of a function to its graph. They will learn how to understand the appropriate domain for a function modeling a situation. Students also have the option to view instructional videos and complete practice quizzes or activities.
The playlist includes:
• 4 links to instructional videos or texts
• 1 link to practice quizzes or activities
• Definitions of key terms, such as function and range
• Visual examples of the linear functions with different ranges and domains
Accompanying Teaching Notes include:
• A review of key terminology
• Links to video tutorials for students struggling with certain parts of the standard, such as mishandling restrictions on the domain of a function | 677.169 | 1 |
What do we offer in college math?
Overview
Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere. College students who have gaps in knowledge can struggle in math, especially within courses that build on previous learning and on placement exams . These students need relevant, high-quality resources that can be accessible at any time.
What we offer to students is a free, online tool, which houses thousands of instructional math videos, exercises, and a personalized learning platform in which students can learn at their own pace through an entire math subject.
Students can access math content either through a personalized mission or our library.
A mission is a personalized math experience through a specific math subject.
Students can use missions to:
learn at their own pace,
master skills that are challenging and appropriate for their level, and
use hints and videos immediately when they are stuck.
For college students in developmental math, our most popular mission is Algebra Basics, which is ideal for anyone looking to prepare for a college placement exam or is in developmental math.
Our library contains thousands of videos and exercises - think of it as a free tutor available to students 24/7!
Students can access this library by simply typing the skill or topic in the search bar. This is typically very useful for students who are either looking for one specific skill,
We believe that coaches are critical to student success.
A coach can be a teacher, a parent, a mentor, or even a peer. No matter who they are, coaches are there to support their students, celebrate their victories, and help them find lessons in defeat.
Our goal is to empower students to take ownership of their own learning and to empower coaches to spend more time doing what they do best: personally interacting with every student, providing guidance and encouragement, and engaging students in collaborative activities.
Both students and coaches have access to dashboards, where they can view learning progress on Khan Academy. Students can track their own progress, and coaches can see how their students are doing.
Every Khan Academy account is both a "learner" account and a "coach" account, so all the features available to learners are also available to you as a coach. The best way to familiarize yourself with our learner features is to experience them for yourself, so feel free to open the "Subjects" menu and start exploring!
If you have questions or need help, visit our Help Center. You can find a link to this resource at the bottom of every page on Khan Academy, under SUPPORT. | 677.169 | 1 |
Numerical Analysis: Analyzing Root Finding Algorithms
In this root finding instructional activity, students compare graphs and shade in regions corresponding to given convergence criteria. They compute the solution of equations of one variable. This two-page instructional activity contains six problems. | 677.169 | 1 |
Written by best-selling author Pat McKeague, this text presents the topic using the same features that have successfully helped so many students learn algebra in his other titles. Approximately half of the book's 91 sections are in the first six chapters on beginning algebra. The material on intermediate algebra follows Chapter 7 | 677.169 | 1 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.