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Synopses & Reviews
Publisher Comments
Mat This book describes ideas of computer algebra and the language of the Mathematica system. It also contains a number of examples, mainly from physics, also from mathematics and chemistry. After reading this book and solving problems in it, the reader will be able to use Mathematica efficiently for solving his/her own problems.
Synopsis
This book uses examples from physical sciences to illustrate the value of Mathematica. After reading this book and solving problems in it, the reader will be able to use Mathematica efficiently for solving his/her own problems.
Synopsis
The basics of computer algebra and the language of Mathematica are described. This title will lead toward an understanding of Mathematica that allows the reader to solve problems in physics, mathematics, and chemistry. Mat
About the Author
Andrey Grozin is a leading researcher at the Budker Institute of Nuclear Physics in Novosibirsk, Russia. | 677.169 | 1 |
Calculators Throw Teachers a New Curve
By JENNIFER 8. LEE
Published: September 2, 1999
LIKE hundreds of thousands of other high school students, Greg Myers, 16, began using a graphing calculator in freshman algebra. Graphing calculators, which bear little resemblance to their 1970's ancestors, are sophisticated devices that can run small computer programs and draw the graph represented by complex equations in an instant. In the last few years, they have become mandatory in many high school math classes and can be used on the SAT and advanced placement exams and other standardized tests.
''I use graphing calculators all the time, and am a big enthusiast,'' Greg said. ''In my math class last year, I was known as the expert on graphing calculators.''
And like many of his high school peers, Greg has found another use for his calculator, a Texas Instruments TI-83. He has gone into business with his brother, Douglas, 17, writing 20 programs that can be fed into graphing calculators to help solve math problems in the SAT exam and SAT II subject tests.
''There is no work involved, you just run the program, plug in the known values, and watch your answers appear,'' says their Web site (members.aol.com/highersat). The brothers are charging $35 for the programs and guarantee that a student's score will rise by 160 points or more. They said they had not had many takers so far. ''Right now, we don't take credit cards,'' Greg said. ''It's got to be check, money order or cash. If things get better, we'll probably start accepting credit cards.''
Graphing calculators, which cost around $100, are lauded as the greatest innovation in math classrooms in a generation. Proponents say they lift students' confidence, cultivate their analytical adeptness and make mathematics more tangible. But they have also given students new ways to one-up their teachers and the rules. In response, adults have been forced to develop more sophisticated defenses, tests and regulations. Even while they solve difficult problems, graphing calculators have changed the political equation in the classroom.
Greg contends that his and his brother's programs are not cheating. ''Cheating is violating regulations unfairly, but our programs do not violate any regulations.''
Graphing calculator programs are allowed on College Board examinations. Teachers and test administrators say that the arrival of the calculators allow questions to focus more on problem solving, rather than plug-and-chug calculations, and they are skeptical about claims like the Myers brothers'.
The two brothers, who attend Princeton Day School, a private school in Princeton, N.J., said that they anticipated a potentially large market for their wares.
Texas Instruments, which dominates the market for high school graphing calculators, estimates that one-quarter to one-half of all high school math students use graphing calculators and that nearly all of them take the SAT I or SAT II. Hewlett-Packard and Casio have a small share of that market.
''These calculators are handheld computers,'' said Ellen Hook of Norfolk, Va., who has been teaching high school math for 20 years. ''They are marvelous. I can do more mathematics better now than I could 15 years ago. Students' critical thinking skills have changed.''
Since their introduction in high schools in the early 1990's, graphing calculators have not only pushed students to expand their mathematical reasoning skills but have also fired up their ingenuity.
Charles Sullivan, a 15-year-old student in Georgetown, Tex., used his Texas Instruments calculator to explore the structure of triangles in trigonometry. The algorithms that computed the intricate relations among cosines, angles and lengths gave him an appreciation for the elegance of math and programming. ''You can understand how math comes together more,'' Charles said.
But once he is convinced that he understands the math, he simply programs his calculator to do the number-crunching for him. ''In math class a lot of the homework is routine, you can do it over and over,'' he said. ''If you know it, you can whip out your calculator, write a program and press Enter a couple of times, and you've got your answer.''
The graphing calculator is also Charles's weapon against boredom. He keeps a number of games, like Space Invaders and Tetris, on his calculator, and he has no qualms about playing them in class.
Texas Instrument calculators are essentially Nintendo Game Boys in disguise -- they have a nearly identical microprocessor with less obtrusive packaging. ''If some kid brought a Game Boy into class, they'd be sent to the office, whereas a graphing calculator can't get you into trouble,'' Charles said.
There are now hundreds of games, including Pac-Man and the Legend of Zelda and versions of Mario Brothers, available for graphing calculators. The majority of them are written by students and are available free for downloading on and other sites on the Internet. The games can then be transferred to the calculators through a cable to the calculators. Students can also exchange games with each other by connecting their calculators, and it is possible to make the calculators communicate through infrared signals. | 677.169 | 1 |
Just the Essentials of Elementary Statistics (with InfoTrac and CD-ROM) / Edition 3
Overview mathematics. The text promotes learning, understanding, and motivation by presenting statistics in a context that relates to personal experiences. Statistics is presented as a useful tool in understanding the world around us through real world applications in areas such as business, economics, engineering, and the physical and natural sciences. | 677.169 | 1 |
Precise Calculator has arbitrary precision and can calculate with complex numbers, fractions, vectors and matrices. Has more than 150 mathematical functions and statistical functions and is programmable (if, goto, print, return, for).
Pc Calculator is a clever note and formula editor combined with an advanced and strong scientific calculator. Being an editor it is extremely user-friendly allowing all possible typing and other errors to be easily corrected and fast recalculated | 677.169 | 1 |
No Kindle device required. Download one of the Free Kindle apps to start reading Kindle books on your smartphone, tablet, and computer. Getting the download link through email is temporarily not available. Please check back later.
1. Ordinary Differential Equations.- 1.0. Ordinary Differential Equations.- 1.1. Example: Global Similarity Transformation, Invariance and Reduction to Quadrature.- 1.2. Simple Examples of Groups of Transformations; Abstract Definition.- 1.3. One-Parameter Group in the Plane.- 1.4. Proof That a One-Parameter Group Essentially Contains Only One Infinitesimal Transformation and Is Determined by It.- 1.5. Transformations; Symbol of the Infinitesimal Transformation U.- 1.6. Invariant Functions and Curves.- 1.7. Important Classes of Transformations.- 1.8. Applications to Differential Equations; Invariant Families of Curves.- 1.9. First-Order Differential Equations Which Admit a Group; Integrating Factor; Commutator.- 1.10. Geometric Interpretation of the Integrating Factor.- 1.11. Determination of First-Order Equations Which Admit a Given Group.- 1.12. One-Parameter Group in Three Variables; More Variables.- 1.13. Extended Transformation in the Plane.- 1.14. A Second Criterion That a First-Order Differential Equation Admits a Group.- 1.15. Construction of All Differential Equations of First-Order Which Admit a Given Group.- 1.16. Criterion That a Second-Order Differential Equation Admits a Group.- 1.17. Construction of All Differential Equations of Second-Order Which Admit a Given Group.- 1.18. Examples of Application of the Method.- 2. Partial Differential Equations.- 2.0. Partial Differential Equations.- 2.1. Formulation of Invariance for the Special Case of One dependent and Two Independent Variables.- 2.2. Formulation of Invariance in General.- 2.3. Fundamental Solution of the Heat Equation; Dimensional Analysis.- 2.4. Fundamental Solutions of Heat Equation Global Affinity.- 2.5. The Relationship Between the Use of Dimensional Analysis and Stretching Groups to Reduce the Number of Variables of a Partial Differential Equation.- 2.6. Use of Group Invariance to Obtain New Solutions from Given Solutions.- 2.7. The General Similarity Solution of the Heat Equation.- 2.8. Applications of the General Similarity Solution of the Heat Equation,.- 2.9. -Axially-Symmetric Wave Equation.- 2.10. Similarity Solutions of the One-Dimensional Fokker-Planck Equation.- 2.11. The Green's Function for an Instantaneous Line Particle Source Diffusing in a Gravitational Field and Under the Influence of a Linear Shear Wind ― An Example of a P.D.E. in Three Variables Invariant Under a Two-Parameter Group.- 2.12. Infinite Parameter Groups ― Derivation of the Poisson Kernel.- 2.13. Far Field of Transonic Flow.- 2.14. Nonlinear and Other Examples.- 2.15. Construction of Partial Differential Equations Invariant Under a Given Multi-parameter Group.- Appendix. Solution of Quasilinear First-Order Partial Differential Equations.- Bibliography. Part 1.- Bibliography. Part 2. | 677.169 | 1 |
Lessons for Polynomials
Exponential Functions - … of functions different from polynomials. With exponential functions the exponent varies while the base remains the same. With polynomial functions the base … User Rating:
Get Connected With Ohm's Law - … may lie a significant distance off the curve. More sophisticated polynomial equations can be derived which pass through all the data … may lie a significant distance off the curve. More sophisticated polynomial equations can be derived which pass through all the data … User Rating:Grade Level: 6-8
Leap Frog Review Game - … -defined functions, including step functions and absolute value functions. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing … . Students will be able to determine limits of rational and polynomial functions. Guiding Questions: What are the guiding questions for this … User Rating:Grade Level: 9-12 | 677.169 | 1 |
0883857 develops the mathematical skills necessary for the investigation of the increasingly pervasive codes used to identify everything from credit cards to books to supermarket goods. An important part of such codes is the check digit, which can point out when an error has been made, either accidentally in transmission, or deliberately in an attempt at fraud. Mathematics plays a crucial role in designing and verifying such check digit schemes. Beginning undergraduates and pre-university students should find this book both accessible and fun. In the process of describing the identification number schemes, the author introduces many fundamental areas of discrete mathematics, including modular arithmetic and group theory, with emphasis on their applications. A unique point of the book is the way it offers students the possibility of writing papers summarizing a given topic to improve their understanding. It also contains pointers into the literature for further | 677.169 | 1 |
, written by Mitchel E. Schultz, provides students and instructors with hundreds of practice problems for self-study, homework assignments, tests and review. The book is organized to correlate chapter by chapter with the textbook. Each chapter contains a number of solved illustrative problems demonstrating step-by-step how representative problems on a particular topic are solved. Following the solved problems are sets of problems for students to solve. | 677.169 | 1 |
Description: Calculus isn't just a required math class to weed out would-be science majors-it's a useful way to understand the patterns in physics, economics, and the natural world. With its distinctive mix of serious educational content and Japanese-style comics, The Manga Guide to Calculus will entertain you while it helps you understand the key concepts of calculus (and ace those exams).
Our story begins as Noriko, a recent liberal arts grad, arrives at a branch office of the Asagake Newspaper to start her career as a journalist. With the help of her overbearing and math-minded boss Kakeru, she's finally able to do some real reporting. But Noriko soon discovers the mathematical functions behind all the stories she struggles to cover.
How to integrate and differentiate trigonometric and other complicated functions
Multivariate calculus and partial differentiation
Taylor expansions
Reluctant calculus students of all abilities will enjoy following along with Noriko as she learns calculus from Kakeru's quirky stories and examples. This charming and easy-to-read guide also includes an appendix with answers to the book's many useful exercises.
This EduManga book is a translation from a bestselling series in Japan, co-published with Ohmsha, Ltd. of Tokyo, | 677.169 | 1 |
Subroutine Libraries
Created in the 1960s as an add-on to languages like Fortran, subroutine libraries were a common feature of early technical computing. But the release of Mathematica in 1988 ushered in a new era, in which technical computing could immediately be done in a single unified system, without having to assemble collections of subroutines.
After over two decades of growth, Mathematica now contains by far the world's largest integrated web of algorithms—including efficient versions of essentially every corner of functionality found in every standard subroutine library. And through new generations of algorithms made possible by its immediate ability to call on symbolic, geometric, discrete and other methods, Mathematica's algorithms routinely outperform the traditional algorithms found in subroutine libraries.
Mathematica dramatically simplifies access to sophisticated numerical methods—not only by providing an integrated interactive environment, but also through its unique methodology of automatic algorithm selection, which robustly selects optimal methods for particular problems, while allowing experts to choose specific named methods. | 677.169 | 1 |
SquareTrade 3-year Office Protection Plan ($50-$75Graphing calculator designed especially for grades 6-8. The TI-73 Explorer provides a larger screen alternative for teachers who feel limited by the two-line display of other fraction and scientific models.
Amazon.com
The Texas Instruments TI73 graphing calculator is designed for middle-grade students. It has a large screen to help students see patterns and analyze data. It features stacked fractions and data analysis functions that allow students to easily view and edit numeric and alphanumeric data in the list editor. They will be able to plot data in statistics plots including pie charts, pictographs, bar charts, scatter plots, histograms, and more.
This calculator is upgradable, allowing classrooms to keep up with changing curriculum for long-term value. Students and teachers can also download software from the Texas Instruments Web site.
The 64 x 96 screen allows students to explore data, simultaneously see entries and results, and visualize patterns. The TI73 has 32 KB of RAM with 25 KB available for the user and 64 KB for application space.
This calculator is made especially durable with an impact-resistant slide case. It comes with a unit-to-unit link cable, a manual, and a one-year limited warranty.
Top Customer Reviews
This is a great calculator for a Middle School/High School Student. There are many functions on this calculator that are really useful. Also, there are mini math games, such as probability, etc. Overall, this is a great calculator if you're looking for a useful, long-lasting calculator because this is one of the best calculators around that I have used. Overall, I would recommend this product, as it has a very good quality.
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This calculator is fine, and TI is certainly the industry standard in graphing calculators...so not much competition there. But I would definitely just pay the extra money for the TI-83 (I had a Plus...but being 22 years old, I am a dinosaur, so can't comment on the Silver edition and what not). Assuming the student doesn't break it, this is the standard through high school and college, so why buy an $80 calculator now, only to purchase another one for $100 in 2 years? It's best to get used to your calculator so you can be efficient on exams, etc...and at some point in high school, the 83 will definitely be much more useful than the 73, if not required.
I had an 83 from 6th grade until my first year of college, when I dropped and broke it. Upgraded to an 89...it helped me out a few times on projects when I didn't feel like calculating out long integrals or derivatives by hand, but didn't help me out on any tests or anything, if that's what you're looking for (most college exams are taken sans calculator - I majored in math).
If your student is at middle school age, I would advise getting the 83 for: 1) increased functionality over the 73, 2) less complicated interface than the 89, 3) most universally used, 4) interesting (but not overly complicated) programming interface, if the student might be into that. I used to program small applications on my 83 in middle school, but when I bought the 89 in college I never took the time to relearn the interface.
Probably best to buy a TI-83 now, and if desired upgrade to the 89 when you hit college, or just stick with the 83 (like most of your fellow students will)
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The TI 73 is the best. At least in my opinion, it is the best for all the people in Middle School. If you are in Middle school at this moment and you need a graphing calculator for math or science, don't go out and spend alot of money on another kind of calculator. The TI 73 is fit for the job!
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Not only did I get my product faster than I expected to, I got it with fresh batteries and replacement batteries. I needed this calculator for college but did not want to buy the TI-84 model because of how expensive it was. I have been very happy this this and the merchant that old this.
So far, this has been great for my Precalculus class. I paid $50 to more that $100 less for it than my classmates paid for theirs. I'm happy so far! (I've only been using it for three weeks...) If I need to revise this opinion, I will... but as of this moment I'm happy with it.
My daughter is a 7th grade student and this is the Calculator her Teacher recommends, it is also the one her School uses.My daughter loves this Calculator for School and home use and would highly recommend it for Students. | 677.169 | 1 |
This course deals with the theoretical treatment of the foundations
of calculus. Topics covered include the real number system, number sequences
and series, continuity, differentiation, Riemann integration, and sequences
& series of functions.
OBJECTIVES:
The purpose of this course is to provide students an opportunity for
a rigorous treatment of analysis. The intention is to effect a transition
from elementary calculus where a considerable part of the effort is necessarily
devoted to mastering the technical aspects of differentiation and integration.
The emphasis thus shifts to the development of concepts and methods of
proof. The student should be adequately prepared by this course to continue
with other branches of analysis such as complex variable theory and topology.
CONTENT:
1. Properties of the Real Numbers
2.1 The complete ordered field definition
2.2 Denseness of the rationals and irrationals
2.3 Countable and uncountable sets
2.4 The completeness property
8.1 Pointwise and uniform convergence of sequences
of functions 8.2 Consequences of uniform convergence 8.3 Convergence and uniform convergence of series
of functions 8.4 Differentiation and integration of series of
functions | 677.169 | 1 |
A Tour of the Calculus
Overview meditation on the theme of continuity. Charts and graphs throughout.
From the Hardcover edition.
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Editorial Reviews
Library Journal
Berlinski (Black Mischief: The Mechanics of Modern Science, LJ 2/15/86) presents an unconventional work on the foundations of calculus. It is in part an informal history of the subject, the author inrerweaves the historical fragments with expository sections that explain the concepts from a modern viewpoint. He gives special attention (very appropriately) to the concept of limits and to several of the fundamental theorems that underpin calculus. He also shows how differential calculus deals with rates of change and how integral calculus works to determine areas under curves. Writing in a breezingly informal style, the author includes a plethora of humorous asides as well as a number of clearly fictitious anecdotes. At times his prose gets a bit too ripe, but the overall effect is to make the book quite readable. The work should be especially useful for providing perspective to college and advanced high school students currently learning calculus. Recommended for all public and college libraries.-Jack W. Weigel, Univ. of Michigan Lib., Ann Arbor
Gilbert Taylor
Even those who flailed through calculus class sense the power and perfection of that branch of mathematics, and Berlinski rekindles the interest of lapsed students in this pleasing excursion through graphs and equations. Berlinski's goal is to explain the mystery of motion and the area and volume of irregular shapes, issues that gave rise to Leibnitz and Newton's invention of calculus. He makes his points one concept at a time, but not so dryly as asking and answering, "What is a function?" No, with dashes of biography or images of his walking around old Prague (to illustrate continuity), Berlinski tangibly grounds the abstract notions, so that attentive readers can ease into and grasp the several full-blown proofs he sets forth, as of the mean-value theorem. Though the math-shy won't necessarily jump to the blackboard to begin differentiating and integrating polynomial equations, Berlinski's animated presentation should tempt them to sit forward and appreciate the elegance of calculus--and perhaps banish recollections of its exam-time terrors.
Most Helpful Customer Reviews
...read this book! Undoubtedly the best book on mathematics I've read. Hugely entertaining in its whimsical style. Read it if you want to know exactly WHY you are studying calculus.
Guest
More than 1 year ago
This book is for ANYONE interested in math. I am 14 and I LOVED this book. Now I am looking for some real calculus textbooks. A must read.
Guest
More than 1 year ago
I have read, and re-read 'A Tour of the Calculus' several times; each time gleaning something new are more subtle than the last. Berlinski has extracted and distilled the essence of the alculus, the continuum, and indeed that of 'real' numbers and functions. I was inspired by this book to revisit elementary analytic functions, this time from an elevated view. How function approximation leads to orthogonal function spaces to fourier series/transform (a special case) to wavelet transforms (the true general case). An outstanding work by Berlinski; at times I laughed out loud when reading the anectdotes, especial about the waiter in Prague - 'nicht da'. Doesn't exist... I heartily recommend 'A Tour of the Calculus'; if you don't find it enlightening and entertaining, you're probably a college professor at a public university :o).
Anonymous
More than 1 year ago
Jorufo
More than 1 year ago
I LOVE math. But this book was more prose than interesting content.
Hlance50
More than 1 year ago
As an old but devoted student of mathematics and science, I was easily drawn to this book by title and synopsis. Neither hinted at what a master of language, of storytelling, of visual creation Mr. Berlinski is. In addition to a wonderful tour, a revisiting of the calculus I enjoyed in my youth, he provided the gift of pleasurable excursions in history, biography, and philosophy. I was brokenhearted when the last page was turned.
Anonymous
More than 1 year ago
Guest
More than 1 year ago
In college thirty years ago, I took a notorious 'gut' course, 'math for poets'. I never thought I would end up reading a book about math written BY a poet...Berlinski is absolutely fabulous, explaining the relationship between calculus and real world applications with great literacy and humor. Make no mistake - this book is NOT a simplification of calculus (I doubt such a thing would be possible). Rather, it is, as its title states, a 'tour', a fanciful, insightful, reasonably accessible introduction to calculus and an appreciation of its power and beauty. And that beauty is what this book is all about. | 677.169 | 1 |
GRE Subject Mathematics
The test consists of approximately 66 multiple-choice questions drawn from courses commonly offered at the undergraduate level.
Approximately 50 percent of the questions involve calculus and its applications — subject matter that can be assumed to be common to the backgrounds of almost all mathematics majors.
About 25 percent of the questions in the test are in elementary algebra, linear algebra, abstract algebra and number theory. The remaining questions deal with other areas of mathematics currently studied by undergraduates in many institutions.
The following content descriptions may assist students in preparing for the test. The percents given are estimates; actual percents will vary somewhat from one edition of the test to another.
CALCULUS — 50%
Material learned in the usual sequence of elementary calculus courses — differential and integral calculus of one and of several variables — includes calculus-based applications and connections with coordinate geometry, trigonometry, differential equations and other branches of mathematics.
ALGEBRA — 25%
Elementary algebra: basic algebraic techniques and manipulations acquired in high school and used throughout mathematics
Other topics: general topology, geometry, complex variables, probability and statistics, and numerical analysis
The above descriptions of topics covered in the test should not be considered exhaustive; it is necessary to understand many other related concepts. Prospective test takers should be aware that questions requiring no more than a good precalculus background may be quite challenging; such questions can be among the most difficult questions on the test. In general, the questions are intended not only to test recall of information but also to assess test takers' understanding of fundamental concepts and the ability to apply those concepts in various situations | 677.169 | 1 |
Clone this wiki locally
Using Modules
Depending upon the type of exercise that you're writing you'll likely need to load one or more utility modules in order to render the exercise correctly. This can be done by specifying a data-require="..." attribute on the HTML element of the page.
There are an ever-increasing number of modules (you can find them in the utils/ directory) but the ones that you'll most-likely need are:
There are also utils for angles, calculus, exponents, kinematics, probability, etc.
We just started working on documenting all of the utility functions. A great way to help out and become more familiar
with the modules is to help us expand the documentation by editing the reference wiki. Feel free to adjust
formatting, re-organize, whatever makes it easier for everyone to find the utility modules they need. | 677.169 | 1 |
Depends. Many years ago when I was working at ChemDept of Warsaw University we had a deal with Math Dept - we were giving them differential equations and they were trying to solve them. If they did - they were listed as authors of electrochemistry papers, not knowing the difference between electrode and solution
I was also working with a mathematician who was involved in using FEM for simulation of diffusion, and he had some quite interesting results. One day he came to the lab and asked what is this silvery liquid in the bottle. Mercury? So it is liquid? Wow! Problem was nobody knew where he should get his PhD from - for sure not as a chemist, and for mathematicians his work was not worth much. It was too practical, and his papers were full of number with decimal digits. Horrible.
So - sometimes you can do interesting things not knowing what you are doing in chemical terms. But to understand what you are doing you better start with GenChem101.
Do you, or anyone else, have any recommendations for books that are more mathematically intense. I have a pretty strong background in mathematics though I have never taken a chemistry class before.
University Chemistry by Peter Siska is the most mathematically intense general chemistry book I have ever read. At least calc1, maybe calc2 mastery necessary to understand. Thus, I'm not a fan of it at all, but it would be what you're looking for.
There are couple of basic chemistry books in amazon that are pretty useful for those that would like to get the interest in Chemistry back. This is what i recommend my students as well as their parents (parents in Asia are very anxious about their child's education). Read through the books listed below and find them pretty easy to digest.
It's pretty good, goes over the concepts well, and is decently math based with a few errors.
I'm using the 6th edition, and I'm not too crazy about it. I can't put my finger on it, but it seems to be missing something. Slightly off topic, but they're supposed to be coming out with a 7th edition before the next semester.
I find it very easy to read and very informative, although it has some incorrect facts such as flawed reaction mechanisms, but your professor should be able to point you to where those are. Absolutely get the study manual - it helps a lot.
General Chemistry by Linus Pauling seems a bit more advanced than what we covered in Gen Chem. It is quite a nice book to have around though, I've read through a lot of it. In school, we used Zumdahl and Zumdahl's "Chemistry". The book itself wasn't bad, but what really made it shine was the companion website. It has videos for every chapter explaining each important topic from the ground up. It was especially helpful for the later chapters on ligand field theory and main group elements. It also has videos of fun reactions dealing with each section of the book. I still get on there from time to time... | 677.169 | 1 |
Find a GriffithFrom fractions and ratios, to integers, decimals, factorization, negative numbers, natural numbers. All that good stuff that makes mathematics work! This subject helps build a foundation in algebra and trigonometry to prepare you for calculus, which uses these concepts and expects a certain level of competence in them. | 677.169 | 1 |
Details about Barron's FCAT High School Math:
This updated manual prepares students in Florida high schools to pass the required state-administered test in math. It presents a diagnostic test and two full-length practice tests modeled on the most recent FCAT math test. All questions are answered and explained. The author also presents an extensive math review that covers current Sunshine State Standards. They include: Real and Complex Number Systems, Relations and Functions, Linear Equations and Inequalities, Polynomials, Rational Expressions and Equations, Radical Expressions and Equations, Mathematical Reasoning and Problem Solving, and others.
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Rent Barron's FCAT High School Math 2nd edition today, or search our site for other textbooks by Pamela Windspirit | 677.169 | 1 |
Introduction to graph theory
Browse related Subjects ...
Read More introduction to the subject for non-mathematicians. The opening chapters provide a basic foundation course, containing definitions and examples, connectedness, Eulerian and Hamiltonian paths and cycles, and trees, with a range of applications. This is followed by two chapters on planar graphs and colouring, with special reference to the four-colour theorem. The next chapter deals with transversal theory and connectivity, with applications to network flows. A final chapter on matroid theory ties together material from earlier chapters, and an appendix discusses algorithms and their efficiency 0582249937 | 677.169 | 1 |
It was a long time ago, but I believe my O-level maths included surds and lots of geometric theorems as well as much more algebra than now. The modern Free Standing Additional Maths qualification is similar in content to O-level. We had a different exam called Additional Maths which counted as an O-level but was usually taken in the lower sixth; it roughly corresponds to a modern AS.
Another difference now is that A-level Physics has less maths. I remember reading a few years ago that a physics teacher took his A-level class to a university open day, where they were treated to a typical first-year lecture. He was shocked at how quickly the lecturer moved from description to maths. A physics teacher of my day would not have been shocked by this; he would have expected it as he would have been doing exactly the same with his A-level class. Physics without maths isn't physics.
The big jump in my day was from 40 minute school lessons with supplied notes to hour long lectures with no notes. Nowadays notes are provided and module 'Learning Outcomes' tell you exactly what you need to learn.
Our lecturers are fairly good with notes (or lack of) - many keep them minimal (or non-existent) so that we're forced to attend if we want a decent grade.
Physics can be rather fun, but some of the assignments aren't - attached is an example: the average score was in the low 40s.
I definitely agree WRT A-level Physics. Ideally, for students taking maths and physics, they'd teach the entire A-level maths course in 1 year, then go through the physics in the second year, with the advantage of understanding what's going on with the equations, instead of being told that it works, so don't ask too many questions
Those assignments look rather like the stuff which used to appear in the A-level S paper - uses A-level content but requires deeper understanding rather than merely plugging numbers into formulae. Is that a school or university assignment?
Only my thoughts, where is the innovation...the free (Constructive) thinkers? There are plenty of (destructive)thinkers..who is going to drive the human race forward..It seems we do well if we manage to survive in this money driven world...I'm thinking of the innovation and free thinking..it seems to be held down and controlled...Ie find a job and earn what you can..
Think about invention...rather than survival..Education for what purpose?Well,
The way I see it is the first kind is important..the second kind who understand the first kind and refer to imagination gives the drive for discovery..ie the science fiction Flat TV...the idea was not possible to put in practice..however it was put in scifi<<<the idea was the driving force for the real thing..lets look a bit further..beam be up scotty<<never possible we assume. Travel faster than light speed also not possible we assume.
I have to confess that I've never been taught about surds, nor even seen the term before. Could you please enlighten me | 677.169 | 1 |
Introductory Mathematics 2
Description
This course aims to consolidate and extend the topics covered in Introductory Mathematics 1. The course includes basic skills and their application to problem solving in the topics of linear function, graphing, probability and statistics.
Availability
Callaghan
Semester 2 - 2016
Ourimbah
Semester 2 - 2016
On-Line from Callaghan
Semester 2 - 2016
Learning Outcomes
On successful completion of the course students will be able to:
1. Students will be proficient in their understanding of appropriate mathematical techniques when commencing undergraduate study.
2. Students will display competency, working individually and in groups.
3. Students will have acquired critical reasoning and problem solving skills in order to solve mathematical problems.
Content
The course content may include:
Graphing linear relationships: The number plane, line y = mx + b, gradient, intercept, line of best fit, application to other variables with linear relationships. | 677.169 | 1 |
Description Students will read, write, and interpret mathematical information by measuring whole numbers and extending skills in fractions, decimals, ratios, and percents. Students will use basic patterns, data, algebraic concepts, measurement, geometry, and computational skills to solve 1-3 step theme based contextualized word problems. This course will integrate these skills into contextualized units. Students completing this course with a 70% or better will meet the requirements of High School 21+ to earn 1 credit of high school math equivalency. All students who are under 18 years of age must have a signed release form from their school district. Students 16-17 years of age must first be admitted to the College following the Alternative Education Program (AEP) Eligibility or the underage admissions policy, which is available in the High School Programs office. Prerequisite: Grade of C or higher in ABE 013, CASAS Score 220, or instructor permission.
Intended
Skills and Attitude Outcomes
A. Students demonstrate proficiency at a level of 70% or better in ratio and proportion, percents, and geometry.
B. Students complete all Washington State Core Competencies for computation, levels 3-4. | 677.169 | 1 |
Course Description and Purpose: This course is designed for students interested in preparing for college level math courses. Students will be exposed to topics involving functions, trigonometry, and other concepts to help prepare them for calculus.
Required Material: Graphing Calculator- We recommend the TI 84+. If you are not able to purchase a new one, there are other options available.
Power Standards:
Students will identify and apply various strategies and different skill sets to solve problems.
Students will compare and contrast real, complex, vector, polar and parametric numbers.
Students will develop expressions, equations, and inequalities and use a variety of techniques to simplify/solve them.
Students will create, analyze, and interpret functions in order to compare relationships between their variables.
Students will create a model to predict appropriate behavior.
Students will apply their knowledge of geometry to prove and utilize trigonometric identities and applications.
Students will distinguish between counting methods to compute
desired quantities.
Assessment & Grading
Formative Assessments:
10% of the overall grade
May include homework assignments, in class assignments, and quizzes
Summative Assessments:
90% of the overall grade
May include unit tests and projects
Meaning of Grade
97
93
90
A+
A
A-
Student has demonstrated a detailed understanding of the course content, skills, and processes. The student can apply knowledge and skills to new situations.
87
83
80
B+
B
B-
Student has demonstrated a complete understanding of the course content, skills, and processes. The student can use their knowledge and skills in a variety of ways.
77
73
70
C+
C
C-
Student has demonstrated a basic understanding of power standards.
67
63
60
D+
D
D-
Student's understanding of power standards show many misconceptions or is incomplete. Student makes many errors in performing skills and processes.
Below 60
F
Student's level of mastery is insufficient and shows little evidence of understanding power standards based on course material.
Reassessment Policy:
All homework for that assessment must be completed prior to the retake of that material.
Student must meet with the teacher to review the first assessment errors prior to the retake. This meeting must take place outside the traditional class time. | 677.169 | 1 |
From angles to functions to identities - solve trig equations with ease Got a grasp on the terms and concepts you need to know, but get lost halfway through a problem or worse yet, not know where to begin? No fear - this hands-on-guide focuses on helping you solve the many types of trigonometry equations you encounter in a focused, step-by-step... more...
If you're preparing for The American Institute of Professional Bookkeepers' (AIPB) bookkeeping certification test, you need an easy- to-follow test-preparation guide that gets you up to speed quickly in all of the bookkeeping basics, from setting up a company's books and recording transactions to managing employee payroll, handling government paperwork,... more...
From genetics to ecology — the easy way to score higher in biology Are you a student baffled by biology? You're not alone. With the help of Biology Workbook For Dummies you'll quickly and painlessly get a grip on complex biology concepts and unlock the mysteries of this fascinating and ever-evolving field of study. Whether used as a complement... more...
Gruber's Complete SAT Math Workbook includes shortcuts and tips, strategies and practice tests. It also features the exclusive Gruber system that has raised actual SAT scores by more than 600 points! more...
Helpful instruction and plenty of practice for your child to master the basics of vocabulary Having a strong grasp of vocabulary is essential for your child to read and write with confidence. Practice Makes Perfect: Mastering Vocabulary gives your child bite-sized explanations of the subject, with engaging exercises that keep her or him motivated... more...
Balance the books -- without all the headaches! Trying to get certified and become an accountant? Own a small business but need a little help balancing your books? Don't worry! This hands-on guide provides the learning and vital practice you need to master important accounting concepts and basics. Perfect as a companion workbook for Accounting For... more...
Helpful instruction and plenty of practice for your child to understand the basics of vocabulary Understanding vocabulary is essential for your child to read and write with confidence. Practice Makes Perfect: Exploring Vocabulary gives your child bite-sized explanations of the subject, with engaging exercises that keep her or him motivated and... more...
Helpful instruction and plenty of practice for your child to understand the basics of multiplication and division Understanding multiplying and dividing is essential for your child to do math problems with confidence. Practice Makes Perfect: Multiplication and Division gives your child bite-sized explanations of the subjects, with engaging exercises... more... | 677.169 | 1 |
...
Show More points and in a handy appendix. The Basics of Practical Optimization presents step-by-step solutions for five prototypical examples that fit the general optimization model, along with instruction on using numerical methods to solve models and making informed use of the results. It also includes information on how to optimize while adjusting the method to accommodate various practical concerns; three fundamentally different approaches to optimizing functions under constraints; and ways to handle the special case when the variables are integers. The author provides four levels of learn-by-doing activities through the book: Exercises meant to be attempted as they are encountered and that are short enough for in-class use; Problems for lengthier in-class work or homework; Computational Problems for homework or a computer lab session; and Implementations usable as collaborative activities in the computer lab over extended periods of time The accompanying Web site offers the Mathematica notebooks that support the Implementations. Audience: This textbook is appropriate for undergraduate students who have taken a multivariable calculus course. Contents: List of Figures; List of Tables; Preface; Chapter 1: Modeling; Chapter 2: Impractical Optimization; Chapter 3: Basic Practical Optimization; Chapter 4: Some Practical Modifications; Chapter 5: How Methods Are Ranked; Chapter 6: Constraints; Chapter 7: More Practical Modifications; Chapter 8: Integer Variables; Chapter 9: Other Methods; Appendix of Asides; Bibliography; Index | 677.169 | 1 |
Overview
• UNIQUE! Full-color format highlights key information on setting up problems, understanding parts of equations, moving decimal points, and more. • Spiral bound format with plenty of white space allows you to use the text as a workbook in which you can write your answers and work out problems. • Consistent chapter formats make it easy to retain information and identify important content. • Chapter objectives emphasize what you should learn from each chapter and how your knowledge applies to patient care. • Key terms defined at the beginning of each chapter help you understand new vocabulary in the text. • Chapter overviews introduce you to the topics discussed in the chapter. • Example problems demonstrate and label each step to getting a solution and show you how to solve similar problems. • Practice the Skill problems incorporated within the chapter for in-class discussion allow you to practice what you've learned before receiving homework assignments. • Math in the Real World boxes include word problems that apply your knowledge to everyday life as well as common healthcare situations. • Strategy boxes demonstrate the steps to solving topic problems and provide a helpful example for solving more problems. • Human Error boxes include hints on common errors and show you how to double-check your answers. • Math Etiquette boxes help you solve problems by presenting proper math rules. • Chapter quizzes allow you to assess your learning and identify areas for further study. | 677.169 | 1 |
Find a Hunters Creek Village, TX AlThomas and received an A in the course. Linear Algebra is the study of matrices and their properties. The applications for linear algebra are far reaching whether you want to continue studying advanced algebra or computer science | 677.169 | 1 |
Popular paid resources
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This is a powerpoint covering all areas of Core 2. It contains brief notes by way of an explanation, model answers to questions and a question or two for the students to do; all of the questions come with answers that you can display when ready. The slide show comes with a progress grid (regularly referred to in the presentation) so that students can mark their progress from start to finish and pinpoint any areas that may need extra work with a "red/amber/green" system that they fill in. It's what I use in my lessons before setting tasks from worksheets or text books to practice. If there are any silly errors then let me know so I can correct them but I have been through the entire presentation with students who are keen to point out mistakes! | 677.169 | 1 |
Shelley's Mathematics Articles - Shelley Walsh
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Sieve of Eratosthenes - Program in C
A class assignment for Mathematics 216: Introduction to Mathematical Computation (with C) at the Univ. of Utah. The new part of the C language learned was the array. Problems: Design a program to make a list of primes less than N = 100 using Eratosthenes'
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Statistical Forecasting - Robert Nau
Lecture notes and teaching materials for a course on statistical forecasting, with particular focus on regression and time series models. Output from Statgraphics and RegressIt illustrate sections such as "Get to know your data" (famous forecasting quotes,Theron J. Hitchman
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Trigonometric Integrals - David Hart
An HTML version of a Maple worksheet used as an alternative to transparencies in a Calculus class lesson at Indiana University. How do you integrate the trigonometric functions? Includes discussion, formulae, diagrams, and exercises.
...more>> | 677.169 | 1 |
College Algebra
Credits: 3Catalog #20804212
College Algebra includes fundamental topics covered in Intermediate Algebra with a more careful look at the mathematical details and a greater emphasis on the concept of function. It covers quadratic, polynomial, rational, exponential and logarithmic functions, equations and inequalities; the use of matrices and determinants in solving linear systems of equations, solving non-linear systems; sequences and series.
Course Offerings
last updated: 03:01:18Hybrid courses combine the flexibility of online learning with in-class meetings. In-class time is reduced for hybrid courses because a significant amount of the course is taught online. Online expectations might include: online discussions, small group work, self-testing exercises, and audio or video lectures. To be successful in a Hybrid course, you should:
1. Be highly motivated, self-disciplined, and technologically-literate.
2. Plan for increased online activity as you add online course learning to the already expected outside homework requirements.
3. Go to the Blackboard site, and ensure you can log in (using your Madison College username and password) approximately one week prior to the start of the class. If you are experiencing trouble with Blackboard, please contact the Student Help Desk at (608) 243-4444.
4. Activate your Madison College email account ( and check it at least one week prior to the start of the class.
This course may make use of an online homework system. Your instructor may choose to use a variety of teaching methods and tools including 'online math software, active learning principles, supplemental video lessons, etc..' Please contact your instructor | 677.169 | 1 |
Structured as a dialogue between a mathematician and a physicist, Symmetry and Quantum Mechanics unites the mathematical topics of this field into a compelling and physically-motivated narrative that focuses on the central role of symmetry.
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The second edition of this book has a new title that more accurately reflects the table of contents. Over the past few years, many new results have been proven in the field of partial differential equations. This edition takes those new results into account, in particular the study of nonautonomous………
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In knot theory, diagrams of a given canonical genus can be described by means of a finite number of patterns ("generators"). Diagram Genus, Generators and Applications presents a self-contained account of the canonical genus: the genus of knot diagrams. The author explores recent research on the…
Submanifolds and Holonomy, Second Edition explores recent progress in the submanifold geometry of space forms, including new methods based on the holonomy of the normal connection. This second edition reflects many developments that have occurred since the publication of its popular predecessor.…
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This book provides an introduction to functional analysis for non-experts in mathematics. As such, it is distinct from most other books on the subject that are intended for mathematicians. Concepts are explained concisely with visual materials, making it accessible for those unfamiliar with…
An Introduction to the Fundamentals
Ordinary Differential Equations: An Introduction to the Fundamentals is a rigorous yet remarkably accessible textbook ideal for an introductory course in ordinary differential equations. Providing a useful resource both in and out of the classroom, the text:
Employs a unique expository style that…
Functions of a Complex Variable provides all the material for a course on the theory of functions of a complex variable at the senior undergraduate and beginning graduate level. Also suitable for self-study, the book covers every topic essential to training students in complex analysis. It also… | 677.169 | 1 |
ACT Math Prep
Everything you get from a traditional test prep book PLUS one-on-one instruction covering all the math on the ACT.**"I just love YourTeacher and the way you explain things. I felt like I was in a classroom instead of just looking at examples."Diane"I am needing prep work for my daughter's ACT. It's been very helpful and I do like how it's set up. The lessons are very clear and understandable - I wish I had this when I was in school!!"Ann"My daughter, who's a junior, is using Your Teacher to prepare for the ACT in the spring. She loves the interactive, go at your on pace style of Your Teacher. I absolutely love the program and certainly would and have recommended it."Sonya**Need more than practice problems to get ready for the ACT…YourTeacher's ACT Math Prep Course covers the exact math you need to know to ace the ACT. Our app includes everything you would get in a traditional ACT test prep course (i.e. practice problems), but ALSO includes the one-on-one instruction you need to truly learn math.Our lessons include:-Multiple video example problems(similar to how a teacher starts class at the board by explaining the examples from the textbook)-Interactive practice problems with built-in support(similar to how a teacher assigns practice and walks around the class providing help)-A Challenge Problem(similar to how a teacher assigns a higher level problem which students must work on their own to prove mastery)-Extra problem worksheets(similar to how a teacher assigns additional problems for homework)-Review notes(similar to how a teacher provides summary handouts or refers you to your textbook)******ACT Math Prep App - Scope and SequenceYourTeacher's ACT Math Prep App covers the exact math you need to ace the ACT. CHAPTER 1: INTEGERSAddition and SubtractionMultiplication and DivisionOrder of OperationsEvaluationAbsolute ValueCHAPTER 2: FRACTIONSDivisibility RulesFactors and PrimesPrime FactorizationMultiples and Least Common MultiplesGreatest Common FactorIntroduction to FractionsEquivalent Fractions (Part I)Lowest TermsEquivalent Fractions (Part II)Improper Fractions and Mixed NumbersComparing Proper FractionsComparing Mixed Numbers and Improper FractionsLeast Common MultipleAddition and SubtractionMultiplication and DivisionOrder of OperationsEvaluationAbsolute ValueCHAPTER 3: DECIMALSUnderstanding DecimalsComparing DecimalsRounding DecimalsLong DivisionDividing Decimals by Whole NumbersDividing Decimals by DecimalsPowers of TenConverting from Scientific to Standard NotationConverting from Standard to Scientific NotationCHAPTER 4: EXPRESSIONS AND EQUATIONSCombining Like TermsDistributive PropertyDistributive / Like TermsOne-Step EquationsTwo-Step EquationsEquations with FractionsEquations Involving DistributiveVariable on Both SidesVariable on Both Sides / FractionsVariable on Both Sides / DistributiveEquations with DecimalsEquations with Fractional SolutionsBeginning FormulasAdvanced FormulasCHAPTER 5: WORD PROBLEMSModeling Two-Step EquationsNumber ProblemsConsecutive Integer ProblemsGeometry ProblemsValue ProblemsIntroductory Motion ProblemsAdvanced Motion ProblemsCHAPTER 6: INEQUALITIES, ABSOLUTE VALUE, FUNCTIONS, & GRAPHINGCHAPTER 7: LINEAR EQUATIONSCHAPTER 8: SYSTEMS OF EQUATIONSCHAPTER 9: EXPONENTS & POLYNOMIALSCHAPTER 10: FACTORINGCHAPTER 11: RATIONAL EXPRESSIONS & EQUATIONSCHAPTER 12: RADICALS & QUADRATICSCHAPTER 13: PROBABILITY & STATISTICSCHAPTER 14: INTRODUCTION TO GEOMETRYCHAPTER 15: PARALLEL LINES & POLYGONSCHAPTER 16: RATIO, PROPORTION, & PERCENTCHAPTER 17: RIGHT TRIANGLESCHAPTER 18: CIRCLESCHAPTER 19: MEASUREMENTCHAPTER 20: ADVANCED AREA & VOLUME(Wifi or 3G connection required)ACT is the registered trademark of ACT, Inc. YourTeacher has no affiliation with ACT, Inc., and the YourTeacher ACT app is not approved or endorsed by ACT | 677.169 | 1 |
Precalculus,the authors encourage graphical, numerical, and algebraic modeling of functions as well as a focus on problem solving, conceptual understanding, and facility with technology. They have created a book that is designed for instructors and written for students making this the most effective precalculus text available today. Contents: P. Prerequisites 1. Functions and Graphs 2. Polynomial, Power, and Rational Functions 3. Exponential, Logistic, and Logarithmic Functions 4. Trigonometric Functions 5. Analytic Trigonometry 6. Applications of Trigonometry 7. Systems and Matrices 8. Analytic Geometry in Two and Three Dimensions 9. Discrete Mathematics 10. An Introduction to Calculus: Limits, Derivatives, and Integrals Appendix A: Algebra Review Appendix B: Key Formulas Appendix C: Logic | 677.169 | 1 |
step-by-step tutorials and guided inquiry activities that are designed to help students discover physical properties and concepts by developing understanding from mathematics. Problem-solving techniques and equation-solving are also discussed. Most of this material is written for upper level classical mechanics courses, but some are meant for other courses. Connections are made with the Intermediate Mechanics Tutorials by Ambrose and Wittmann.
Supplementary homework sets are available for some labs. All materials are available in Word and PDF formats.
Disclaimer: ComPADRE offers citation styles as a guide only. We cannot offer interpretations about citations as this is an automated procedure. Please refer to the style manuals in the Citation Source Information area for clarifications. | 677.169 | 1 |
Itasca, IL AlgebraEdward P.
...Finally, algebra 2 typically ends with complex numbers, logarithmic and exponential functions. When | 677.169 | 1 |
Palos Verdes Peninsula CalculusElena B.
...No new concept is totally independent but an extension of a previous one. Students should not memorize formulas but understand where formulas are coming from. In this process, the brain gets trained for future problem solving situations | 677.169 | 1 |
Precise Calculator has arbitrary precision and can calculate with complex numbers, fractions, vectors and matrices. Has more than 150 mathematical functions and statistical functions and is programmable (if, goto, print, return, for). | 677.169 | 1 |
11/03/2011
A course in combinatorics online
A course in combinatorics book download
a valuable book. van. Mathematics Books: A Course in Combinatorics (Paperback) Review & Description. ' The Times Higher Education Supplement Book Description This major textbook, a product of many years' teaching, will appeal to all teachers. administer such a course This book has been particularly designed to serve as a training manual for program- 6 Combinatorics. a course in combinatorics manual PDF Download Download a course in combinatorics manual. 129. '. a course in Combinatorics eBook Downloads a course in Combinatorics free PDF ebook downloads. Paperback. A+Course+in+Combinatorics - Scribd This page intentionally left blank A course in combinatorics This is the second edition of a popular book on combinatorics, a subject dealing with ways of arranging. Amazon.com: A Course in Combinatorics (9780521006019): J. 9780521006019 - AbeBooks: 0521006015 Book Description: Cambridge University Press. Alibris has A Course in Combinatorics and other books by J H Van Lint, R M Wilson, including new & used copies, rare, out-of-print signed editions, and more. A Course in Combinatorics - Cambridge University Press This is the second edition of a popular book on combinatorics, a subject dealing with ways of arranging and distributing objects, and which involves ideas from | 677.169 | 1 |
was written with the goal of having students succeed in this course, and gain a foundation to succeed in further mathematics courses. To that end, the authors have written a text with a theme (showing the connections between the zeros, x-intercepts, and solutions), with a series of side-by-side features (designed to show examples being solved algebraically and graphically), and with the knowledge that many students are using graphing technology to help them learn the key concepts in this course (and so the book automatically comes bundled with a free graphing calculator manual). Thus, the approach of this text is more interactive than most texts and the authors feel that, accordingly, more students will succeed in this course. | 677.169 | 1 |
Details about Basic College Mathematics:
KEY BENEFIT: The Bittinger Worktext Series changed the face of developmental education with the introduction of objective-based worktexts that presented math one concept at a time. This approach allowed students to understand the rationale behind each concept before practicing the associated skills and then moving on to the next topic. With this revision, Marv Bittinger continues to focus on building success through conceptual understanding, while also supporting students with quality applications, exercises, and new review and study materials to help students apply and retain their knowledge. KEY TOPICS: Whole Numbers; Fraction Notation: Multiplication and Division; Fraction Notation and Mixed Numerals; Decimal Notation; Ratio and Proportion; Percent Notation; Data, Graphs, and Statistics; Measurement; Geometry; Real Numbers; Algebra: Solving Equations and Problems MARKET: For all readers interested in basic college mathematics.
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Rent Basic College Mathematics 11th edition today, or search our site for other textbooks by Marvin L. Bittinger. Every textbook comes with a 21-day "Any Reason" guarantee. Published by Pearson. | 677.169 | 1 |
This book is a well-organized text with carefully constructed examples, a full quota of exercises with solutions, and an emphasis that is algebraic rather than geometric. The book is Sage-enabled with approximately 90 examples of Sage code spread throughout. The book sections can be loaded into Sage as worksheets so that the code can be evaluated immediately; however, it is not necessary to use Sage in order to make use of this textbook.
The HTML version is a fully featured electronic text with cross references that expand in place. It works well on an iPad or any device with Internet access. The Sage examples are executed by the Sage single cell server over the Internet, and students can experiment with their own examples as they read without needing to install Sage or log into a Sage server. Most students will prefer this version over both the pdf and print versions.
As stated in the preface of the book, the dual aims are "to teach the fundamental concepts and techniques of matrix algebra and abstract vector spaces, and to teach the techniques associated with understanding the definitions and theorems forming a coherent area of mathematics." | 677.169 | 1 |
Details about Using and Understanding Mathematics: A Quantitative Reasoning Approach:
With all of the topics needed to fulfil a general education requirement, this text, one of the few for the emerging Quantitative Literacy/Quantitative Reasoning Course, helps to reduce mathematical anxiety and focuses on the practicality of mathematics in college, career and life.
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Rent Using and Understanding Mathematics: A Quantitative Reasoning Approach 3rd edition today, or search our site for other textbooks by Jeffrey O. Bennett. Every textbook comes with a 21-day "Any Reason" guarantee. Published by Pearson.
Need help ASAP? We have you covered with 24/7 instant online tutoring. Connect with one of our tutors now. | 677.169 | 1 |
MATLAB for Engineers Explained
Overview number of exercises, tips, and solutions mean that the book can be read with or without a computer. Recent development in MATLAB to advance programming is described using realistic examples in order to prepare students for larger programming projects. The development of Matlab programming and the rapid increase in the use of MATLAB in engineering courses makes this a valuable self-study guide for both engineering students and practising engineers. The reader will find that this time-less material can be used throughout their education and into their career.
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Editorial Reviews
From the Publisher
From the reviews:
"The book consists of three parts: an initiation in matlab, more advanced programming in matlab, and some elaborated applications. … The text gives a bottom-up learning-by-example approach. … Thus the reader is forced to sit at the computer and do experiments, which is in my opinion the best and fastest way to learn matlab. The excellent help tool of matlab should do the rest. … The command summaries in the appendices make it … a substitute for the matlab manuals." (Adhemar Bultheel, Bulletin of the Belgian Mathematical Society, Vol. 12 (1), 2005) | 677.169 | 1 |
Math.NET aims to provide a self contained clean framework for symbolic mathematical (Computer Algebra System) and numerical/scientific computations, including a parser and support for linear algebra, complex differential analysis, system solving and more | 677.169 | 1 |
Derived from the author?s course on the subject, Elements of Differential Topology explores the vast and elegant theories in topology developed by Morse, Thom, Smale, Whitney, Milnor, and others. It begins with differential and integral calculus, leads you through the intricacies of manifold theory, and concludes with discussions on algebraic topology,... more...
The authors show that there are underlying mathematical reasons for why games and puzzles are challenging (and perhaps why they are so much fun). They also show that games and puzzles can serve as powerful models of computation?quite different from the usual models of automata and circuits?offering a new way of thinking about computation. The appendices... more...
This updated and revised edition of a widely acclaimed and successful text for undergraduates examines topology of recent compact surfaces through the development of simple ideas in plane geometry. Containing over 171 diagrams, the approach allows for a straightforward treatment of its subject area. It is particularly attractive for its wealth of applications... more...
Students See How Far a Little Math Can Take Them in Their Daily Lives Mathematics for the Liberal Arts teaches everyday mathematics topics to non-math majors at the undergraduate level. Through numerous examples and more than 600 exercises, students learn how to use math seamlessly in a variety of practical areas, from conversion factors, statistics,... more...
Many parallels between complex dynamics and hyperbolic geometry have emerged in the past decade. Building on work of Sullivan and Thurston, this book gives a unified treatment of the construction of fixed-points for renormalization and the construction of hyperbolic 3- manifolds fibering over the circle. Both subjects are studied via geometric... more...
Recent success with the four-dimensional Poincaré conjecture has revived interest in low-dimensional topology, especially the three-dimensional Poincaré conjecture and other aspects of the problems of classifying three-dimensional manifolds. These problems have a driving force, and have generated a great body of research, as well as insight.... more...
This work presents fundamental ideas and concepts through simple examples, and subsequently develops them in a systematic way. Important terminology is clearly defined. The work is organized according to the ?learn & check? model, with practice exercises at every important juncture along with suggested solutions at the end of each chapter.... more...
Geometric Topology contains the proceedings of the 1977 Georgia Topology Conference, held at the University of Georgia on August 1977. The book is comprised of contributions from leading experts in the field of geometric topology.These contributions are grouped into four sections: low dimensional manifolds, topology of manifolds, shape theory and... more... | 677.169 | 1 |
Synopses & Reviews
Publisher Comments
Hailed by The New York Times Book Review as "nothing less than a major contribution to the scientific culture of this world," thisSynopsis
ThisAbout the Author
The Russian Equation
Representative of the tremendous impact which Russian mathematicians have had on the Dover list since the Sputnik era is this outstanding book edited by A. D. Aleksandrov and others.
Critical Acclaim for Mathematics: Its Content, Methods and Meaning:
"In effect, these volumes present a do-it-yourself course for the person who would like to know what the chief fields of modern mathematics are all about but who does not aspire to be a professional mathematician or a professional user of mathematics. The coverage is extremely wide, including such important areas as linear algebra, group theory, functional analysis, ordinary and partial differential equations, the theory of functions of real and complex variables, and related subjects. . . . What makes these volumes so readable as compared with usual mathematics textbooks is the emphasis here upon basic concepts and results rather than upon the intricate and wearying proofs that make such demands in conventional textbooks and courses. There are proofs in these volumes, but usually they are presented only for the most important results, and even then to emphasize key areas and to illustrate the kind of methodology employed. . . . It is hard to imagine that any intelligent American with a curious mind and some good recollection of his high school and college mathematics would not find many entrancing discoveries in the intellectual gold mine that is this work." — The New York Times Book Review
"An excellent reference set for bright high school students and beginning college students . . . also of value to their teachers for lucid discussions and many good elementary examples in both familiar and unfamiliar branches. The intelligentsia of laymen who care to tackle more than today's popular magazine articles on mathematics will find many rewarding introductions to subjects of current interest." — The Mathematics Teacher
"Whether a physicist wishes to know what a Lie algebra is or how it is related to a Lie group, or an undergraduate would like to begin the study of homology, or a crystallographer is interested in Fedorov groups, or an engineer in probability, or any scientist in computing machines, he will find here a connected, lucid account." — Science
Table of Contents
Volume 1. Part 1
Chapter 1. A general view of mathematics (A.D. Aleksandrov)
1. The characteristic features of mathematics
2. Arithmetic
3. Geometry
4. Arithmetic and geometry
5. The age of elementary mathematics
6. Mathematics of variable magnitudes
7. Contemporary mathematics
Suggested reading
Chapter 2. Analysis (M.A. Lavrent'ev and S.M. Nikol'skii)
1. Introduction
2. Function
3. Limits
4. Continuous functions
5. Derivative
6. Rules for differentiation
7. Maximum and minimum; investigation of the graphs of functions
8. Increment and differential of a function
9. Taylor's formula
10. Integral
11. Indefinite integrals; the technique of integration
12. Functions of several variables
13. Generalizations of the concept of integral
14. Series
Suggested reading
Part 2.
Chapter 3. Analytic Geometry (B. N. Delone)
1. Introduction
2. Descartes' two fundamental concepts
3. Elementary problems
4. Discussion of curves represented by first- and second-degree equations | 677.169 | 1 |
Elements of Algebra
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Read More solve equations of varying degrees, methods of rendering certain formulas rational, and more. In 1771, Joseph-Louis Lagrange included an addendum to the French edition containing topics such as continued fractions and Diophantine equations. This edition of Elements of Algebra was completely re-written using Microsoft Word and its Equation Editor over the course of several months and is not just another scanned copy of John Hewlett's original English language translation. This new edition contains Euler's Part I (Containing the Analysis of Determinate Quantities) and Part II (Containing the Analysis of Indeterminate Quantities), Lagrange's Additions as well as all of the footnotes by Johann Bernoulli and others. While much of the text remains the same as that edition, several changes were made to make it more accessible to the modern reader: 1. The words "shew" and "shewn" were replaced by "show" and "shown," respectively. 2. The original notation for continued fractions has been replaced with the more modern notation. 3. Several German-to-English and French-to-English translation issues were resolved. 4. While the order is the same as John Hewlett's English language translation, in order to give the work more space, each chapter now begins on its own page. 5. Several inline formulas were moved out-of-line to give the work more space. 6. Several errors in the formulas were discovered and corrected. 7. The word "formulae" was replace by "formulas." 8. The footnotes were moved to a Notes section at the end of the book. 9. The overuse of the comma in several places was brought up to modern standards. It is my hope that by creating this new "modern" edition, a renewed interest will be generated for a work which certainly does not deserve to be forgotten, written by a brilliant mathematician considered "the master of us all | 677.169 | 1 |
books.google.com - An entertaining and captivating way to learn the fundamentals of using algorithms to solve problemsThe algorithmic approach to solving problems in computer technology is an essential tool. With this unique book, algorithm guru Roland Backhouse shares his four decades of experience to teach the fundamental... Problem Solving
Algorithmic Problem Solving
An entertaining and captivating way to learn the fundamentals of using algorithms to solve problems
The algorithmic approach to solving problems in computer technology is an essential tool. With this unique book, algorithm guru Roland Backhouse shares his four decades of experience to teach the fundamental principles of using algorithms to solve problems. Using fun and well-known puzzles to gradually introduce different aspects of algorithms in mathematics and computing. Backhouse presents you with a readable, entertaining, and energetic book that will motivate and challenge you to open your mind to the algorithmic nature of problem solving.
Provides a novel approach to the mathematics of problem solving focusing on the algorithmic nature of problem solving
Uses popular and entertaining puzzles to teach you different aspects of using algorithms to solve mathematical and computing challenges
Features a theory section that supports each of the puzzles presented throughout the book
Assumes only an elementary understanding of mathematics
Let Roland Backhouse and his four decades of experience show you how you can solve challenging problems with algorithms!
About the author (2011)
Roland Backhouse leads the Foundations of Programming research group at the University of Nottingham. He returned to the UK in 1999 after spending 13 years in the Netherlands, 9 of which as professor at Eindhoven University of Technology (the institution that pioneered the style of reasoning and construction of programs that forms the basis for this book). | 677.169 | 1 |
In Algebra 1 we also study graphical methods in order to visualize functions as straight lines or parabolas. Further we learn about factorization and the solutions of quadratic equations. Seeing many advanced students who struggle with algebra 1 concepts makes me feel good about my algebra 1 students because I help them to learn it properly from the beginning | 677.169 | 1 |
This first course in differential geometry presents the fundamentals of the metric differential geometry of curves and surfaces in a Euclidean space of three dimensions. Written by an outstanding teacher and mathematician, it explains the material in the most effective way, using vector notation and technique. It also provides an introduction to the... more...
Twelve essays take a playful approach to the subject, exploring how to play poker over the telephone without the possibility of cheating, how to distinguish plausible fallacies from unbelievable facts, and how to cope mathematically with contorted worms, drunken tennis players, and snakes that eat their own tails. Former columnist for Scientific... more...
This volume lays the mathematical foundations for the theory of differential games, developing a rigorous mathematical framework with existence theorems. It begins with a precise definition of a differential game and advances to considerations of games of fixed duration, games of pursuit and evasion, the computation of saddle points, games of survival,... more...
A problem-oriented text for evaluating statistical procedures through decision and game theory. First-year graduates in statistics, computer experts and others will find this highly respected work best introduction to growing field. more...
"Mind-exercising and thought-provoking."? New Scientist If playing games is natural for humans, analyzing games is equally natural for mathematicians. Even the simplest of games involves the fundamentals of mathematics, such as figuring out the best move or the odds of a certain chance event. This entertaining and wide-ranging guide demonstrates... more...
A noted research mathematician explores decision making in the absence of perfect information. His clear presentation of the mathematical theory of games of strategy encompasses applications to many fields, including economics, military, business, and operations research. No advanced algebra or non-elementary calculus occurs in most of the proofs.... more...
Suitable for advanced undergraduates and graduate students, this text surveys the classical theory of the calculus of variations. It takes the approach most appropriate for applications to problems of optimizing the behavior of engineering systems. Two of these problem areas have strongly influenced this presentation: the design of the control systems... more...
Anyone with a knowledge of basic mathematics will find this an accessible and informative introduction to game theory. It opens with the theory of two-person zero-sum games, two-person non-zero sum games, and n-person games, at a level between nonmathematical introductory books and technical mathematical game theory books. Succeeding sections focus... more...
N-person game theory provides a logical framework for analyzing contests in which there are more than two players or sets of conflicting interests-anything from a hand of poker to the tangled web of international relations. In this sequel to his Two-Person Game Theory, Dr. Rapoport provides a fascinating and lucid introduction to the theory,... more...
Many illuminating and instructive examples of the applications of game theoretic models to problems in political science appear in this volume, which requires minimal mathematical background. 1975 edition. 24 figures. more... | 677.169 | 1 |
Richard M. Felder Department of Chemical Engineering North Carolina State University
MAPLE is a powerful mathematical software package. It
can be used to obtain both symbolic and numerical solutions of
problems in arithmetic, algebra, and calculus and to generate
plots of the solutions it generates.
This is the first of a set of tutorials that outline how to use
MAPLE. In it, we outline basic MAPLE syntax and show how to solve sets of
algebraic equations, generate data plots, and generate an annotated printout
of the results. Realize, however, that we are barely tapping the power of this
tool to perform almost any type of mathematical analysis you might ever need to
perform. Once you have mastered the material in this tutorial, you may go on to
the second one, which covers differentiation, integration, and solving ordinary
differential equations.
If this is your first experience with MAPLE, simply reading the tutorial
will do you very little good. If you want to be able to use MAPLE to solve
mathematics problems, start the program with the commands in the "Getting
Started" section and then carry out each step given in the
subsequent sections. Bullets will mark the instructions you should follow and
the commands you should type.
Open MAPLE. If you are on EOS (a UNIX-based operating system at
North Carolina State University)
Log on to your EOS account.
Bring up the application
menu(point to an open space on the desktop and click the middle mouse
button) and choose Math and Statistics --> Maple(4)
(Alternatively, at the command prompt in the Xterm window, type
add maple and then xmaple &)
You should eventually see a large window headed by "MapleV,
Release 4," with a smaller window headed by "Untitled(1)
- [Server 1]." The smaller window is the worksheet.
The MAPLE command prompt [ >
will show at the upper left of the worksheet.
When you are finished with the MAPLE session, you will leave the
program by selecting "Exit" under the "File"
menu (upper left of the MAPLE window). (Don't do it now.)
Saving your program
As with any other program application (word processing, spreadsheeting,
etc.), you should get in the habit of saving your work frequently,
so if the system crashes you can recover most of
what you did.
Choose "Save as" under the File menu.
Make "tutorial1.mws" the name of your file.
This name should then appear at the top of your worksheet. As
you work, frequently choose "Save" under the "File"
menu or type CONTROL-s (hold down the CONTROL
or CTRL key and type s) to save your recent work.
Getting help
Go to the "Help" menu (it's at the top
right of the MAPLE window) and select Contents.
You will see a number of topics in black lettering ("What's
New," "Mathematics," etc.) and "Introduction"
and "Worksheet Interface" in green lettering. You
can click on any title with green letters to open that section
of the Help file.
Click on "Introduction" and read through the
examples to get your first look at what MAPLE does.
When
you are finished, click on the upper left corner of the "Intro"
window and select "Close" to get rid of the window.
In the "Contents" window, click on the [+] next
to "Mathematics."
A large number of green titles
will appear, any of which can be clicked on for information about
that topic.
A MAPLE command is a statement of a calculation followed by a semicolon (the
result is displayed on the screen) or a colon (the result is stored but not
displayed). Following are some commands followed by the displayed results. Enter the commands on your worksheet and verify the given results. When you get to "Save the file," select "Save" under the "File" menu or type CTRL-S.
[> 4+5;
9
[> 6^2;
36
[> 3*4 + 7;
19
[> 3*(4+7);
33
Save the file
If you don't include a semicolon or colon at the end of a
command, MAPLE will interpret the next command line as a continuation
of the previous command.
Tip: Once you've done something in a program step (like redefine a
variable), it stays done unless you change it again. Even if you click on an
earlier step and re-execute it, the effect of the change remains.
If a program does not seem to be working, before trying
anything else re-execute all steps from the beginning by clicking on the
Edit menu and selecting Execute -> Worksheet. (You can also select
a set of commands and re-execute them by selecting Execute -> Selection.)
Arithmetic operations
The symbols +, -, *, /, and
^ (or **) denote addition, subtraction, multiplication, division,
and exponentiation (4^2 = 4**2 = 42 = 16). When a
string of operations are specified in a command, MAPLE first does
exponentiations, then multiplications and divisions, then additions
and subtractions. To change the order, use parentheses.
Editing commands.
If you make a mistake in a command (like forgetting
a semicolon) or want to change a command, you can go back and
edit the command with the cursor and mouse as you would a word-processor
text.
Operating on the last result.
Use a quotation mark
(") to refer to the result of the last calculation, and a
double quotation mark ("") to refer to the next-to-last
result. For example,
The second argument in the evalf command (the 3 following
the comma) specifies the number of significant figures you want in
the result. If you omit this command, you will get ten significant
figures.
On-line help
You can get help with MAPLE syntax
by using the HELP menu, as described previously. If you have
a question about a particular command, you can quickly get help
by typing a question mark followed by the command name (no semicolon).
For example,
[> ? evalf
will open a window containing information about what the evalf
function does and how to use it.
Close the help window. Click on the little box at the upper left of the window and select Close.
Defining variables
Use the colon-equal symbol (:=)
to define variables--that is, to assign values to them. Once you
have defined a variable, simply typing its name will show its
value, and using the name in a formula will cause the value to
be substituted.
[> A := 5; B := 2;
A := 5
B := 2
(Notice that you can string commands together on the same line.)
[> A;
5
[> 4*A/B;
10
Save the file
Variable names are case-sensitive: A and a are two completely
different variables.
Specifying the number of digits (significant figures) in
your results.
When you do floating point arithmetic, MAPLE
defaults to 10 significant figures:
[> evalf(5/2);
2.500000000
You can change the default by setting a variable "Digits"
equal to the desired number of significant figures.
[> Digits := 4; (Make sure you capitalize
the word Digits.)
Digits := 4
[> evalf(5/2);
2.500
Built-in functions
MAPLE has many built-in mathematical functions including sin, cos, tan,
exp (exp(x)=ex), ln, and log10. However, if you want to calculate the
value of one of these functions, you must use the evalf command.
[> ln(10);
ln(10)
[> evalf(ln(10), 4);
2.303
Expressions and functions
An expression is a string of constants, variables, and
mathematical operators (+ - * / ^ =
ln sin ...). The following are all expressions:
A function is a relationship for a variable (the dependent
variable in terms of one or more other variables (the independent
variables). The following are each functions:
y(x) = 3*x + 2 [--> y(1) = 5]
z(x,y) = 2*x/y [--> z(5,3) = 10/3]
MAPLE handles expressions and functions in completely different ways, which can
lead to a great deal of confusion when relations like this are encountered:
y = 6*x^2 - 3*x + 4
This could be an expression relating x and y or the
definition of y as a function of x. To do anything to or
with this relation (like substituting a value of x into it, or solving
it for one variable in terms of the other one, or differentiating or
integrating it), you must know whether the relation is an expression or a
function.
In the next two sections we show how expressions and functions are defined and
evaluated. In the remainder of the tutorial we will continue to make the
distinction between the two types of relations. If you forget the difference,
refer back to this section.
Defining expressions and substituting values into them
Use the colon-equal (:=) to define an expression, and the subs
function to substitute a value into it.
[> f := x^2;
f := x2
[> f;
x2
[> subs(x=5, f);
25
Typing [> f(5); will not work for the latter calculation.
If you want to use functional notation like that, you need to
use the next method to define a function.
Defining functions and substituting argument values into them
Use the colon-equal (:=) and the minus-greater
than (->) symbols to define a function, and functional notation
to substitute a value into it.
[> g := x -> 1/(x+1);
g := x -> 1/(x+1)
[> g(x);
1/(x+1)
[> g(4);
1/5
Save the file
Converting expressions into functions and vice versa
We noted that the previously defined expression
f = x2 cannot be treated as a function.
[> f(5);
x(5)^2
You can convert the expression into a function of x, however, by
using the unapply command.
[> f := unapply(f,x);
f := (x -> x)2
[> f(5);
25
You can also take the function f = x2 and convert it back into
an expression.
Equations are expressions, so we can use the colon-equal (:=)
to define equations involving one or more variables.
(This will be the first step in getting MAPLE to solve the equations.)
Suppose equations relating two variables are
2x1 - 5x2 = 12
12x1 + 4x2 = 17
To enter and store these equations for subsequent processing, enter the
commands
[> eq1 := 2*x1 - 5*x2 = 12:
[> eq2 := 12*x1 + 4*x2 = 17:
(Don't forget the multiplication symbols.) We used colons instead
of semicolons in these commands, so MAPLE does not reproduce the
equation definitions.
Save the file.
Solving sets of linear algebraic equations.
Suppose you wish to solve the pair of equations defined above for the
variables x1 and x2. Use
the solve (exact arithmetic) or fsolve (floating point
arithmetic) command as follows:
(Type these commands carefully;
if you make the slightest mistake, like omitting a comma or typing
the wrong kind of brace, MAPLE will give you nothing but heartache.)
[> sol := solve({eq1, eq2}, {x1,x2});
[> sol := fsolve({eq1, eq2}, {x1,x2});
sol := {x2 = -1.618, x1 = 1.956}
(There does not seem to be any way to predict which variable
MAPLE will print first.)
The argument of the solve or fsolve command consists of two sets
of curly braces enclosed in parentheses. (If you only have one
equation in one unknown, you don't need the braces.) The first
set of braces contains the equations to be solved or the symbols
for them, separated by commas; the second set contains the set
of unknown variables to be solved for.
Extracting components of a solution.
The solution set sol generated with the fsolve command in the previous
step contains two components: the equation x1= 1.9559, and the equation x2= -1.6176.
You can extract either of these equations or only their right-hand
sides as follows (the example shows the procedure for the first
element).
[> sol[2];
x1 = 1.956
[> rhs(sol[2]);
1.956
Save the file
You could also use the lhs function to extract the left-hand
side of the equation.
The procedure for solving nonlinear equations
with solve or fsolve is the same as that outlined above for
linear equations. A difference in the possible outcome is that the procedure
used by MAPLE (or any other nonlinear procedure) may not converge
to a solution, even if one exists, or it may converge to a solution
that is mathematically correct but physically meaningless (like
a negative mass fraction).
To improve the odds of success in solving nonlinear equations, specify
in the solve or fsolve command a range of values for each unknown
variable within which the desired solution must fall. The more
you can narrow the search range, the better the chances for convergence.
For example, if a variable x is the solution of the equation
x = e-x,
and you know the solution must be a positive fraction (0<x<1),
you can proceed as follows:
[> eq := x = exp(-x);
eq := x = e(-x)
[> sol := fsolve(eq, x, x=0..1);
sol := .5671
(Since there is only one variable, you could have omitted the
"x=" in front of the range 0..1.)
The procedure is the same if you have simultaneous nonlinear
equations, except that now you need to enclose the equations,
the variables, and ranges for each variable within curly braces.
For example, suppose you want to find the solution of the equations
sin(x+y) -yex = 0
x2 - y = 2
and you know that because of the physical meanings of x
and y the solutions must fall in the ranges -1<x<1
and -2<y<0. The solution program
would be as follows (Type carefully!):
[> f := sin(x+y) - y*exp(x) = 0:
[> g := x^2 - y = 2:
[> solfg := fsolve({f,g}, {x,y}, {x = -1..1, y =
-2..0});
solfg := {x = -.6687, y = -1.553}
Save the file
If you omitted the search ranges for x and y, the
procedure would have converged to a different solution that satisfies
the equations but has no physical significance. (Try it and see.)
If you want to specify a semi-infinite range for a search, you
can use infinity or -infinity as one of the
limits of the range.
Finding roots of polynomials.
Suppose you want to solve the equation
6x4 - 7x3
+ 6x2 - 1 = 0
You know that the equation has four roots (solutions),
which may be real numbers or complex conjugate pairs. You can
find the roots either exactly or with floating-point arithmetic
using the solve or fsolve command. In what follows, I
is the MAPLE symbol for .
[> p := 6*x^4 - 7*x^3 + 6*x^2 - 1 = 0;
p := 6x4 - 7x3+ 6x2 - 1 = 0
[> sol := solve(p, x);
[> solf := fsolve(p, x, complex);
solf := -.3333, .5000 - .8660I, .5000, .5000
+ .8660I
If you had left off "complex" in the last command
the program would have generated only the real roots.
This command makes a wide range of plotting functions available for use. Their
names should appear on the screen in response to this command. We
will use the "plot", "textplot", and "display"
routines in the package.
Plotting tabular data
The specific volumes of saturated steam at several temperatures
are listed below:
T(oC)
9.7
20.4
31.0
41.5
51.1
V(m3/kg)
108.7
56.4
31.1
18.1
11.5
To use MAPLE to generate a plot of V vs. T, we
use the following commands:
The first command generated a list (an array enclosed
in square brackets) consisting of the [T,V] data points, each
point enclosed in its own square brackets. The second command
told MAPLE to plot the points, specifying the labels and ranges
for the abscissa and ordinate axes, and to show but not connect
the data points. (If the "style=point" option had been
omitted, the points would have been connected with line segments.)
There are ways to change the axis labels, the symbols and colors
of the data points, and other characteristics of the plot. We
will hold off on giving you some of these options until we discuss
plotting functions.
The command to generate plots
of two functions on the same graph has the form
[> plot( [f1(x), f2(x)], x=a..b, options); (Don't type
this now)
where
f1 and f2 are either MAPLE
functions or functions you defined using (:= ->) a and b are the desired boundaries of the abscissa
range options are statements of how you want the plots to
appear (details follow).
If you only want to plot one function,
you can omit the square brackets enclosing the function labels.
For example, to generate plots of f(x) = sin(x) and
g(x) = 0.25x-1 vs. x from x=0 to x=4,
the commands would be
(We are omitting the MAPLE output here but you should
see it on your terminal screen.) You could also have added a
range for the ordinate axis (e.g. y=-1..1); by omitting it,
you allowed MAPLE to do its own scaling.
Save the file
Plot options
Following are some but not all of the
options you can include in a plot command. For a more detailed
list, enter [> ? plot and click on the
"plot(options)" link.
What follows is for your information only. Don't do anything
with it now.
color = c, where c may be any
of most common colors and some not so common (coral, sienna, etc.)
To see a list of colors, enter [> ? plot,color (don't
do it now).
style = s, where s may be "point"
or "line" (line is default). "Point" shows
closely spaced points, "line" shows line segments.
When a function is plotted in the line style, the line segments
are usually small enough to make the curve look continuous.
labels = [xlabel, ylabel], where the two
arguments in square brackets are the labels you want for the abscissa
and ordinate. The defaults are the variable names.
linestyle = n, where n is usually
0, 2, or 3 but may be any other integer. The value of n specifies
the type of line you want plotted when style=line. n=0 (default)
is a solid line, n=2 is dots and dashes, and n=3 is dashes.
You can use the "textplot" command to insert text on a plot. In this
case, we will label the two curves on the graph we just generated by inserting
"f(x)" centered at the point (x=0.8, f=0.5) and "g(x)" just above and to the
left of the point (x=2, g=-0.5). We first define a variable to represent the
graph.
In the textplot commands that follow, enclose the texts to be inserted
[f(x) and g(x)] in forward apostrophes (`) (located under the ~ character on
most keyboards), as opposed to ordinary apostrophes (').
You should see the same graph with the plot labels inserted. Notice how f(x)
is centered on the point specified in the first textplot command (this is the
default alignment), and the lower right corner of g(x) is located at the point
specified in the second textplot command. It usually takes several trials to
get the text exactly where you want it.
Insert text in the worksheet.
Click on the bracket ( [ ) to the left of the first command
line ("4+5;"). The bracket should be highlighted.
Click on the Insert menu and select Paragraph -> Before.
A line should open up before the command line.
On the new line, type (without hitting the "Return" key)
MAPLE TUTORIAL I
At the left of the third line from the top of the big MAPLE
window (the "context bar"), find a horizontal window
with the word "Normal" in it and a triangle just to
the right of the word. This is the "style droplist."
The window to the right of the style droplist is the "font
droplist," and the one to the right of that is the "font size
droplist."
Click on the triangle to the right of the style droplist to bring
up a list of predefined styles. Then click on Title. Your
title should now be centered, underlined, and in large type.
Put your name (or names and group number) as the author(s)
of the report.
With the cursor at the end of the title, hit the "Return"
key. (The word "Author" should come up in the style droplist.) Type your
name(s) or, if applicable, group number, colon, and last names separated by
commas. Don't hit "Return" yet.
Change the style of this line from author to Heading 1 (click
on the style droplist button as before). Then use the font size
droplist to change the size from 18 point to 14 point. Then click
on the "Centered Text" button (second from the right
on the context bar).
What you typed should now be boldface and
centered, in larger type than normal size but smaller than in
the title style. Hit the "Return" key. You should now
be back in Normal style.
Define the first section and add a heading
By clicking and dragging the mouse, highlight the region from the "4 + 5" command through the "? evalf" command.
Click on the Format menu and select Indent. A
little square with a minus sign in it should appear.
Click to the right of the little square and type Command syntax and arithmetic
Click on the little square to omit the section (except the
heading) from the screen display. Then click on the square again
(which should now have a plus sign in it) to display the section
again.
Define sections for the remainder of the worksheet
Using the same sequence of steps (highlight region to be indented,
indent using the Format menu, type heading), define a section containing the
commands from "A := 5;" command line to the
"f(5)" command line and the "25" output line that follows it.
Give this section the heading "Defining variables, expressions, and functions." Save.
Define a section containing the commands from " eq1 := ...
"command line to the "solf := ..." command and the output line that follows it, giving it the heading
"Define and solve sets of algebraic equations." Save.
Define a section containing the commands from "with(plots)"
through the final chart. (You may have to do some juggling to highlight this entire area, perhaps starting at the bottom and moving up with the pointer.) Give it the heading "Generate and label plots of tabulated data and functions." Save.
Printing
When your program is complete, you can
edit it to get rid of unwanted command lines, and then print it.
Select "Print" under the "File" menu,
note the name of the file that will be created when you execute
the print command (tutorial1.ps--change it if you want to), and then click on
"Print." | 677.169 | 1 |
Precise Calculator has arbitrary precision and can calculate with complex numbers, fractions, vectors and matrices. Has more than 150 mathematical functions and statistical functions and is programmable (if, goto, print, return, for). | 677.169 | 1 |
The Prentice Hall Algebra 1 curriculum introduces the basic concepts and properties of algebra along with numerous practice and problem solving exercises. Homeschoolers, please note: the textbook references a number of online elements on a website, including videos, online assignments, interactive activities, vocabulary definitions, and more. Access to this website is not included, or available to homeschoolers. The course can be used without these components. Grade 9/Algebra 1 provides instruction on the foundations of algebra, solving equations, solving inequalities, functions, linear functions, systems of equalities and inequalities, exponents and exponential functions, polynomials and factoring, quadratic functions and equations, radical expressions and equations, rational expressions...
Less
Prentice-Halls' Writing & Grammar curriculum helps students develop their language arts skills through an easy-to-follow, three-part division which focuses on writing, grammar/usage/mec hanics, and academic & workplace skills. Web codes provide access to the student website for online videos, quizzes and tests, providing an effective mix of traditional texts and modern technology. Conversational lessons are to-the-point with bullet pointed lists, bolded terms, graphics, and examples used to help highlight important concepts. Integrated exercises provide an immediate opportunity to practice new skills. The teacher's edition featured reduced-size student pages with teacher notes and answers in the large margins. Lesson objectives; time & resource manager notes that include instructions | 677.169 | 1 |
>Does this mean that the student must be able to find the sine of 30 >degrees without anything but pencil and paper before being allowed to >use a technical aid (even a table)?
My daughter came to me with help on trig (she's in IMP) and I was unpleasantly surprised to find thet they didn't have any trig tables. In working with her on some problems I used tables, and let her verify with the calculator.
I DO believe that there is more learning with use of tables than with the calculator, because she got to see the numbers in the context of other sines. How else does she get reinforcement on what reasonable values for the sine of a particular angle are. or begin to understand that small changes in angle lead to small (but not proportionate) changes in sine? I'm sure that the teacher covered or will cover all of this, and will do it well, but since my daughter didn't grasp that topic too well, the reinforcement was useful.
Similarly, I would really like to make her plot (by hand, on graph paper) a few sine curves. I don't believe that seeing the curves on a graphing calculator can possibly teach as well as doing them by hand, and having to THINK about what value to plot at 45 degrees, or where the curve of sin 2x touches the x axis.
Sure, use the calculators for calculating, but I think only very few students are likely to sit with them and experiment for long enough to gain the same understanding that's to be had from even a small visit to the trig tables. | 677.169 | 1 |
Proofs without words are generally pictures or diagrams that help the reader see why a particular mathematical statement may be true, and how one could begin to go about proving it. While in some proofs without words an equation or two may appear to help guide that process, the emphasis is clearly on providing visual clues to stimulate mathematical thought. The proofs in this collection are arranged by topic into five chapters: Geometry and algebra; Trigonometry, calculus and analytic geometry; Inequalities; Integer sums; and Sequences and series. Teachers will find that many of the proofs in this collection are well suited for classroom discussion and for helping students to think visually in mathematics.
Editorial Reviews
Book Description
Proofs without words are generally pictures or diagrams that help the reader see why a particular mathematical statement may be true, and how one could begin to go about proving it. Teachers will find that many of the proofs in this collection are well suited for classroom discussion and for helping students to think visually in mathematics.
Top Customer Reviews
Famous mathematicians have often emphasized the role of visual intuition; e.g., Hilbert: "Who does not always use along with the double inequality a > b > c the picture of three points following one another on a straight line as the geometrical picture of the idea "between"? Who does not make use of drawings of segments and rectangles enclosed in one another, when it is required to prove with perfect rigor a difficult theorem on the continuity of functions or the existence of points of condensation?" (from his famous address at the 1900 International Congress). This book is a collection of well over 100 one-page proofs, collected from various sources. The topics range from number theory to calculus, and most of them require no advanced mathematics. Typically there is a statement of a result, with a labelled diagram showing how it is "proved"; in some cases there are a few auxiliary equations along with the picture. These are not simple, often requiring quite a bit of thought before the "Aha!" moment. Working through them is a valuable exercise for the student of mathematics--having seen, e.g., six different visual proofs of the Pythagorean theorem, one comes to really *understand* the result, not just "follow the logic". I have not encountered any better way than this book to "see" how mathematical truth is discovered and proved. It can be valuable as a supplement to courses through precalculus and elementary calculus. Perhaps one of its best uses is to inspire teachers to present results in a more lively way then "definition-theorem-proof" or "just memorize it".
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How many of you remember doing geometry proofs in High School? How many of you enjoyed writing them? I don't know about you but I've always preferred pictures to words when it comes to understanding how something works.
This is a wonderful book that provides visual insights into how one might go about proving mathematical theorems. The Pythagorean Theorem has always been a mystery to me. How are the squares of the sides of a right triangle related to its hypotenuse? "Proof Without Words" has five clever illustrations that guide readers in writing their own proofs.
If you ever doubted that algebra and geometry were related, the diagrams demonstrating how to compute sums of series will produce aha! experiences.
Writing proofs when one is guided by visual cues is a much more fulfilling endeavor than stringing together dry facts from memory. This book delivers much fulfillment in exploring theorems in geometry, algebra, trigonometry, sequences, and other aspects of Math.
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Wow! When I found these books I immediately fell in love. I'm a visual thinker and I love visual proofs. These books have no words but have hundreds of proofs. One proof per page. I wish I had more time to work through all these proofs. Some of the proofs take a few seconds to understand but many others take hours. These books are the real vitamins for the brain.
I've placed this book #20 in my Top 100 Programming, Computer and Science books list:
Proof without words really opened my insights into how some of the proofs of formula came about. Sometimes we just take for granted the formula and do not discover how they came about. Through graphics and pictures we are able to visualize better.
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Genre
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ISBN 978-0-07-353235-6
MHID 0-07-353235-5
L
i
n
e
a
r
A
l
g
e
b
r
a
Introduction to Linear Algebra with Applications by Jim DeFranza and Daniel Gagliardi provides
the proper balance between computation, problem solving, and abstraction that will equip students with
the necessary skills and problem solving strategies to allow for a greater understanding and appreciation
of linear algebra and its numerous applications.
Introduction to Linear Algebra with Applications provides students with the necessary tools for success:
Abstract theory is essential to understanding how linear
algebra is applied.
Each concept is fully developed presenting natural connections
between topics giving students a working knowledge of the theory and techniques for each module
covered.
Applications have been carefully chosen to highlight the utility of linear algebra in
order to see the relevancy of the subject matter in other areas of science as well as in mathematics.
Ranging from routine to more challenging, each exercise set extends the concepts
or techniques by asking the student to construct complete arguments. End of chapter True/False
questions help students connect concepts and facts presented in the chapter.
Examples are designed to develop intuition and prepare students to think more
conceptually about new topics as they are introduced.
Students are introduced to the study of linear algebra in a sequential and thorough
manner through an engaging writing style gaining a clear understanding of the theory essential for
applying linear algebra to mathematics or other elds of science.
Summaries conclude each section with important facts and techniques providing
students with easy access to the material needed to master the exercise sets.
Linear Algebra
Linear Algebra
DeFranza
M
D
k=1
a
jk
c
ki
= (AC)
ji
= ((AC)
t
)
ij
The proofs of parts 1, 3, and 4 are left as exercises.
DEFINITION 5 Symmetric Matrix An n ×n matrix is symmetric provided that A
t
= A.
EXAMPLE 5 Find all 2 ×2 matrices that are symmetric.
Solution Let
A =
_
a b
c d
_
Then A is symmetric if and only if
A =
_
a b
c d
_
=
_
a c
b d
_
= A
t
which holds if and only if b = c. So a 2 ×2 matrix is symmetric if and only if the
matrix has the form _
a b
b d
_
Fact Summary
Let A, B, and C be matrices.
1. The definitions of matrix addition and scalar multiplication satisfy many of
the properties enjoyed by real numbers. This allows algebra to be carried
out with matrices.
Confirming Pages
1.3 Matrix Algebra 37
2. When AB is defined, the ij entry of the product matrix is the dot product
of the ith row vector of A with the jth column vector of B.
3. Matrix multiplication does not in general commute. Even when AB and
BA are both defined, it is possible for AB = BA.
4. The distributive properties hold. That is, A(B +C) = AB +AC and
(B +C)A = BA+CA.
5. (A +B)
t
= A
t
+B
t
, (AB)
t
= B
t
A
t
, (A
t
)
t
= A, (cA)
t
= cA
t
6. The matrix A =
a b
c d
is symmetric if and only if b = c.
Exercise Set 1.3
In Exercises 1–4, use the matrices
A =
k=1
a
1k
C
1k
Similar to the situation for 3 ×3 matrices, the determinant of any square matrix
can be found by expanding along any row or column.
THEOREM 12 Let A be an n ×n matrix. Then the determinant of A equals the cofactor expansion
along any row or any column of the matrix. That is, for every i = 1, . . . , n and
j = 1, . . . , n,
det(A) = a
i1
C
i1
+a
i2
C
i2
+· · · +a
in
C
in
=
n
k=1
a
ik
C
ik
and
det(A) = a
1j
C
1j
+a
2j
C
2j
+· · · +a
nj
C
nj
=
n
k=1
a
kj
C
kj
For certain square matrices the computation of the determinant is simplified. One
such class of matrices is the square triangular matrices.
DEFINITION 5 Triangular Matrices An m×n matrix is upper triangular if a
ij
= 0, for all
i > j, and is lower triangular if a
ij
= 0, for all i < j. A square matrix is a
diagonal matrix if a
ij
= 0, for all i = j.
Revised Confirming Pages
1.6 Determinants 57
Some examples of upper triangular matrices are
Since the determinant on the right is n ×n and upper triangular, by the inductive
hypothesis
det(A) = (−1)
2n+2
(a
n+1,n+1
)(a
11
a
22
· · · a
nn
)
= a
11
a
22
· · · a
nn
a
n+1,n+1
Properties of Determinants
Determinants for large matrices can be time-consuming to compute, so any properties
of determinants that reduce the number of computations are useful. Theorem 14 shows
how row operations affect the determinant.
Confirming Pages
58 Chapter 1 Systems of Linear Equations and Matrices
THEOREM 14 Let A be a square matrix.
1. If two rows of A are interchanged to produce a matrix B, then det(B) =
−det(A).
2. If a multiple of one row of A is added to another row to produce a matrix B,
then det(B) = det(A).
3. If a row of A is multiplied by a real number α to produce a matrix B, then
det(B) = αdet(A).
Proof (1) The proof is by induction on n. For the case n = 2 let
A =
_
a b
c d
_
Then det(A) = ad −bc. If the two rows of A are interchanged to give the matrix
B =
_
c d
a b
_
then det(B) = bc −ad = −det(A).
Assume that the result holds for n ×n matrices and A is an (n +1) ×(n +1)
matrix. Let B be the matrix obtained by interchanging rows i and j of A. Expanding
the determinant of A along row i and of B along row j, we have
det(A) = a
i1
C
i1
+a
i2
C
i2
+· · · +a
in
C
in
and
det(B) = a
j1
D
j1
+a
j2
D
j2
+· · · +a
jn
D
jn
= a
i1
D
j1
+a
i2
D
j2
+· · · +a
in
D
jn
where C
ij
and D
ij
are the cofactors of A and B, respectively. To obtain the result
there are two cases. If the signs of the cofactors C
ij
and D
ij
are the same, then
they differ by one row interchanged. If the signs of the cofactors C
ij
and D
ij
are
opposite, then they differ by two rows interchanged. In either case, by the inductive
hypothesis, we have
det(B) = −det(A)
The proofs of parts 2 and 3 are left as exercises.
We note that in Theorem 14 the same results hold for the similar column oper-
ations. To highlight the usefulness of this theorem, recall that by Theorem 13, the
determinant of a triangular matrix is the product of the diagonal entries. So an alter-
native approach to finding the determinant of a matrix A is to row-reduce A to
triangular form and apply Theorem 14 to record the effect on the determinant. This
method is illustrated in Example 3.
EXAMPLE 3 Find the determinant of the matrix
A =
_
¸
¸
_
0 1 3 −1
2 4 −6 1
0 3 9 2
−2 −4 1 −3
_
¸
¸
_
Confirming Pages
1.6 Determinants 59
Solution Since column 1 has two zeros, an expansion along this column will involve the
fewest computations. Also by Theorem 14, if row 2 is added to row 4, then the
determinant is unchanged and
det(A) =
¸
¸
¸
¸
¸
¸
¸
¸
0 1 3 −1
2 4 −6 1
0 3 9 2
0 0 −5 −2
¸
¸
¸
¸
¸
¸
¸
¸
Expansion along the first column gives
det(A) = −2
¸
¸
¸
¸
¸
¸
1 3 −1
3 9 2
0 −5 −2
¸
¸
¸
¸
¸
¸
We next perform the operation −3R
1
+R
2
−→ R
2
, leaving the determinant again
unchanged, so that
det(A) = −2
¸
¸
¸
¸
¸
¸
1 3 −1
0 0 5
0 −5 −2
¸
¸
¸
¸
¸
¸
Now, interchanging the second and third rows gives
det(A) = (−2)(−1)
¸
¸
¸
¸
¸
¸
1 3 −1
0 −5 −2
0 0 5
¸
¸
¸
¸
¸
¸
This last matrix is triangular, thus by Theorem 13,
det(A) = (−2)(−1)[(1)(−5)(5)]
= −50
Theorem 15 lists additional useful properties of the determinant.
THEOREM 15 Let A and B be n ×n matrices and α a real number.
1. The determinant computation is multiplicative. That is,
det(AB) = det(A) det(B)
2. det(αA) = α
n
det(A)
3. det(A
t
) = det(A)
4. If A has a row (or column) of all zeros, then det(A) = 0.
5. If A has two equal rows (or columns), then det(A) = 0.
6. If A has a row (or column) that is a multiple of another row (or column), then
det(A) = 0.
Confirming Pages
60 Chapter 1 Systems of Linear Equations and Matrices
EXAMPLE 4 Let A =
_
1 2
3 −2
_
and B =
_
1 −1
1 4
_
. Verify Theorem 15, part 1.
Solution In this case the product is
AB =
_
3 7
1 −11
_
so that det(AB) = −33 −7 = −40. We also have det(A) det(B) = (−8)(5) = −40.
Properties of the determinant given in Theorem 15 can be used to establish the
connection between the determinant and the invertibility of a square matrix.
THEOREM 16 A square matrix A is invertible if and only if det(A) = 0.
Proof If the matrix A is invertible, then by Theorem 15,
1 = det(I) = det(AA
−1
) = det(A) det(A
−1
)
Since the product of two real numbers is zero if and only if at least one of them
is zero, we have det(A) = 0 [also det(A
−1
) = 0].
To establish the converse, we will prove the contrapositive statement. Assume
that A is not invertible. By the remarks at the end of Sec. 1.4, the matrix A is
row equivalent to a matrix R with a row of zeros. Hence, by Theorem 14, there is
some real number k = 0 such that det(A) = k det(R), and therefore by Theorem 15,
part 4,
det(A) = k det(R) = k(0) = 0
COROLLARY 1 Let A be an invertible matrix. Then
det(A
−1
) =
1
det(A)
Proof If A is invertible, then as in the proof of Theorem 16, det(A) = 0,
det(A
−1
) = 0, and
det(A) det(A
−1
) = 1
Therefore,
det(A
−1
) =
1
det(A)
The final theorem of this section summarizes the connections between inverses,
determinants, and linear systems.
THEOREM 17 Let A be a square matrix. Then the following statements are equivalent.
1. The matrix A is invertible.
2. The linear system Ax = b has a unique solution for every vector b.
Confirming Pages
1.6 Determinants 61
3. The homogeneous linear system Ax = 0 has only the trivial solution.
4. The matrix A is row equivalent to the identity matrix.
5. The determinant of the matrix A is nonzero.
Determinants can be used to find the equation of a conic section passing through
The graph of the equation
(x −h)
2
a
2
+
(y −k)
2
b
2
= 1
is an ellipse with center
(h, k), horizontal axis of
length 2a, and vertical
axis of length 2b.
specified points. In the 17th century, Johannes Kepler's observations of the orbits of
planets about the sun led to the conjecture that these orbits are elliptical. It was Isaac
Newton who, later in the same century, proved Kepler's conjecture. The graph of an
equation of the form
Ax
2
+Bxy +Cy
2
+Dx +Ey +F = 0
is a conic section. Essentially, the graphs of conic sections are circles, ellipses, hyper-
bolas, or parabolas.
EXAMPLE 5
An astronomer who wants to determine the approximate orbit of an object travel-
ing about the sun sets up a coordinate system in the plane of the orbit with the
sun at the origin. Five observations of the location of the object are then made
and are approximated to be (0, 0.31), (1, 1), (1.5, 1.21), (2, 1.31), and (2.5, 1).
Use these measurements to find the equation of the ellipse that approximates the
orbit.
Solution We need to find the equation of an ellipse in the form
Ax
2
+Bxy +Cy
2
+Dx +Ey +F = 0
Each data point must satisfy this equation; for example, since the point (2, 1.31) is
on the graph of the conic section,
A(2)
2
+B(2)(1.31) +C(1.31)
2
+D(2) +E(1.31) +F = 0
so
4A +2.62B +1.7161C +2D +1.31E +F = 0
Substituting the five points in the general equation, we obtain the 5 ×6 linear
system (with coefficients rounded to two decimal places)
_
¸
¸
¸
¸
_
¸
¸
¸
¸
_
0.1C + 0.31E +F =0
A+ B + C + D + E +F =0
4A+2.62B +1.72C + 2D +1.31E +F =0
2.25A+1.82B +1.46C +1.5D+1.21E +F =0
6.25A+ 2.5B + C +2.5D+ E +F =0
Since the equation Ax
2
+Bxy +Cy
2
+Dx +Ey +F = 0 describing the
ellipse passing through the five given points has infinitely many solutions, by
Theorem 17, we have
Revised Confirming Pages
62 Chapter 1 Systems of Linear Equations and Matrices
= 0
Expanding the determinant gives us the equation
−0.014868x
2
+0.0348xy −0.039y
2
+0.017238x −0.003y +0.00483 = 0
The graph of the orbit is shown in Fig. 2.
x
y
Figure 2
Cramer's Rule
Determinants can also be used to solve linear systems. To illustrate the technique
consider the 2 ×2 linear system
ax +by = u
cx +dy = v
with ad −bc = 0. By Theorem 17, the linear system has a unique solution.
To eliminate the variable y, we multiply the first equation by d and the second
equation by b, and then we subtract the two equations. This gives
adx +bdy −(bcx +bdy) = du −bv
Simplifying, we have
(ad −bc)x = du −bv so that x =
du −bv
ad −bc
Using a similar procedure, we can solve for y.
y =
av −cu
ad −bc
Using determinants, we can write the solution as
x =
2
}
be ordered bases for V. Now let v be a vector in V, and suppose that the coordinates
of v relative to B are given by
[v]
B
=
_
x
1
x
2
_
that is v = x
1
v
1
+x
2
v
2
To determine the coordinates of v relative to B
, we first write v
1
and v
2
in terms of
the vectors v
1
and v
2
. Since B
is a basis, there are scalars a
1
, a
2
, b
1
, and b
2
such
that
v
1
= a
1
v
B
)
−1
Fact Summary
Let V be a vector space with dim(V ) = n.
1. In ޒ
n
, the coordinates of a vector with respect to the standard basis are the
components of the vector.
2. Given any two ordered bases for V, a transition matrix can be used to
change the coordinates of a vector relative to one basis to the coordinates
relative to the other basis.
3. If B and B
are two ordered bases for V, the transition matrix from B to B
is the matrix [I]
B
B
whose column vectors are the coordinates of the basis
vectors of B relative to the basis B
, . . . , y
(n)
, where n is a fixed positive integer, is called an ordinary
differential equation of order n. We will henceforth drop the qualifier ordinary since
none of the equations we investigate will involve partial derivatives. Also, for obvious
reasons we will narrow the scope of our discussion and consider only equations of a
certain type.
Confirming Pages
186 Chapter 3 Vector Spaces
The Exponential Model
One of the simplest kinds of differential equations is the first-order equation given by
y
= ky
where k is a real number. This equation is used to model quantities that exhibit
exponential growth or decay and is based on the assumption that the rate of change
of the quantity present at any time t is directly proportional to the quantity present at
time t. A solution to a differential equation is a function y = f (t ) that satisfies the
equation, that is, results in an identity when substituted for y in the original equation.
To solve this equation, we write it as
y
y
= k
and integrate both sides of the equation with respect to the independent variable to
obtain
ln y =
_
y
= 3y. Then any function of
the form y(t ) = Ce
3t
is a solution. Since the parameter C in the solution is arbitrary,
the solution produces a family of functions all of which satisfy the differential equation.
For this reason y(t ) = Ce
3t
is called the general solution to y
= 3y.
In certain cases a physical constraint imposes a condition on the solution that
allows for the identification of a particular solution. If, for example, in the previous
problem it is required that y = 2 when t = 0, then 2 = Ce
3(0)
, so that C = 2. This is
called an initial condition. A differential equation together with an initial condition
is called an initial-value problem. The solution to the previous initial-value problem
is given by
y(t ) = 2e
3t
From a linear algebra perspective we can think of the general solution to the
differential equation y
= ky as the span, over ޒ, of the vector e
kt
which describes a
one-dimensional subspace of the vector space of differentiable functions on the real
line.
Second-Order Differential Equations with Constant
Coefficients
We now extend the differential equation of the previous subsection to second-order
and consider equations of the form
y
+ay
+by = 0
Motivated by the solution to the exponential model, we check to see if there are any
solutions of the form y = e
rx
, for some real number r. After computing the first and
Confirming Pages
3.5 Application: Differential Equations 187
second derivatives y
= re
rx
and y
= r
2
e
rx
, we see that y = e
rx
is a solution of the
second-order equation if and only if
r
2
e
rx
+are
rx
+be
rx
= 0
that is,
e
rx
(r
2
+ar +b) = 0
Since e
rx
> 0 for every choice of r and x, we know e
rx
is a solution of y
+ay
+
by = 0 if and only if
r
2
+ar +b = 0
This equation is called the auxiliary equation. As this equation is quadratic there
are three possibilities for the roots r
1
and r
2
. This in turn yields three possible vari-
ations for the solution of the differential equation. The auxiliary equation can have
two distinct real roots, one real root, or two distinct complex roots. These cases are
considered in order.
Case 1 The roots r
1
and r
2
are real and distinct.
In this case there are two solutions, given by
y
1
(x) = e
r
1
x
and y
2
(x) = e
r
2
x
EXAMPLE 1 Find two distinct solutions to the differential equation y
+5y = 0 is given
by r
2
−2r +5 = 0. Applying the quadratic formula gives the complex roots r
1
=
1 +2i and r
2
= 1 −2i. The two solutions to the differential equation are then given
by
y
1
(x) = e
x
cos 2x and y
2
(x) = e
x
sin 2x
In what follows we require Theorem 16 on existence and uniqueness for second-
order linear differential equations. A proof can be found in any text on ordinary
differential equations.
THEOREM 16 Let p(x), q(x), and f (x) be continuous functions on the interval I. If x
0
is in I,
then the initial-value problem
y
+p(x)y
+q(x)y = f (x) y(x
0
) = y
0
y
(x
0
) = y
0
has a unique solution on I.
Fundamental Sets of Solutions
With solutions in hand for each one of these cases, we now consider the question
as to whether there are other solutions to equations of this type, and if so, how they
can be described. The simple (but elegant) answer, to which the remainder of this
section is devoted, is found by using linear algebra. We will see that in each case
the functions y
1
(x) and y
2
(x) form a basis for the vector space of solutions to the
equation y
+ay
+by = 0. Accordingly, every solution y(x) to this equation can be
written as a linear combination y(x) = c
1
y
1
(x) +c
2
y
2
(x).
Toward this end, for a positive integer n ≥ 0, let V = C
(n)
(I) be the vector space
of all functions that are n times differentiable on the real interval I. If n = 0, then
C
(0)
(I) denotes the set of all continuous functions on I. We first show that the solution
set to the differential equation y
(x) = 0
Taken together, these equations form a linear system of two equations in the two
variables c
1
and c
2
. Observe that the determinant of the corresponding coefficient
matrix is W
_
f, g
¸
(x). Hence, if W[f, g](x) is nonzero for some x
0
∈ I, then by
Confirming Pages
190 Chapter 3 Vector Spaces
Theorem 17 of Sec. 1.6 we know that c
1
= c
2
= 0. Accordingly, f (x) and g(x)
are linearly independent.
The function W[f, g] of Theorem 18 is called the Wronskian of f and g. The
Wronskian, and the result of Theorem 18, can be extended to any finite set of functions
that have continuous derivatives up to order n.
If y
1
and y
2
are solutions to the differential equation y
+ay
+by = 0, then
Abel's formula for the Wronskian gives the next result.
THEOREM 19 Let y
1
(x) and y
2
(x) be solutions to the differential equation y
+ay
+by = 0.
The functions y
1
and y
2
are linearly independent if and only if W[y
1
, y
2
](x) = 0
for all x in I.
At this point we are now ready to show that any two linearly independent solutions
to the differential equation y
+ay
+by = 0 span the subspace of solutions.
THEOREM 20 Fundamental Set of Solutions Suppose that y
1
(x) and y
2
(x) are two linearly
independent solutions, on the interval I, to the differential equation
y
+ay
+by = 0
Then every solution can be written as a linear combination of y
1
(x) and y
2
(x).
Proof Let y(x) be a particular solution to the initial-value problem
y
2
(x)
Now substituting x
0
into both of these equations and using the initial conditions
above, we obtain the linear system of two equations in the two variables c
1
and c
2
given by
_
c
1
y
1
(x
0
) +c
2
y
2
(x
0
) = y
0
c
1
y
1
(x
0
) +c
2
y
2
(x
0
) = y
0
Observe that the determinant of the coefficient matrix is the Wronskian
W[y
1
, y
2
](x
0
). Since y
1
(x) and y
2
(x) are linearly independent, then by Theorem 19,
the determinant of the coefficient matrix is nonzero. Consequently, by Theorem 17
Confirming Pages
3.5 Application: Differential Equations 191
of Sec. 1.6, there exist unique numbers c
1
and c
2
that provide a solution for the
linear system. Define the function g by
g(x) = c
1
y
1
(x) +c
2
y
2
(x)
Then g(x) is also a solution to the original initial-value problem. By the uniqueness
part of Theorem 16,
y(x) = g(x) = c
1
y
1
(x) +c
2
y
2
(x)
as claimed.
The linearly independent solutions y
1
(x) and y
2
(x) of Theorem 20 are called
a fundamental set of solutions. In light of this theorem, the fundamental set
{y
1
(x), y
2
(x)} is a basis for the subspace S of solutions to y
+ay
+by = 0. As
there are two of them, dim(S) = 2.
We now return to the specific cases for the solutions to y
= 0 which reduces to case 1.
There are many physical applications of second-order differential equations with
constant coefficients. Two important areas are in mechanical and electrical oscillations.
Confirming Pages
192 Chapter 3 Vector Spaces
A fundamental problem in mechanics is the motion of an object on a vibrating spring.
The motion of the object is described by the solution of an initial-value problem of
the form
my
+cy
+ky = f (x) y(0) = A y
(0) = B
where m is the mass of the object attached to the spring, c is the damping coefficient,
k is the stiffness of the spring, and f (x) represents some external force. If there are
no external forces acting on the system, then f (x) = 0.
EXAMPLE 4
Let the mass of an object attached to a spring be m = 1, and the spring constant
k = 4. Solve the three initial-value problems describing the position of the object
attached to the spring with no external forces; initial conditions y(0) = 2, y
(0) = 0;
and damping coefficients c equaling 2, 4, and 5.
Solution The differential equation describing the position of the object is given by
y
(x) is the derivative of p(x).
a. Show that T is a linear transformation.
b. Find the image of the polynomial p(x) = 3x
3
+2x
2
−x +2.
c. Describe the polynomials in P
3
that are mapped to the zero vector of P
2
.
Solution First observe that if p(x) is in P
3
, then it has the form
p(x) = ax
3
+bx
2
+cx +d
so that
T (p(x)) = p
(x) = 3ax
2
+2bx +c
Since p
(x) is in P
2
, then T is a map from P
3
into P
2
.
a. To show that T is linear, let p(x) and q(x) be polynomials of degree 3 or less,
and let k be a scalar. Recall from calculus that the derivative of a sum is the
sum of the derivatives, and that the derivative of a scalar times a function is
the scalar times the derivative of the function. Consequently,
T (kp(x) +q(x)) =
d
dx
(kp(x) +q(x))
=
d
dx
(kp(x)) +
d
dx
(q(x))
= kp
(x)
Find the null space and range of T.
Solution Recall that the derivative of a constant polynomial is 0. Since these are the only
polynomials for which the derivative is 0, we know that N(T ) is the set of constant
polynomials in P
4
. We claim that the range of T is all of P
3
. To see this, let
q(x) = ax
3
+bx
2
+cx +d be an arbitrary element of P
3
. A polynomial p(x)
whose derivative is q(x) is found by using the antiderivative. That is, to find p(x),
we integrate q(x) to obtain
p(x) =
_
q(x) dx =
_
(ax
3
+bx
2
+cx +d) dx =
a
4
x
4
+
b
3
x
3
+
c
2
x
2
+dx +e
which is an element of P
4
, with p
(x)
Find the dimension of the range of T, and give a description of the range.
Solution Let B = {1, x, x
2
, x
3
, x
4
} be the standard basis for P
4
. Since p(x) is in N(T ) if and
only if its degree is 0 or 1, the null space is the subspace of P
4
consisting of polyno-
mials with degree 1 or less. Hence, {1, x} is a basis for N(T ), and dim(N(T )) = 2.
Since dim(P
4
) = 5, by Theorem 5 we have
2 +dim(R(T )) = 5 so dim(R(T )) = 3
Then as in the proof of Theorem 5, we have
_
T (x
2
), T (x
3
), T (x
4
)
_
=
_
2, 6x, 12x
2
_
is a basis for R(T ). Observe that R(T ) is just the subspace P
2
of P
4
.
Matrices
In Sec. 3.2 we defined the column space of a matrix A, denoted by col(A), as the
span of its column vectors. We also defined the null space of the m×n matrix A as
the set of all vectors x in ޒ
n
such that Ax = 0. We further examine these notions here
in the context of linear transformations. In particular, let A be an m×n matrix and
let T: ޒ
n
−→ޒ
m
be the linear transformation defined by
T (v) = Av
This last equation can be written in vector form as
T (v) = v
1
A
1
+v
2
A
2
+· · · +v
n
A
n
where A
i
are the column vectors of A, and v
i
are the components of v for 1 ≤ i ≤ n.
In this way we see that the range of T, which is a subspace of ޒ
m
, is equal to the
column space of A, that is,
R(T ) = col(A)
The dimension of the column space of A is called the column rank of A. We also have
N(T ) = {v ∈ ޒ
n
| Av = 0} = N(A)
The dimension of N(A) is called the nullity of A. Applying Theorem 5, we have
column rank(A) +nullity(A) = n
Another subspace of ޒ
n
associated with the matrix A is the row space of A,
denoted by row(A), and is the span of the row vectors of A. Since the transpose
operation maps the row vectors of A to the column vectors of A
t
, the row space of
A is the same as the column space of A
t
, that is,
row(A) = col(A
t
)
Confirming Pages
222 Chapter 4 Linear Transformations
By using the algorithm for finding a basis, given in Sec. 3.3, a basis for col(A) can
be found by row reduction. In particular, the columns with the leading 1s in the row-
reduced form of A correspond to the column vectors of A needed for a basis of col(A).
Hence, the column rank of A is equal to the number of leading 1s in the row-reduced
form of A. On the other hand, row-reducing A eliminates row vectors that are linear
combinations of the others, so that the nonzero row vectors of the reduced form of A
form a basis for row(A). Hence, the row rank is equal to the number of leading 1s
in the reduced form of A. We have now established Theorem 6.
THEOREM 6 The row rank and the column rank of a matrix A are equal.
We can now define the rank of a matrix A as dim(row(A)) or dim(col(A)).
Again by Theorem 5, we have
rank(A) +nullity(A) = n
Linear Systems
When the nullity of a matrix A is known, the above formula can sometimes be used to
determine whether the linear system Ax = b is consistent. For example, suppose that
a linear system consists of 20 equations each with 22 variables. Further suppose that a
basis for the null space of the 20 ×22 coefficient matrix consists of two vectors. That
is, every solution to the homogeneous linear system Ax = 0 is a linear combination
of two linearly independent vectors in ޒ
22
. Then nullity(A) = 2, so that
dim(col(A)) = rank(A) = 22 −nullity(A) = 20
But the only subspace of ޒ
20
with dimension 20 is ޒ
20
itself. Hence, col(A) = ޒ
20
,
and consequently every vector b in ޒ
20
is a linear combination of the columns of A.
That is, the linear system Ax = b is consistent for every vector b in ޒ
20
. In general,
if A is an m×n matrix, nullity(A) = r, and dim(col) = n −r = m, then the linear
system Ax = b is consistent for every vector b in ޒ
m
.
We now add several more items to the list of equivalences given in Theorem 9
of Sec. 2.3, connecting solutions of the linear system Ax = b and properties of the
coefficient matrix A.
THEOREM 7 Let A be an n ×n matrix. Then the following statements are equivalent.
1. The matrix A is invertible.
2. The linear system Ax = b has a unique solution for every vector b.
3. The homogeneous linear system Ax = 0 has only the trivial solution.
4. The matrix A is row equivalent to the identity matrix.
Confirming Pages
4.2 The Null Space and Range 223
5. The determinant of the matrix A is nonzero.
6. The column vectors of A are linearly independent.
7. The column vectors of A span ޒ
n
.
8. The column vectors of A are a basis for ޒ
n
.
9. rank(A) = n
10. R(A) = col(A) = ޒ
n
11. N(A) = {0}
12. row(A) = ޒ
n
13. The number of pivot columns of the reduced row echelon form of A is n.
Fact Summary
Let V and W be vector spaces and T a linear transformation from V into W.
1. The null space N(T ) is a subspace of V, and the the range R(T ) is a
subspace of W.
2. If B = {v
1
, . . . , v
n
} is a basis for V , then
R(T ) = span{T (v
1
), . . . , T(v
n
)}
3. If V and W are finite dimensional vector spaces, then
dim(V ) = dim(R(T )) +dim(N(T ))
4. If A is an m×n matrix, then
rank(A) +nullity(A) = n
5. If A is an m×n matrix, then the rank of A is the number of leading 1s in
the row-reduced form of A.
6. If A is an n ×n invertible matrix, in addition to Theorem 9 of Sec. 2.3, we
know that rank(A) = n, R(A) = col(A) = ޒ
n
, N(A) = {0}, and the
number of leading 1s in the row echelon form of A is n.
Exercise Set 4.2
In Exercises 1–4, define a linear operator T: ޒ
2
→ޒ
2
by
T
__
x
y
__
=
_
x −2y
−2x +4y
_
Determine whether the vector v is in N(T ).
1. v =
_
0
0
_
Confirming Pages
224 Chapter 4 Linear Transformations
2. v =
_
2
1
_
3. v =
_
1
3
_
4. v =
_
1
2
1
4
_
In Exercises 5–8, define a linear operator
T: P
3
→P
3
by
T (p(x)) = xp
(x)
a. Describe the range of T.
b. Find dim(R(T )).
c. Find dim(N(T )).
38. Define a linear operator T: P
n
→P
n
by
T (p(x)) =
d
k
dx
k
(p(x))
where 1 ≤ k ≤ n. Show dim(N(T )) = k.
39. Suppose T: ޒ
4
→ޒ
6
is a linear transformation.
a. If dim(N(T )) = 2, then find dim(R(T )).
b. If dim(R(T )) = 3, then find dim(N(T )).
40. Show that if T: V →V is a linear operator such
that R(T ) = N(T ), then dim(V) is even.
41. Let
A =
_
1 0
0 −1
_
Define T: M
2×2
→M
2×2
by
T (B) = AB −BA
Find a basis for the null space of T.
42. Define T: M
n×n
→M
n×n
by T (A) = A
t
. Show
that R(T ) = M
n×n
.
43. Define T: M
n×n
→M
n×n
by T (A) = A +A
t
.
a. Find R(T ).
b. Find N(T ).
44. Define T: M
n×n
→M
n×n
by T (A) = A −A
t
.
a. Find R(T ).
b. Find N(T ).
45. Let A be a fixed n ×n matrix, and define
T: M
n×n
→M
n×n
by T (B) = AB. When does
R(T ) = M
n×n
?
46. Let A be a fixed n ×n diagonal matrix, and
define T: ޒ
n
→ޒ
n
by T (v) = Av.
a. Show dim(R(T )) is the number of nonzero
entries on the diagonal of A.
b. Find dim(N(T )). How is it related to the
diagonal terms of the matrix A?
4.3
ß
Isomorphisms
Many of the vector spaces that we have discussed are, from an algebraic perspective,
the same. In this section we show how an isomorphism, which is a special kind of
linear transformation, can be used to establish a correspondence between two vector
spaces. Essential to this discussion are the concepts of one-to-one and onto mappings.
For a more detailed description see App. A, Sec. A.2.
DEFINITION 1 One-to-One and Onto Let V and W be vector spaces and T: V →W a
mapping.
1. The mapping T is called one-to-one (or injective) if u = v implies that T (u) =
T (v). That is, distinct elements of V must have distinct images in W.
2. The mapping T is called onto (or surjective) if T (V) = W. That is, the range
of T is W.
A mapping is called bijective if it is both injective and surjective.
When we are trying to show that a mapping is one-to-one, a useful equiva-
lent formulation comes from the contrapositive statement. That is, T is one-to-one if
T (u) = T (v) implies that u = v. To show that a mapping is onto, we must show that
if w is an arbitrary element of W, then there is some element v ∈ V with T (v) = w.
Confirming Pages
4.3 Isomorphisms 227
EXAMPLE 1 Let T: ޒ
2
→ޒ
2
be the mapping defined by T (v) = Av, with
A =
1 1
−1 0
Show that T is one-to-one and onto.
Solution First, to show that T is one-to-one, let
u =
(x). Find the matrix
representation [T ]
B
1
relative to the basis
B
1
= {1, x, x
2
} and the matrix representation
[T ]
B
2
relative to B
2
= {1, x, 1 +x
2
}. Find the
transition matrix P = [I]
B
1
B
2
, and use Theorem 15
to show directly that the matrices [T ]
B
1
and [T ]
B
2
are similar.
17. Show that if A and B are similar matrices and B
and C are similar matrices, then A and C are
similar matrices.
18. Show that if A and B are similar matrices, then
det(A) = det(B).
19. Show that if A and B are similar matrices, then
tr(A) = tr(B).
20. Show that if A and B are similar matrices, then
A
t
and B
t
are similar matrices.
21. Show that if A and B are similar matrices, then
A
n
and B
n
are similar matrices for each positive
integer n.
22. Show that if A and B are similar matrices and λ
is any scalar, then det(A −λI) = det(B −λI).
4.6
ß
Application: Computer Graphics
The rapid development of increasingly more powerful computers has led to the explo-
sive growth of digital media. Computer-generated visual content is ubiquitous, found
in almost every arena from advertising and entertainment to science and medicine.
The branch of computer science known as computer graphics is devoted to the study
of the generation and manipulation of digital images. Computer graphics are based
on displaying two- or three-dimensional objects in two-dimensional space. Images
displayed on a computer screen are stored in memory using data items called pixels,
which is short for picture elements. A single picture can be comprised of millions
of pixels, which collectively determine the image. Each pixel contains informa-
tion on how to color the corresponding point on a computer screen, as shown in
Fig. 1. If an image contains curves or lines, the pixels which describe the object
may be connected by a mathematical formula. The saddle shown in Fig. 1 is an
example.
Figure 1
Confirming Pages
256 Chapter 4 Linear Transformations
Graphics Operations in ޒ
2
To manipulate images, computer programmers use linear transformations. Most of the
examples we consider in this section use linear operators on ޒ
2
. One of the proper-
ties of linear transformations that is especially useful to our work here is that linear
transformations map lines to lines, and hence polygons to polygons. (See Exercise 10
of the Review Exercises for Chapter 4.) Therefore, to visualize the result of a linear
transformation on a polygon, we only need to transform the vertices. Connecting the
images of the vertices then gives the transformed polygon.
Scaling and Shearing
A transformation on an object that results in a horizontal contraction or dilation
(stretching) is called a horizontal scaling. For example, let T be the triangle shown
in Fig. 2 with vertices (1, 1), (2, 1), and
_
3
2
, 3
_
. Suppose that we wish to perform
a horizontal scaling of T by a factor of 3. The transformed triangle T
, as shown in Fig. 3.
In general, a horizontal scaling by a factor k is given by the linear transformation
S
h
defined by
S
h
__
x
y
__
=
_
kx
y
_
Confirming Pages
4.6 Application: Computer Graphics 257
The matrix representation of S
h
, relative to the standard basis for ޒ
2
, is given by
[S
h
]
B
=
_
k 0
0 1
_
Similarly, a vertical scaling is given by the linear operator
S
v
__
x
y
__
=
_
x
ky
_
The matrix representation of S
v
, relative to the standard basis for ޒ
2
, is given by
[S
v
]
B
=
_
1 0
0 k
_
If both components are multiplied by the same number k, then the result is called a
uniform scaling. In all the above cases, if k > 1, then the transformation is called a
dilation, or stretching; and if 0 < k < 1, then the operator is a contraction.
EXAMPLE 1
Let T denote the triangle with vertices given by the vectors
v
1
=
_
0
1
_
v
2
=
_
2
1
_
v
3
=
_
1
3
_
as shown in Fig. 4.
a. Stretch the triangle horizontally by a factor of 2.
b. Contract the triangle vertically by a factor of 3.
c. Stretch the triangle horizontally by a factor of 2, and contract the triangle
vertically by a factor of 3.
Solution a. To stretch the triangle horizontally by a factor of 2, we apply the matrix
_
2 0
0 1
_
to each vertex to obtain
v
is obtained from triangle T through a horizontal scaling by a factor of
3, followed by a vertical scaling by a factor of 2, without changing the left vertex
(1, 1). The scalings alone will move the point (1, 1) to (3, 2). One way to correct this
is to first translate the triangle so that the left vertex is located at the origin, perform
the scaling, and then translate back. The matrix to perform all these operations is
the product of the matrices for each transformation. The matrix is given by
_
_
1 0 1
0 1 1
0 0 1
_
_
_
_
1 0 0
0 2 0
0 0 1
_
_
_
_
3 0 0
0 1 0
0 0 1
_
_
_
_
1 0 −1
0 1 −1
0 0 1
_
_
=
_
_
3 0 −2
0 2 −1
0 0 1
_
_
Notice that
_
_
1 0 1
0 1 1
0 0 1
_
_
=
_
_
1 0 −1
0 1 −1
0 0 1
_
_
−1
that is, the matrix representation for translation by
_
1
1
_
is the inverse of the matrix
representation for translation by
_
−1
−1
_
.
Rotation
Another common graphics operation is a rotation through an angle θ. See Fig. 12. To
describe how a point is rotated, let (x, y) be the coordinates of a point in ޒ
2
and θ a
real number. From trigonometry, the new coordinates (x
, y
) of a point (x, y) rotated
x
y
x
y
Rotation by 45
◦
Figure 12 by θ rad about the origin are given by
x
(x)
then q(x) = x
2
is in R(T ).
37. The linear operator T: ޒ
3
→ޒ
3
defined by
T
_
_
_
_
x
y
z
_
_
_
_
=
_
_
3 −3 0
1 2 1
3 −1 1
_
_
_
_
x
y
z
_
_
is an isomorphism.
38. If A is an m×n matrix and T: ޒ
n
→ޒ
m
is
defined by
T (v) = Av
then the range of T is the set of all linear
combinations of the column vectors of A.
39. If A is an m×n matrix with m > n and
T: ޒ
n
→ޒ
m
is defined by
T (v) = Av
then T cannot be one-to-one.
40. If A is an m×n matrix with m > n and
T: ޒ
n
→ޒ
m
is defined by
T (v) = Av
then T cannot be onto.
Confirming Pages
C H A P T E R
Eigenvalues and
Eigenvectors
CHAPTER OUTLINE 5.1 Eigenvalues and Eigenvectors 276
5.2 Diagonalization 287
5.3 Application: Systems of Linear Differential Equations 300
5.4 Application: Markov Chains 310
A
Markov chain is a mathematical model used
to describe a random process that, at any
given time t = 1, 2, 3, . . . , is in one of a finite
number of states. Between the times t and t +1
the process moves from state j to state i with
a probability p
ij
. Markov processes are also
memoryless; that is, the next state of the sys-
tem depends only on the current state. As an
example, consider a city C with surrounding res-
idential areas N, S, E, and W. Residents can
move between any two locations or stay in their
current location, with fixed probabilities. In this
case a state is the location of a resident at any
given time. The state diagram shown in Fig. 1
describes the situation with the probabilities of
moving from one location to another shown in the
corresponding transition matrix A = (p
ij
). For
example, entry p
34
= 0.2 is the probability that
U.S. Geological Survery/DAL
C
N
S E
W
Figure 1 a resident in region E moves to region S. Since a resident is assumed to be living
in one of the five regions, the probability of being in one of these regions is 1, and
hence each column sum of A is equal to 1. A square matrix with each entry between
C N S E W
C
N
S
E
W
_
¸
¸
_
0.3 0.2 0.4 0.1 0.1
0.2 0.4 0.1 0.2 0.1
0.1 0.2 0.2 0.2 0.1
0.2 0.1 0.2 0.3 0.2
0.2 0.1 0.1 0.2 0.5
_
¸
¸
_
0 and 1 and column sums all equal to 1 is called a stochastic matrix. The initial
distribution of the population is called the initial probability vector. Assume that the
initial population distribution is given by the vector
v =
_
¸
¸
¸
¸
_
0.3
0.2
0.1
0.2
0.2
_
¸
¸
¸
¸
_
275
Confirming Pages
276 Chapter 5 Eigenvalues and Eigenvectors
Then the population distribution after one time step is Av, after two time steps is
A(Av) = A
2
v, and so on. For example, after 10 time steps, the population distribution
(rounded to two decimal places) is
A
10
v =
_
¸
¸
¸
¸
_
0.21
0.20
0.16
0.20
0.23
_
¸
¸
¸
¸
_
Notice that the sum of the entries of the population distribution vector is equal to 1.
Starting with some initial distribution vector, the long-term behavior of the Markov
chain, that is, A
n
v as n tends to infinity, gives the limiting population distribution
in the five regions into the future. When A
n
v approaches a distribution vector s
as n tends toward infinity, we say that s is the steady-state vector. If a transition
matrix for a Markov chain is a stochastic matrix with positive terms, then for any
initial probability vector v, there is a unique steady-state vector s. Moreover, if s is
the steady-state vector, then As = s. Finding the steady-state vector is equivalent to
solving the matrix equation
Ax = λx
with λ = 1. In general, if there is a scalar λ and a nonzero vector v such that Av = λv,
then λ is called an eigenvalue for the matrix A and v is an eigenvector corresponding
to the eigenvalue λ. In our Markov chain example, the steady-state vector corresponds
to the eigenvalue λ = 1 for the transition matrix A.
In the last decade the growth in the power of modern computers has, quite mirac-
ulously, made it possible to compute the eigenvalues of a matrix with rows and
columns in the billions. Google's page rank algorithm is essentially a Markov chain
with transition matrix consisting of numerical weights for each site on the World Wide
Web used as a measure of its relative importance within the set. The algorithm was
developed by Larry Page and Sergey Brin, the founders of Google.
For any n ×n matrix A, there exists at least one number-vector pair λ, v such that
Av = λv (although λ may be a complex number). That is, the product of A and v is
a scaling of the vector v. Many applications require finding such number-vector pairs.
5.1
ß
Eigenvalues and Eigenvectors
One of the most important problems in linear algebra is the eigenvalue problem. It
can be stated thus: If A is an n ×n matrix, does there exist a nonzero vector v such
that Av is a scalar multiple of v?
DEFINITION 1 Eigenvalue andEigenvector Let A be an n ×n matrix. A number λ is called
an eigenvalue of A provided that there exists a nonzero vector v in ޒ
n
such that
Av = λv
Confirming Pages
5.1 Eigenvalues and Eigenvectors 277
Every nonzero vector satisfying this equation is called an eigenvector of A corre-
sponding to the eigenvalue λ.
The zero vector is a trivial solution to the eigenvalue equation for any number λ
and is not considered as an eigenvector.
As an illustration, let
A =
_
1 2
0 −1
_
Observe that _
1 2
0 −1
_ _
1
0
_
=
_
1
0
_
so v
1
=
_
1
0
_
is an eigenvector of A corresponding to the eigenvalue λ
1
= 1. We
also have _
1 2
0 −1
_ _
1
−1
_
=
_
−1
1
_
= −1
_
1
−1
_
so v
2
=
_
1
−1
_
is another eigenvector of A corresponding to the eigenvalue λ
2
= −1.
In Example 1 we show how to find eigenvalues and eigenvectors for a 2 ×2
matrix.
EXAMPLE 1
Let
A =
_
0 1
1 0
_
a. Find the eigenvalues of A.
b. Find the eigenvectors corresponding to each of the eigenvalues found in
part (a).
Solution a. The number λ is an eigenvalue of A if there is a nonzero vector v =
_
x
y
_
such that
_
0 1
1 0
_ _
x
y
_
= λ
_
x
y
_
which is equivalent to
_
y
x
_
=
_
λx
λy
_
This matrix equation is equivalent to the homogeneous linear system
_
−λx + y = 0
x − λy = 0
By Theorem 17 of Sec. 1.6, the linear system has a nontrivial solution if and
only if
¸
¸
¸
¸
−λ 1
1 −λ
¸
¸
¸
¸
= 0
Confirming Pages
278 Chapter 5 Eigenvalues and Eigenvectors
Consequently, λ is an eigenvalue of A if and only if
λ
2
−1 = 0 so that λ
1
= 1 and λ
2
= −1
b. For λ
1
= 1, a vector v
1
=
_
x
y
_
is an eigenvector if
_
0 1
1 0
_ _
x
y
_
=
_
x
y
_
This yields the linear system
_
−x +y =0
x −y =0
with solution set S =
__
t
t
_¸
¸
¸
¸
t ∈ ޒ
_
Thus, any vector of the form v
1
=
_
t
t
_
, for t = 0, is an eigenvector corre-
sponding to the eigenvalue λ
1
= 1. In a similar way, we find that any vector
of the form v
2
=
_
t
−t
_
, for t = 0, is an eigenvector of A corresponding to
the eigenvalue λ
2
= −1. Specific eigenvectors of A can be found by choosing
any value for t so that neither v
1
nor v
2
is the zero vector. For example, letting
t = 1, we know that
v
1
=
_
1
1
_
is an eigenvector corresponding to λ
1
= 1 and
v
2
=
_
1
−1
_
is an eigenvector corresponding to λ
2
= −1.
Geometric Interpretation of Eigenvalues and Eigenvectors
A nonzero vector v is an eigenvector of a matrix A only when Av is a scaling of
the vector v. For example, let A =
_
1 −1
2 4
_
. Using the techniques just introduced,
the eigenvalues of A are λ
1
= 2 and λ
2
= 3 with corresponding eigenvectors
v
1
=
_
1
−1
_
and v
2
=
_
1
−2
_
, respectively. Observe that
Av
1
=
_
1 −1
2 4
_ _
1
−1
_
=
_
2
−2
_
= 2v
1
and
Av
2
=
_
1 −1
2 4
_ _
1
−2
_
=
_
3
−6
_
= 3v
2
Confirming Pages
5.1 Eigenvalues and Eigenvectors 279
In Fig. 2, we provide sketches of the vectors v
1
, v
2
, Av
1
, and Av
2
to underscore that
the action of A on each of its eigenvectors is a scaling. Observe that this is not the
case for an arbitrary vector. For example, if v =
_
1
1
_
, then
Av =
_
1 −1
2 4
_ _
1
1
_
=
_
0
6
_
Eigenspaces
Notice that in Example 1, for each of the eigenvalues there are infinitely many eigen-
vectors. This is the case in general. To show this, let v be an eigenvector of the matrix
x
y
Ϫ5
Ϫ6
5
5
v
1
Av
1 v
2
Av
2
Figure 2 A corresponding to the eigenvalue λ. If c is any nonzero real number, then
A(cv) = cA(v) = c(λv) = λ(cv)
so cv is another eigenvector associated with the eigenvalue λ. Notice that all eigen-
vectors corresponding to an eigenvalue are parallel but can have opposite directions.
Building on the procedure used in Example 1, we now describe a general method
for finding eigenvalues and eigenvectors. If A is an n ×n matrix, then
Av = λv
for some number λ if and only if
Av −λv = 0 that is (A −λI)v = Av −λIv = 0
Again by Theorem 17, of Sec. 1.6, this equation has a nontrivial solution if and only
if
det(A −λI) = 0
We summarize this result in Theorem 1.
THEOREM 1 The number λ is an eigenvalue of the matrix A if and only if
det(A −λI) = 0
The equation det(A −λI) = 0 is called the characteristic equation of the matrix
A, and the expression det(A −λI) is called the characteristic polynomial of A. If
A is an n ×n matrix and λ is an eigenvalue of A, then the set
V
λ
= {v ∈ ޒ
n
| Av = λv}
is called the eigenspace of A corresponding to λ. Notice that V
λ
is the union of the
set of eigenvectors corresponding to λ and the zero vector.
We have already shown that V
λ
is closed under scalar multiplication. Therefore,
to show that V
λ
is a subspace of ޒ
n
, we need to show that it is also closed under
addition. To see this, let u and v be vectors in V
λ
; that is, Au = λu and Av = λv for
a particular eigenvalue λ. Then
A(u +v) = Au +Av = λu +λv = λ(u +v)
Confirming Pages
280 Chapter 5 Eigenvalues and Eigenvectors
Alternatively, the set
V
λ
= {v ∈ ޒ
n
| Av = λv} = {v ∈ ޒ
n
| (A −λI)v = 0} = N(A −λI)
Since V
λ
is the null space of the matrix A −λI, by Theorem 3 of Sec. 4.2 it is a
subspace of ޒ
n
.
EXAMPLE 2
Find the eigenvalues and corresponding eigenvectors of
A =
_
2 −12
1 −5
_
Give a description of the eigenspace corresponding to each eigenvalue.
Solution By Theorem 1 to find the eigenvalues, we solve the characteristic equation
det(A −λI) =
¸
¸
¸
¸
2 −λ −12
1 −5 −λ
¸
¸
¸
¸
= (2 −λ)(−5 −λ) −(1)(−12)
= λ
2
+3λ +2
= (λ +1)(λ +2) = 0
Thus, the eigenvalues are λ
1
= −1 and λ
2
= −2. To find the eigenvectors, we need
to find all nonzero vectors in the null spaces of A −λ
1
I and A −λ
2
I. First, for
λ
1
= −1,
A −λ
1
I = A +I =
_
2 −12
1 −5
_
+
_
1 0
0 1
_
=
_
3 −12
1 −4
_
The null space of A +I is found by row-reducing the augmented matrix
_
3 −12 0
1 −4 0
_
to
_
1 −4 0
0 0 0
_
The solution set for this linear system is given by S =
__
4t
t
_¸
¸
¸
¸
t ∈ ޒ
_
. Choosing
t = 1, we obtain the eigenvector v
1
=
_
4
1
_
. Hence, the eigenspace corresponding
to λ
1
= −1 is
V
λ
1
=
_
t
_
4
1
_¸
¸
¸
¸
t is any real number
_
For λ
2
= −2,
A −λ
2
I =
_
4 −12
1 −3
_
In a similar way we find that the vector v
2
=
_
3
1
_
is an eigenvector corresponding
to λ
2
= −2. The corresponding eigenspace is
Confirming Pages
5.1 Eigenvalues and Eigenvectors 281
V
λ
2
=
_
t
_
3
1
_¸
¸
¸
¸
t is any real number
_
The eigenspaces V
λ
1
and V
λ
2
are lines in the direction of the eigenvectors
_
4
1
_
and
_
3
1
_
, respectively. The images of the eigenspaces, after multiplication by
A, are the same lines, since the direction vectors A
_
4
1
_
and A
_
3
1
_
are scalar
multiples of
_
4
1
_
and
_
3
1
_
, respectively.
In Example 3 we illustrate how the eigenspace associated with a single eigenvalue
can have dimension greater than 1.
EXAMPLE 3
Find the eigenvalues of
A =
_
¸
¸
_
1 0 0 0
0 1 5 −10
1 0 2 0
1 0 0 3
_
¸
¸
_
and find a basis for each of the corresponding eigenspaces.
Solution The characteristic equation of A is
det(A −λI) =
¸
¸
¸
¸
¸
¸
¸
¸
1 −λ 0 0 0
0 1 −λ 5 −10
1 0 2 −λ 0
1 0 0 3 −λ
¸
¸
¸
¸
¸
¸
¸
¸
= (λ −1)
2
(λ −2)(λ −3) = 0
Thus, the eigenvalues are
λ
1
= 1 λ
2
= 2 and λ
3
= 3
Since the exponent of the factor λ −1 is 2, we say that the eigenvalue λ
1
= 1 has
algebraic multiplicity 2. To find the eigenspace for λ
1
= 1, we reduce the matrix
A −(1)I =
_
¸
¸
_
0 0 0 0
0 0 5 −10
1 0 1 0
1 0 0 2
_
¸
¸
_
to
_
¸
¸
_
1 0 0 2
0 0 1 −2
0 0 0 0
0 0 0 0
_
¸
¸
_
Confirming Pages
282 Chapter 5 Eigenvalues and Eigenvectors
Hence, the eigenspace corresponding to λ
1
= 1 is
V
1
=
_
¸
¸
_
¸
¸
_
s
_
¸
¸
_
0
1
0
0
_
¸
¸
_
+t
_
¸
¸
_
−2
0
2
1
_
¸
¸
_
¸
¸
¸
¸
¸
¸
¸
¸
s, t ∈ ޒ
_
¸
¸
_
¸
¸
_
Observe that the two vectors
_
¸
¸
_
0
1
0
0
_
¸
¸
_
and
_
¸
¸
_
−2
0
2
1
_
¸
¸
_
are linearly independent and hence form a basis for V
λ
1
. Since dim(V
λ
1
) = 2, we
say that λ
1
has geometric multiplicity equal to 2. Alternatively, we can write
V
λ
1
= span
_
¸
¸
_
¸
¸
_
_
¸
¸
_
0
1
0
0
_
¸
¸
_
,
_
¸
¸
_
−2
0
2
1
_
¸
¸
_
_
¸
¸
_
¸
¸
_
Similarly, the eigenspaces corresponding to λ
2
= 2 and λ
3
= 3 are, respectively,
V
λ
2
= span
_
¸
¸
_
¸
¸
_
_
¸
¸
_
0
5
1
0
_
¸
¸
_
_
¸
¸
_
¸
¸
_
and V
λ
3
= span
_
¸
¸
_
¸
¸
_
_
¸
¸
_
0
−5
0
1
_
¸
¸
_
_
¸
¸
_
¸
¸
_
In Example 3 the algebraic and geometric multiplicities of each eigenvalue are
equal. This is not the case in general. For example, if
A =
_
1 1
0 1
_
then the characteristic equation is (λ −1)
2
= 0. Thus, λ = 1 has algebraic multiplicity
2. However,
V
λ
=
__
t
0
_¸
¸
¸
¸
t ∈ ޒ
_
so λ = 1 has geometric multiplicity 1.
Although eigenvectors are always nonzero, an eigenvalue can be zero. Also, as
mentioned at the beginning of this section, eigenvalues can be complex numbers.
These cases are illustrated in Example 4.
EXAMPLE 4
Find the eigenvalues of
A =
_
_
0 0 0
0 0 −1
0 1 0
_
_
Confirming Pages
5.1 Eigenvalues and Eigenvectors 283
Solution The characteristic equation is
det(A −λI) =
¸
¸
¸
¸
¸
¸
−λ 0 0
0 −λ −1
0 1 −λ
¸
¸
¸
¸
¸
¸
= −λ
3
−λ = −λ(λ
2
+1) = 0
Thus, the eigenvalues are λ
1
= 0, λ
2
= i, and λ
3
= −i. The corresponding eigen-
vectors are _
_
1
0
0
_
_
_
_
0
1
−i
_
_
and
_
_
0
1
i
_
_
A fact that will be useful in the next section has to do with the eigenvalues of a
square triangular matrix. For example, let
A =
_
2 4
0 −3
_
Since det(A −λI) = 0 if and only if (2 −λ)(−3 −λ) = 0, we see that the eigenval-
ues of A are precisely the diagonal entries of A. In general, we have the following
result.
PROPOSITION 1 The eigenvalues of an n ×n triangular matrix are the numbers on the diagonal.
Proof Let A be an n ×n triangular matrix. By Theorem 13 of Sec. 1.6, the
characteristic polynomial is given by
det(A −λI) = (a
11
−λ)(a
22
−λ) · · · (a
nn
−λ)
Hence, det(A −λI) = 0 if and only if λ
1
= a
11
, λ
2
= a
22
, . . . , λ
n
= a
nn
.
Eigenvalues and Eigenvectors of Linear Operators
The definitions of eigenvalues and eigenvectors can be extended to linear operators.
DEFINITION 2 Eigenvalue andEigenvector of a Linear Operator Let V be a vector space
and T:V −→V be a linear operator. A number λ is an eigenvalue of T provided
that there is a nonzero vector v in V such that T (v) = λv. Every nonzero vector that
satisfies this equation is an eigenvector of T corresponding to the eigenvalue λ.
As an illustration define T: P
2
→P
2
by
T (ax
2
+bx +c) = (−a +b +c)x
2
+(−b −2c)x −2b −c
Observe that
T (−x
2
+x +1) = 3x
2
−3x −3 = −3(−x
2
+x +1)
Confirming Pages
284 Chapter 5 Eigenvalues and Eigenvectors
so p(x) = −x
2
+x +1 is an eigenvector of T corresponding to the eigenvalue
λ = −3.
Example 5 is from ordinary differential equations.
EXAMPLE 5 Interpret the solutions to the equation
f
(x) = kf (x)
as an eigenvalue problem of a linear operator.
Solution Let D denote the collection of all real-valued functions of one variable that have
derivatives of all orders. Examples of such functions are polynomials, the trigono-
metric functions sin(x) and cos(x), and the natural exponential function e
x
on ޒ.
Define a linear operator T: D −→D by
T (f (x)) = f
(x)
Then λ is an eigenvalue of T if there is a function f (x), not identically zero, such
that T (f (x)) = λf (x). That is, f (x) satisfies the differential equation
f
(x) = λf (x)
Nonzero solutions to this differential equation are eigenvectors of the operator T,
called eigenfunctions, corresponding to the eigenvalue λ. The general solution to
this equation is given by
f (x) = ke
λx
where k is an arbitrary constant. This class of functions is a model for exponential
growth and decay with extensive applications.
Fact Summary
Let A be an n ×n matrix.
1. The number λ is an eigenvalue of A if and only if det(A −λI) = 0.
2. The expression det(A −λI) is a polynomial of degree n.
3. If λ is an eigenvalue of A and c is a nonzero scalar, then cλ is another
eigenvalue of A.
4. If λ is an eigenvalue of A, then the eigenspace
V
λ
= {v ∈ ޒ
n
| Av = λv}
is a subspace of ޒ
n
.
Confirming Pages
5.1 Eigenvalues and Eigenvectors 285
5. The eigenspace corresponding to λ is the null space of the matrix A −λI.
6. The eigenvalues of a square triangular matrix are the diagonal entries.
Exercise Set 5.1
In Exercises 1–6, a matrix A and an eigenvector v are
given. Find the corresponding eigenvalue directly by
solving Av = λv.
1. A =
_
3 0
1 3
_
v =
_
0
1
_
2. A =
_
−1 1
0 −2
_
v =
_
−1
1
_
3. A =
_
_
−3 2 3
−1 −2 1
−3 2 3
_
_
v =
_
_
1
0
1
_
_
4. A =
_
_
1 0 1
3 2 0
3 0 −1
_
_
v =
_
_
−
4
3
1
4
_
_
5. A =
_
¸
¸
_
1 0 1 1
0 1 0 0
1 1 0 0
0 1 0 1
_
¸
¸
_
v =
_
¸
¸
_
−1
0
−1
1
_
¸
¸
_
6. A =
_
¸
¸
_
1 1 1 0
−1 −1 0 −1
−1 1 0 1
0 −1 −1 0
_
¸
¸
_
v =
_
¸
¸
_
0
1
−1
0
_
¸
¸
_
In Exercises 7–16, a matrix A is given.
a. Find the characteristic equation for A.
b. Find the eigenvalues of A.
c. Find the eigenvectors corresponding to each
eigenvalue.
d. Verify the result of part (c) by showing that
Av
i
= λ
i
v
i
.
7. A =
_
−2 2
3 −3
_
8. A =
_
−2 −1
−1 −2
_
9. A =
_
1 −2
0 1
_
10. A =
_
0 2
−1 −3
_
11. A =
_
_
−1 0 1
0 1 0
0 2 −1
_
_
12. A =
_
_
0 2 0
0 −1 1
0 0 1
_
_
13. A =
_
_
2 1 2
0 2 −1
0 1 0
_
_
14. A =
_
_
1 1 1
0 1 0
0 0 1
_
_
15. A =
_
¸
¸
_
−1 0 0 0
0 2 0 0
0 0 −2 0
0 0 0 4
_
¸
¸
_
Confirming Pages
286 Chapter 5 Eigenvalues and Eigenvectors
16. A =
_
¸
¸
_
3 2 3 −1
0 1 2 1
0 0 2 0
0 0 0 −1
_
¸
¸
_
17. Show that if λ
2
+bλ +c is the characteristic
polynomial of the 2 ×2 matrix A, then
b = −tr(A) and c = det(A).
18. Let A be an invertible matrix. Show that if λ is
an eigenvalue of A, then 1/λ is an eigenvalue of
A
−1
.
19. Let A be an n ×n matrix. Show that A is not
invertible if and only if λ = 0 is an eigenvalue
of A.
20. Let V be a vector space with dim(V) = n and
T: V −→V a linear operator. If λ is an
eigenvalue of T with geometric multiplicity n,
then show that every nonzero vector of V is an
eigenvector.
21. Let A be an idempotent matrix. Show that if λ is
an eigenvalue of A, then λ = 0 or λ = 1.
22. Show that A and A
t
have the same eigenvalues.
Give an example to show A and A
t
can have
different eigenvectors.
23. Show that if there is a positive integer n such that
A
n
= 0, then λ = 0 is the only eigenvalue of A.
24. Let A =
_
1 0
0 −1
_
. Define an operator
T: M
2×2
→M
2×2
by
T (B) = AB −BA
a. Show that e =
_
0 1
0 0
_
is an eigenvector
corresponding to the eigenvalue λ = 2.
b. Show that f =
_
0 0
1 0
_
is an eigenvector
corresponding to the eigenvalue λ = −2.
25. Let A and B be n ×n matrices with A invertible.
Show that AB and BA have the same eigenvalues.
26. Show that no such matrices A and B exist such
that
AB −BA = I
27. Show that the eigenvalues of a square triangular
matrix are the diagonal entries of the matrix.
28. Let λ be an eigenvalue of A. Use mathematical
induction to show that for all n in the set of all
natural numbers ގ, if λ is an eigenvalue of A,
then λ
n
is an eigenvalue of A
n
. What can be said
about corresponding eigenvectors?
29. Let C = B
−1
AB. Show that if v is an eigenvector
of C corresponding to the eigenvalue λ, then Bv
is an eigenvector of A corresponding to λ.
30. Let A be an n ×n matrix and suppose v
1
, . . . , v
m
are eigenvectors of A. If S = span{v
1
, . . . , v
m
},
show that if v ∈ S, then Av ∈ S.
31. Let T: ޒ
2
→ޒ
2
be the linear operator that
reflects a vector through the x axis. Find the
eigenvalues and corresponding eigenvectors for T.
32. Define a linear operator T: ޒ
2
→ޒ
2
by
T
_
x
y
_
=
_
y
x
_
Show that the only eigenvalues of T are λ = ±1.
Find the corresponding eigenvectors.
33. Define a linear operator T: ޒ
2
→ޒ
2
by
T
_
x
y
_
=
_
cos θ −sin θ
sin θ cos θ
_ _
x
y
_
That is, the action of T is a counterclockwise
rotation of a vector by a nonnegative angle θ.
Argue that if θ = 0, π, then T has no real
eigenvalues; if θ = 0, then λ = 1 is an eigenvalue;
and if θ = π, then λ = −1 is an eigenvalue.
34. Let D denote the function space of all real-valued
functions that have two derivatives, and define a
linear operator T on D by
T (f ) = f
−2f
−3f
a. Show that for each k, the function f (x) = e
kx
is an eigenfunction for the operator T.
b. Find the corresponding eigenvalues for each
eigenfunction f (x) = e
kx
.
Confirming Pages
5.2 Diagonalization 287
c. Find two nonzero functions f such that
f
= {v
1
, v
2
, v
3
} and
P =
_
_
1 1 −5
1 1 4
−2 1 7
_
_
Then
[T ]
B
=
1
9
_
_
−1 4 −3
5 1 3
−1 1 0
_
_
_
_
3 −1 2
2 0 2
1 3 0
_
_
_
_
1 1 −5
1 1 4
−2 1 7
_
_
=
_
_
−2 0 0
0 4 0
0 0 1
_
_
Fact Summary
Let A be an n ×n matrix.
1. If A is diagonalizable, then A = PDP
−1
or equivalently D = P
−1
AP. The
matrix D is a diagonal matrix with diagonal entries the eigenvalues of A.
The matrix P is invertible whose column vectors are the corresponding
eigenvectors.
2. If A is diagonalizable, then the diagonalizing matrix P is not unique. If the
columns of P are permuted, then the diagonal entries of D are permuted in
the same way.
3. The matrix A is diagonalizable if and only if A has n linearly independent
eigenvectors.
4. If A has n distinct eigenvalues, then A is diagonalizable.
Revised Confirming Pages
298 Chapter 5 Eigenvalues and Eigenvectors
5. Every 2 ×2 real symmetric matrix is diagonalizable and has real
eigenvalues.
6. Similar matrices have the same eigenvalues.
7. If A is diagonalizable, then the algebraic multiplicity for each eigenvalue is
equal to the dimension of the corresponding eigenspace (the geometric
multiplicity). The common sum of these multiplicities is n.
8. Let T: V −→V be a linear operator on a finite dimensional vector
space V. If V has an ordered basis B consisting of eigenvectors of T, then
[T ]
B
is a diagonal matrix.
9. Let T: V −→V be a linear operator and B
1
and B
2
ordered bases for V.
Then [T ]
B
1
and [T ]
B
2
have the same eigenvalues.
Exercise Set 5.2
In Exercises 1–4, show that A is diagonalizable, using
the matrix P.
1. A =
(x).
Show that T is diagonalizable.
372x
1
+2x
2
+2x
3
−x
1
+2x
2
+x
3
x
1
−x
2
_
_
Show that T is not diagonalizable.
384x
1
+2x
2
+4x
3
4x
1
+2x
2
+4x
3
4x
3
_
_
Show that T is diagonalizable.
39. Let T be a linear operator on a finite dimensional
vector space, A the matrix for T relative to a
basis B
1
, and B the matrix for T relative to a
basis B
2
. Show that A is diagonalizable if and
only if B is diagonalizable.
5.3
ß
Application: Systems of Linear Differential
Equations
In Sec. 3.5 we considered only a single differential equation where the solution
involved a single function. However, in many modeling applications, an equation
that involves the derivatives of only one function is not sufficient. It is more likely
that the rate of change of a variable quantity will be linked to other functions outside
itself. This is the fundamental idea behind the notion of a dynamical system. One of
the most familiar examples of this is the predator-prey model. For example, suppose
we wish to create a model to predict the number of foxes and rabbits in some habitat.
The growth rate of the foxes is dependent on not only the number of foxes but also the
number of rabbits in their territory. Likewise, the growth rate of the rabbit population
in part is dependent on their current number, but is obviously mitigated by the number
of foxes in their midst. The mathematical model required to describe this relationship
is a system of differential equations of the form
_
y
1
(t ) = f (t, y
1
, y
2
)
y
2
(t ) = g(t, y
1
, y
2
)
In this section we consider systems of linear differential equations. Problems such as
predator-prey problems involve systems of nonlinear differential equations.
Uncoupled Systems
At the beginning of Sec. 3.5 we saw that the differential equation given by
y
= ay
has the solution y(t ) = Ce
at
, where C = y(0). An extension of this to two dimensions
is the system of differential equations
_
y
1
= ay
1
y
2
= by
2
Confirming Pages
5.3 Application: Systems of Linear Differential Equations 301
where a and b are constants and y
1
and y
2
are functions of a common variable t . This
system is called uncoupled since y
1
and y
2
depend only on y
1
and y
2
, respectively.
The general solution of the system is found by solving each equation separately and
is given by
y
1
(t ) = C
1
e
at
and y
2
(t ) = C
2
e
bt
where C
1
= y
1
(0) and C
2
= y
2
(0).
The previous system of two differential equations can also be written in matrix
form. To do this, define
y
= Ay
The matrix form of the solution is given by
y(t ) =
_
e
at
0
0 e
bt
_
y(0)
where y(0) =
_
y
1
(0)
y
2
(0)
_
As an illustration, consider the system of differential equations
_
y
1
= −y
1
y
2
= 2y
2
In matrix form the system is written as
y
= Ay =
_
−1 0
0 2
_
y
The solution to the system is
y =
_
e
−t
0
0 e
2t
_
y(0)
that is,
y
1
(t ) = y
1
(0)e
−t
and y
2
(t ) = y
2
(0)e
2t
The Phase Plane
In the case of a single differential equation, it is possible to sketch particular solutions
in the plane to see explicitly how y(t ) depends on the independent variable t . However,
for a system of two differential equations, the solutions are vectors which depend on
a common parameter t , which is usually time. A particular solution can be viewed as
a parameterized curve or trajectory in the plane, called the phase plane. Shown in
Fig. 1 are trajectories for several particular solutions of the system
_
y
1
= −y
1
y
2
= 2y
2
Confirming Pages
302 Chapter 5 Eigenvalues and Eigenvectors
Figure 1
The vectors shown in Fig. 1 comprise the direction field for the system and
describe the motion along a trajectory for increasing t . This sketch is called the phase
portrait for the system. Phase portraits are usually drawn without the direction field.
We have done so here to give a more complete picture of the system and its solutions.
Diagonalization
In the previous example, the matrix A is diagonal, as this is the case for any uncoupled
system of differential equations. We now consider more general systems of the form
y
= Ay
for which A is not a diagonal matrix, but is diagonalizable with real distinct eigen-
values. To solve problems of this type, our strategy is to reduce the system y
= Ay
to one that is uncoupled.
To develop this idea, let A be a 2 ×2 diagonalizable matrix with distinct real
eigenvalues. Consider the system of differential equations given by
y
= Ay
Since A is diagonalizable, then by Theorem 2 of Sec. 5.2 there is a diagonal matrix
D and an invertible matrix P such that
D = P
−1
AP
The diagonal matrix D is given by
D =
_
λ
1
0
0 λ
2
_
where λ
1
and λ
2
are the eigenvalues of A. The column vectors of P are the corre-
sponding eigenvectors. To uncouple the system y
2
(t ) are, respectively, the rates of change for the amount of
salt in tank 1 and tank 2. To develop a system of equations, note that for each
tank
Rate of change of salt = rate in −rate out
Since the volume of brine in each tank remains constant, for tank 1, the rate in
is
5
100
y
2
(t ) while the rate out is
5
50
y
1
(t ). For tank 2, the rate in is
5
50
y
1
(t ) while
the rate out is
5
100
y
2
(t ). The system of differential equations is then given by
_
y
1
(t ) =
5
100
y
2
(t ) −
5
50
y
1
(t )
y
2
(t ) =
5
50
y
1
(t ) −
5
100
y
2
(t )
that is,
_
y
1
(t ) =−
1
10
y
1
(t ) +
1
20
y
2
(t )
y
2
(t ) =
1
10
y
1
(t ) −
1
20
y
2
(t )
Since the initial amounts of salt in tank 1 and tank 2 are 8 and 0 lb, respectively,
the initial conditions on the system are y
1
(0) = 8 and y
2
(0) = 0.
b. The system of equations in matrix form is given by
y
3
= 2y
1
− 4y
2
+ 7y
3
y
1
(0) = 2 y
2
(0) = −1 y
3
(0) = 0
9. Suppose that two brine storage tanks are
connected with two pipes used to exchange
solutions between them. The first pipe allows
water from tank 1 to enter tank 2 at a rate of
1 gal/min. The second pipe reverses the process,
allowing water to flow from tank 2 to tank 1, also
at a rate of 1 gal/min. Initially, the first tank
contains a well-mixed solution of 12 lb of salt in
60 gal of water, while the second tank contains
120 gal of pure water.
a. Find the linear system of differential equations
to describe the amount of salt in each tank at
time t .
b. Solve the system of equations by reducing it to
an uncoupled system.
c. Determine the amount of salt in each tank as t
increases to infinity and explain the result.
10. On a cold winter night when the outside
temperature is 0 degrees Fahrenheit (0
o
F) at
9:00 p.m. the furnace in a two-story home fails.
Suppose the rates of heat flow between the
upstairs, downstairs, and outside are as shown in
the figure. Further suppose the temperature of the
first floor is 70
◦
F and that of the second floor is
60
◦
F when the furnace fails.
1
0
2
0.1
0.1
0.5
0.5
0.2
0.2
a. Use the balance law
Net rate of change = rate in − rate out
to set up an initial-value problem to model the
heat flow.
b. Solve the initial-value problem found in
part (a).
c. Compute how long it takes for each floor to
reach 32
◦
F.
5.4
ß
Application: Markov Chains
In probability theory a Markov process refers to a type of mathematical model used
to analyze a sequence of random events. A critical factor when computing the proba-
bilities of a succession of events is whether the events are dependent on one another.
For example, each toss of a fair coin is an independent event as the coin has no mem-
ory of a previous toss. A Markov process is useful in describing the tendencies of
conditionally dependent random events, where the likelihood of each event depends
on what happened previously.
As an illustration, we consider a simple weather model based on the two obser-
vations that
Table 1
Sunny Cloudy
Sunny 0.7 0.5
Cloudy 0.3 0.5
1. If today is sunny, then there is a 70 percent chance that tomorrow will be sunny.
2. If today is cloudy, then there is a 50 percent chance that tomorrow will be cloudy.
The conditional probabilities for the weather tomorrow, given the weather for
today, are given in Table 1.
The column headings in Table 1 describe today's weather, and the row headings
the weather for tomorrow. For example, the probability that a sunny day today is
Confirming Pages
5.4 Application: Markov Chains 311
followed by another sunny day tomorrow is 0.7, and the probability that a sunny day
today is followed by a cloudy day tomorrow is 0.3. Notice that the column sums are
both 1, since, for example, it is certain that a sunny day today is followed by either
a sunny day or a cloudy day tomorrow.
In a Markov process, these observations are applied iteratively, giving us the
ability to entertain questions such as, If today is sunny, what is the probability that it
will be sunny one week from today?
State Vectors and Transition Matrices
To develop the Markov process required to make predictions about the weather using
the observations above, we start with a vector v =
_
v
1
v
2
_
whose components are the
probabilities for the current weather conditions. In particular, let v
1
be the probability
that today is sunny and v
2
the probability that today is cloudy. Each day the com-
ponents of v change in accordance with the probabilities, listed in Table 1, giving us
the current state of the weather. In a Markov process, the vector v is called a state
vector, and a sequence of state vectors a Markov chain. Using Table 1, the state
vector v
=
_
v
1
v
2
_
for the weather tomorrow has components
v
1
= 0.7v
1
+0.5v
2
and v
2
= 0.3v
1
+0.5v
2
That is, the probability v
1
of a sunny day tomorrow is 0.7 times the probability of a
sunny day today plus 0.5 times the probability of a cloudy day today. Likewise, the
probability v
2
of a cloudy day tomorrow is 0.3 times the probability of a sunny day
today plus 0.5 times the probability of a cloudy day today. For example, if today is
sunny, then v
1
= 1 and v
2
= 0 so that
v
1
= 0.7(1) +0.5(0) = 0.7 and v
2
= 0.3(1) +0.5(0) = 0.3
which is in agreement with the observations above. Observe that if we let T be the
matrix
T =
_
0.7 0.5
0.3 0.5
_
then the relationship above between v and v
can be written using matrix multiplica-
tion as
_
v
1
v
2
_
=
_
0.7 0.5
0.3 0.5
_ _
v
1
v
2
_
In a Markov chain, the matrix used to move from one state to the next is called
the transition matrix. If n is the number of possible states, then the transition matrix
T is an n ×n matrix where the ij entry is the probability of moving from state j
to state i. In the above example t
12
= 0.5 gives the probability that a cloudy day
is followed by one that is sunny. A vector with positive entries whose sum is 1 is
called a probability vector. A matrix whose column vectors are probability vectors
is called a stochastic matrix. The transition matrix T given above is an example of
a stochastic matrix.
Confirming Pages
312 Chapter 5 Eigenvalues and Eigenvectors
Returning to the weather example, to predict the weather 2 days forward, we
apply the transition matrix T to the vector v
2
_
=
_
0.64 0.60
0.36 0.40
_ _
1
0
_
=
_
0.64
0.36
_
In general, after n days the state vector for the weather is given by
T
n
v =
_
0.7 0.5
0.3 0.5
_
n
_
v
1
v
2
_
To answer the question posed earlier about the weather one week after a sunny day,
we compute
_
0.7 0.5
0.3 0.5
_
7
_
1
0
_
=
_
0.625 0.625
0.375 0.375
_ _
1
0
_
=
_
0.625
0.375
_
That is, if today is sunny, then the probability that it will be sunny one week after
today is 0.625, and the probability it will be cloudy is 0.375.
Diagonalizing the Transition Matrix
As we have just seen, determining future states in a Markov process involves comput-
ing powers of the transition matrix. To facilitate the computations, we use the methods
of Sec. 5.2 to diagonalize the transition matrix. To illustrate, we again consider the
transition matrix
T =
_
7
10
5
10
3
10
5
10
_
of the weather example above. Observe that T has distinct eigenvalues given by
λ
1
= 1 and λ
2
=
2
10
with corresponding eigenvectors
v
1
=
_
5
3
1
_
and v
2
=
_
−1
1
_
Confirming Pages
5.4 Application: Markov Chains 313
For reasons that will soon be clear, we scale v
1
(by the reciprocal of the sum of its
components) so that it becomes a probability vector. Observe that this new vector
´v
1
=
_
5
8
3
8
_
is also an eigenvector since it is in the eigenspace V
λ
1
. Since the 2 ×2 transition
matrix has two distinct eigenvalues, by Corollary 1 of Sec. 5.2, T is diagonalizable
and, by Theorem 2 of Sec. 5.2, can be written as
T = PDP
−1
=
_
5
8
−1
3
8
1
_
_
1 0
0
2
10
_ _
1 1
−
3
8
5
8
_
By Exercise 27 of Sec. 5.2, the powers of T are given by
T
n
= PD
n
P
−1
= P
_
1
n
0
0
_
2
10
_
n
_
P
−1
As mentioned above, this gives us an easier way to compute the state vector for
large values of n. Another benefit from this representation is that the matrix D
n
approaches
_
1 0
0 0
_
as n gets large. This suggests that the eigenvector corresponding to λ = 1 is useful in
determining the limiting proportion of sunny days to cloudy days far into the future.
Steady-State Vector
Given an initial state vector v, of interest is the long-run behavior of this vector in a
Markov chain, that is, the tendency of the vector T
n
v for large n. If for any initial
state vector v there is some vector s such that T
n
v approaches s, then s is called a
steady-state vector for the Markov process.
In our weather model we saw that the transition matrix T has an eigenvalue λ = 1
and a corresponding probability eigenvector given by
´v
1
=
_
5
8
3
8
_
=
_
0.625
0.375
_
Confirming Pages
314 Chapter 5 Eigenvalues and Eigenvectors
We claim that this vector is a steady-state vector for the weather model. As verification,
let u be an initial probability vector, say, u =
_
0.4
0.6
_
. We then compute
T
10
u =
_
0.6249999954
0.3750000046
_
and T
20
u =
_
0.6250000002
0.3750000002
_
which suggests that T
n
u approaches ´v
1
. That this is in fact the case is stated in
Theorem 6. Before doing so, we note that a regular transition matrix T is a transition
matrix such that for some n, all the entries of T
n
are positive.
THEOREM 6 If a Markov chain has a regular stochastic transition matrix T, then there is a
unique probability vector s with T s = s. Moreover, s is the steady-state vector for
any initial probability vector.
EXAMPLE 1
A group insurance plan allows three different options for participants, plan A, B,
or C. Suppose that the percentages of the total number of participants enrolled in
each plan are 25 percent, 30 percent, and 45 percent, respectively. Also, from past
experience assume that participants change plans as shown in the table.
A B C
A 0.75 0.25 0.2
B 0.15 0.45 0.4
C 0.1 0.3 0.4
a. Find the percent of participants enrolled in each plan after 5 years.
b. Find the steady-state vector for the system.
Solution Let T be the matrix given by
T =
_
_
0.75 0.25 0.2
0.15 0.45 0.4
0.1 0.3 0.4
_
_
a. The number of participants enrolled in each plan after 5 years is approximated
by the vector
T
5
v =
_
_
0.49776 0.46048 0.45608
0.28464 0.30432 0.30664
0.21760 0.23520 0.23728
_
_
_
_
0.25
0.30
0.45
_
_
=
_
_
0.47
0.30
0.22
_
_
so approximately 47 percent will be enrolled in plan A, 30 percent in plan B,
and 22 percent in plan C.
b. The steady-state vector for the system is the probability eigenvector corre-
sponding to the eigenvalue λ = 1, that is,
s =
_
_
0.48
0.30
0.22
_
_
Revised Confirming Pages
5.4 Application: Markov Chains 315
Exercise Set 5.4
1. Each year it is estimated that 15 percent of the
population in a city moves to the surrounding
suburbs and 8 percent of people living in the
suburbs move to the city. Currently, the total
population of the city and surrounding suburbs is
2 million people with 1.4 million living in the
city.
a. Write the transition matrix for the Markov
chain describing the migration pattern.
b. Compute the expected population after 10
years.
c. Find the steady-state probability vector.
2. After opening a new mass transit system, the
transit authority studied the user patterns to try to
determine the number of people who switched
from using an automobile to the system. They
estimated that each year 30 percent of those who
tried the mass transit system decided to go back
to driving and 20 percent switched from driving
to using mass transit. Suppose that the population
remains constant and that initially 35 percent of
the commuters use mass transit.
a. Write the transition matrix for the Markov
chain describing the system.
b. Compute the expected number of commuters
who will be using the mass transit system in 2
years. In 5 years.
c. Find the steady-state probability vector.
3. A plant blooms with red, pink, or white flowers.
When a variety with red flowers is cross-bred
with another variety, the probabilities of the new
plant having red, pink, or white flowers are given
in the table.
R P W
R 0.5 0.4 0.1
P 0.4 0.4 0.2
W 0.1 0.2 0.7
Suppose initially there are only plants with pink
flowers which are bred with other varieties with
the same likelihood. Find the probabilities of each
variety occurring after three generations. After 10
generations.
4. A fleet of taxis picks up and delivers commuters
between two nearby cities A and B and the
surrounding suburbs S. The probability of a driver
picking up a passenger in location X and
delivering the passenger to location Y is given in
the table. The taxi company is interested in
knowing on average where the taxis are.
A B S
A 0.6 0.3 0.4
B 0.1 0.4 0.3
S 0.3 0.3 0.3
a. If a taxi is in city A, what is the probability it
will be in location S after three fares?
b. Suppose 30 percent of the taxis are in city A,
35 percent are in city B, and 35 percent are in
the suburbs. Calculate the probability of a taxi
being in location A, B, or S after five fares.
c. Find the steady-state probability vector.
5. An endemic disease that has reached epidemic
proportions takes the lives of one-quarter of those
who are ill each month while one-half of those
who are healthy become ill. Determine whether
the epidemic will be eradicated. If so, estimate
how long it will take.
6. A regional study of smokers revealed that from
one year to the next 55 percent of smokers quit
while 20 percent of nonsmokers either became
new smokers or started smoking again. If 70
percent of the population are smokers, what
fraction will be smoking in 5 years? In 10 years?
In the long run?
7. A frog is confined to sitting on one of four lily
pads. The pads are arranged in a square. Label the
corners of the square A, B, C, and D clockwise.
Each time the frog jumps, the probability of
Confirming Pages
316 Chapter 5 Eigenvalues and Eigenvectors
jumping to an adjacent pad is 1/4, the probability
of jumping to the diagonal pad is 1/6, and the
probability of landing on the same pad is 1/3.
a. Write the transition matrix for the Markov
process.
b. Find the probability state vector after the frog
has made n jumps starting at pad A.
c. Find the steady-state vector.
8. Let the transition matrix for a Markov process be
T =
_
0 1
1 0
_
a. Find the eigenvalues of T .
b. Find T
n
for n ≥ 1. Use T
n
to explain why the
Markov process does have a steady-state
vector.
c. Suppose T is the transition matrix describing
the population distribution at any time for a
constant population where residents can move
between two locations. Describe the interaction
in the population.
9. Show that for all p and q such that 0 < p < 1
and 0 < q < 1, the transition matrix
T =
_
1 −p q
p 1 −q
_
has steady-state probability vector
_
q
p+q
p
p+q
_
10. Suppose the transition matrix T for a Markov
process is a 2 ×2 stochastic matrix that is also
symmetric.
a. Find the eigenvalues for the matrix T.
b. Find the steady-state probability vector for the
Markov process.
ReviewExercises for Chapter 5
1. Let
A =
_
a b
b a
_
for some real numbers a and b.
a. Show that
_
1
1
_
is an eigenvector of A.
b. Find the eigenvalues of A.
c. Find the eigenvectors corresponding to each
eigenvalue found in part (b).
d. Diagonalize the matrix A, using the
eigenvectors found in part (b). That is, find the
matrix P such that P
−1
AP is a diagonal
matrix. Specify the diagonal matrix.
2. Let
A =
_
_
0 0 2
0 2 0
0 0 −1
_
_
a. Find the eigenvalues of A.
b. From your result in part (a) can you conclude
whether A is diagonalizable? Explain.
c. Find the eigenvectors corresponding to each
eigenvalue.
d. Are the eigenvectors found in part (c) linearly
independent? Explain.
e. From your result in part (c) can you conclude
whether A is diagonalizable? Explain.
f. If your answer to part (e) is yes, find a matrix
P that diagonalizes A. Specify the diagonal
matrix D such that D = P
−1
AP.
3. Repeat Exercise 2 with
A =
_
¸
¸
_
1 0 1 0
1 1 1 0
0 0 0 0
1 0 1 0
_
¸
¸
_
4. Let T be a linear operator on a finite dimensional
vector space with a matrix representation
A =
_
_
1 0 0
6 3 2
−3 −1 0
_
_
a. Find the characteristic polynomial for A.
b. Find the eigenvalues of A.
c. Find the dimension of each eigenspace of A.
Confirming Pages
5.4 Application: Markov Chains 317
d. Using part (c), explain why the operator T is
diagonalizable.
e. Find a matrix P and diagonal matrix D such
that D = P
−1
AP.
f. Find two other matrices P
1
and P
2
and
corresponding diagonal matrices D
1
and D
2
such that D
1
= P
−1
1
AP
1
and D
2
= P
−1
2
AP
2
.
5. Let
A =
_
_
0 1 0
0 0 1
−k 3 0
_
_
a. Show the characteristic equation of A is
λ
3
−3λ +k = 0.
b. Sketch the graph of y(λ) = λ
3
−3λ +k for
k < −2, k = 0, and k > 2.
c. Determine the values of k for which the matrix
A has three distinct real eigenvalues.
6. Suppose that B = P
−1
AP and v is an eigenvector
of B corresponding to the eigenvalue λ. Show
that Pv is an eigenvector of A corresponding to
the eigenvalue λ.
7. Suppose that A is an n ×n matrix such that every
row of A has the same sum λ.
a. Show that λ is an eigenvalue of A.
b. Does the same result hold if the sum of every
column of A is equal to λ?
8. Let V be a vector space and T: V −→V a linear
operator. A subspace W of V is invariant under
T if for each vector w in W, the vector T (w) is
in W.
a. Explain why V and {0} are invariant subspaces
of every linear operator on the vector space.
b. Show that if there is a one-dimensional
subspace of V that is invariant under T, then T
has a nonzero eigenvector.
c. Let T be a linear operator on ޒ
2
with matrix
representation relative to the standard basis
given by
A =
_
0 −1
1 0
_
Show that the only invariant subspaces of T
are ޒ
2
and {0}.
9. a. Two linear operators S and T on a vector
space V are said to commute if S(T (v)) =
T (S(v)) for every vector v in V. If S and T are
commuting linear operators on V and λ
0
is an
eigenvalue of T, show that V
λ
0
is invariant
under S, that is, S(V
λ
0
) ⊆ V
λ
0
.
b. Let S and T be commuting linear operators on
an n-dimensional vector space V. Suppose that
T has n distinct eigenvalues. Show that S and
T have a common eigenvector.
c. A pair of linear operators T and S on a vector
space V is called simultaneously diagonalizable
if there is an ordered basis B for V such that
[T ]
B
and [S]
B
are both diagonal. Show that if
S and T are simultaneously diagonalizable
linear operators on an n-dimensional vector
space V, then S and T commute.
d. Show directly that the matrices
A =
_
_
3 0 1
0 2 0
1 0 3
_
_
and
B =
_
_
1 0 −2
0 1 0
−2 0 1
_
_
are simultaneously diagonalizable.
10. The Taylor series expansion (about x = 0) for the
natural exponential function is
e
x
= 1 +x +
1
2!
x
2
+
1
3!
x
3
+· · · =
∞
k=0
1
n!
x
k
If A is an n ×n matrix, we can define the matrix
exponential as
e
A
= I +A +
1
2!
A
2
+
1
3!
A
3
+· · ·
= lim
m→∞
(I +A +
1
2!
A
2
+
1
3!
A
3
+· · · +
1
m!
A
m
)
Confirming Pages
318 Chapter 5 Eigenvalues and Eigenvectors
a. Let D be the diagonal matrix
D =
_
¸
¸
¸
_
λ
1
0 0 . . . 0
0 λ
2
0 . . . 0
.
.
.
.
.
.
.
.
.
.
.
.
0 . . . . . . . . . λ
n
_
¸
¸
¸
_
and find e
D
.
b. Suppose A is diagonalizable and D = P
−1
AP.
Show that e
A
= Pe
D
P
−1
.
c. Use parts (a) and (b) to compute e
A
for the
matrix
A =
_
6 −1
3 2
_
Chapter 5: Chapter Test
In Exercises 1–40, determine whether the statement is
true or false.
1. The matrix
P =
_
1 1
0 1
_
diagonalizes the matrix
A =
_
−1 1
0 −2
_
2. The matrix
A =
_
−1 1
0 −2
_
is similar to the matrix
D =
_
−1 0
0 −1
_
3. The matrix
A =
_
_
−1 0 0
0 1 0
−1 −1 1
_
_
is diagonalizable.
4. The eigenvalues of
A =
_
−1 0
−4 −3
_
are λ
1
= −3 and λ
2
= −1.
5. The characteristic polynomial of
A =
_
_
−1 −1 −1
0 0 −1
2 −2 −1
_
_
is λ
3
+2λ
2
+λ −4.
6. The eigenvectors of
A =
_
−4 0
3 −5
_
are
_
0
1
_
and
_
1
3
_
.
7. The matrix
A =
_
3 −2
2 −1
_
has an eigenvalue λ
1
= 1 and V
λ
1
has
dimension 1.
8. If
A =
¸
¸
¸
¸
¸
¸
−
1
2
√
3
2
√
3
2
1
2
¸
¸
¸
¸
¸
¸
then AA
t
= I.
9. If A is a 2 ×2 matrix with det(A) < 0, then A
has two real eigenvalues.
10. If A is a 2 ×2 matrix that has two distinct
eigenvalues λ
1
and λ
2
, then tr(A) = λ
1
+λ
2
.
11. If A =
_
a b
b a
_
, then the eigenvalues of A are
λ
1
= a +b and λ
2
= b −a.
12. For all integers k the matrix A =
_
1 k
1 1
_
has
only one eigenvalue.
13. If A is a 2 ×2 invertible matrix, then A and A
−1
have the same eigenvalues.
14. If A is similar to B, then tr(A) = tr(B).
15. The matrix A =
_
1 1
0 1
_
is diagonalizable.
Confirming Pages
5.4 Application: Markov Chains 319
16. If A =
_
a b
c d
_
and
a +c = b +d = λ
then λ is an eigenvalue of A.
In Exercises 17–19, let
A =
_
_
1 0 0
0 2 0
0 0 −1
_
_
and
B =
_
_
−1 0 0
0 1 0
0 0 2
_
_
17. The matrices A and B have the same eigenvalues.
18. The matrices A and B are similar.
19. If
P =
_
_
0 1 0
0 0 1
1 0 0
_
_
then B = P
−1
AP.
20. If a 2 ×2 matrix has eigenvectors
_
−1
1
_
and
_
1
−2
_
, then it has the form
_
2α −β α −β
β −2α 2β −α
_
21. The only matrix similar to the identity matrix is
the identity matrix.
22. If λ = 0 is an eigenvalue of A, then the matrix A
is not invertible.
23. If A is diagonalizable, then A is similar to a
unique diagonal matrix.
24. If an n ×n matrix A has only m distinct
eigenvalues with m < n, then A is not
diagonalizable.
25. If an n ×n matrix A has n distinct eigenvalues,
then A is diagonalizable.
26. If an n ×n matrix A has a set of eigenvectors that
is a basis for ޒ
n
, then A is diagonalizable.
27. If an n ×n matrix A is diagonalizable, then A has
n linearly independent eigenvectors.
28. If A and B are n ×n matrices, then AB and BA
have the same eigenvalues.
29. If D is a diagonal matrix and A = PDP
−1
, then
A is diagonalizable.
30. If A is invertible, then A is diagonalizable.
31. If A and B are n ×n invertible matrices, then
AB
−1
and B
−1
A have the same eigenvalues.
32. A 3 ×3 matrix of the form
_
_
a 1 0
0 a 1
0 0 b
_
_
always has fewer than three distinct eigenvalues.
33. If A and B are n ×n diagonalizable matrices with
the same diagonalizing matrix, then AB = BA.
34. If λ is an eigenvalue of the n ×n matrix A, then
the set of all eigenvectors corresponding to λ is a
subspace of ޒ
n
.
35. If each column sum of an n ×n matrix A is a
constant c, then c is an eigenvalue of A.
36. If A and B are similar, then they have the same
characteristic equation.
37. If λ is an eigenvalue of A, then λ
2
is an
eigenvalue of A
2
.
38. If A is a 2 ×2 matrix with characteristic
polynomial λ
2
+λ −6, then the eigenvalues of
A
2
are λ
1
= 4 and λ
2
= 9.
39. Define a linear operator T: P
1
→P
1
, by
T (a +bx) = a +(a +b)x. Then the matrix
representation for A relative to the standard
basis is
A =
_
1 0
1 1
_
and so T is not diagonalizable.
40. If V = span{e
x
, e
−x
} and T: V →V is defined
by T (f (x)) = f
Since B is an orthonormal set, this reduces to
v, v
i
= c
i
v
i
, v
i
= c
i
As this argument can be carried out for any vector in B, then c
i
= v, v
i
for all
i = 1, 2, . . . , n.
In Theorem 6, if the ordered basis B is orthogonal and v is any vector in V, then
the coordinates relative to B are given by
c
i
=
v, v
i
v
i
, v
i
for each i = 1, . . . , n
so that
v =
v, v
1
v
1
, v
1
v
1
+
v, v
2
v
2
, v
2
v
2
+ · · · +
v, v
n
v
n
, v
n
v
n
Fact Summary
All vectors are in an inner product space.
1. An inner product on a vector space is a function that assigns to each pair of
vectors a real number and generalizes the properties of the dot product on
ޒ
n
.
2. The norm of a vector is defined analogously to the definition in ޒ
n
by
|| v || =
√
v, v.
3. An orthogonal set of vectors is linearly independent. Thus, any set of n
orthogonal vectors is a basis for an inner product space of dimension n.
4. When an arbitrary vector is written in terms of the vectors in an orthogonal
basis, the coefficients are given explicitly by an expression in terms of the
inner product. If {v
1
, . . . , v
n
} is the orthogonal basis and v is an arbitrary
vector, then
v =
v, v
1
is a basis for V is due to Corollary 1 of Sec. 6.2.
From Theorem 7, we also know that every finite dimensional vector space has
an orthonormal basis. That is, if B = {w
1
, w
2
, . . . , w
n
} is an orthogonal basis, then
dividing each vector by its length gives the orthonormal basis
B | 677.169 | 1 |
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Mathematics for Secondary School Teachers
Elizabeth G. Bremigan, Ralph J. Bremigan, and John D. Lorch
TEXTBOOK*
Mathematics for Secondary School Teachers discusses topics of central importance in the secondary school mathematics curriculum, including functions, polynomials, trigonometry, exponential and logarithmic functions, number and operation, and measurement. The authors have written a flexible text, through which instructors can emphasize any of the following: Basics: exploration of key pre-college topics from intuitive and rigorous points of view; Connections: exploration of relationships among topics, using tools from college-level mathematics; Extensions: exploration of college-level mathematical topics that have a compelling relationship to pre-college mathematics. Mathematics for Secondary School Teachers provides a balance of discovery learning and direct instruction. Activities and exercises address the range of learning objectives appropriate for future teachers. Introductory discussion questions encourage prospective teachers to take stock of their knowledge of pre-college topics.
Mathematics for Secondary School Teachers is primarily intended as the text for a bridge or capstone course for pre-service secondary school mathematics teachers. It can also be used in alternative licensure programs, as a supplement to a mathematics methods course, as the text for a graduate course for in-service teachers, and as a resource and reference for in-service faculty development.
A hardcover version of this book is available in our regular store.
* As a textbook, Mathematics for Secondary School Teachers does have DRM. Our DRM protected PDFs can be downloaded to three computers. iOS and Android tablets can open secure PDFs using the AWReader app (available in the App Store and the Play 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 |
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Synopses & Reviews
Publisher Comments
'Many mathematics students have trouble the first time they take a course, such as linear algebra, abstract algebra, introductory analysis, or discrete mathematics, in which they are asked to prove various theorems. This textbook will prepare students to make the transition from solving problems to proving theorems by teaching The author shows how complex proofs are built up from these smaller steps, using detailed \"scratchwork\" sections to expose the machinery of proofs about the natural numbers, relations, functions, and infinite sets. Numerous exercises give students the opportunity to construct their own proofs. No background beyond standard high school mathematics is assumed. This book will be useful to anyone interested in logic and proofs: computer scientists, philosophers, linguists, and of course mathematicians.'
Synopsis
Teaches the techniques needed to read and write proofs.
Synopsis
Many students have trouble the first time they take a mathematics course in which proofs play a significant role. This textbook prepares students to make the transition from solving problems to proving theorems, by teaching them the techniques needed to read and write proofs. | 677.169 | 1 |
Carrollton, TX AlgebraWe miss out on a lot if we understand only half of the words being used or read. To improve vocabulary I focus on reading, writing down the words that are unfamiliar, finding the meaning and rereading the sentence or paragraph. I then try reusing the word in a sentence.
Belita M.William G.
...Discrete mathematics is concerned with the relationships among mathematical structures that are discrete instead of continuous. For example, the arithmetic of integers would be a topic in discrete mathematics, whereas the arithmetic of all real numbers is in the realm of continuous mathematics. ... | 677.169 | 1 |
HEWLETT PACKARD HCMS-2902 DisplayThe Texas Instruments TI-30X IIS Calculator combines statistics and advanced scientific functions and is a durable and affordable calculator for the classroom. The two-line display helps students explore math and science concepts in the classroom | 677.169 | 1 |
Find a Bensenville Ge helped students of all ages backgrounds and abilities make studying more efficient. Discrete math is a catch-all term encompassing many diverse areas of mathematics. There is no universal agreement as to what constitutes discrete math | 677.169 | 1 |
30856456
ISBN: 0130856452
Publication Date: 2000
Publisher: Prentice Hall
AUTHOR
Sadun, Lorenzo A.
SUMMARY
Preface The purpose of the book This book was designed as a textbook for a junior-senior level second course in linear algebra at the University of Texas at Austin. In that course, and in this book, I try to show math, physics, and engineering majors the incredible power of linear algebra in the real world. The hope is that, when faced with a linear system (or a nonlinear system that can be reasonably linearized), future engineers will think to decompose the system into modes that they can understand. Usually this is done by diagonalization. Sometimes this is done by decomposing into a convenient orthonormal basis, such as Fourier series. Sometimes a continuous decomposition, into d functions or by Fourier transforms, is called for. The underlying ideas of breaking a vector into modes (the Superposition Principle) and of decoupling a complicated system by a suitable choice of linear coordinates (the Decoupling Principle) appear throughout physics and engineering. My goal is to impress upon students the importance of these principles, while giving them enough tools to use them effectively. There are many existing types of second linear algebra courses, and many books to match, but few if any make this goal a priority. Some courses are theoretical, going in the direction of functional analysis, Lie Groups or abstract algebra. "Applied" second courses tend to be heavily numerical, teaching efficient and robust algorithms for factorizing or diagonalizing matrices. Some courses split the difference, developing matrix theory in depth, proving classification theorems (e.g., Jordan form) and estimates (e.g., Gershegorin's Theorem). While each of these courses is well-suited for its chosen audience, none give a prospective physicist or engineer substantial insight into how or why to apply linear algebra at all. Notes to the instructor The readers of this book are assumed to have taken an introductory linear algebra class, and hence to be familiar with basic matrix operations such as row reduction, matrix multiplication and inversion, and taking determinants. The reader is also assumed to have had some exposure to vector spaces and linear transformations. This material is reviewed in Chapters 2 and 3, pretty much from the beginning, but a student who has never seen an abstract vector space will have trouble keeping up. The subject of Chapter 4, eigenvalues, is typically covered quite hastily at the end of a first course (if at all), so I work under the assumption that readers do not have any prior knowledge of eigenvalues. The key concept of these introductory chapters is that a basis makes a vector space look likeR n (or sometimesC n ) and makes linear transformations look like matrices. Some bases make the conversion process simple, while others make the end results simple. The standard basis inR n makes coordinates easy to find, but may result in an operator being represented by an ugly matrix. A basis of eigenvectors, on the other hand, makes the operator appear simple but makes finding the coordinates of a vector difficult. To handle problems in linear algebra, one must be adept in coordinatization and in performing change-of-basis operations, both for vectors and for operators. One premise of this book is that standard software packages (e.g., MATLAB, Maple or Mathematica) make it easy to diagonalize matrices without any knowledge of sophisticated numerical algorithms. This frees us to consider theuseof diagonalization, and some general features of important classes of operators (e.g., Hermitian or unitary operators). Diagonalization, by computer or by hand, gives a set of coordinates in which a problem, even a problem with an infinite number of degrees of freedom, decouples into a collection of independent scalar equations. This is what I call the Decoupling Principle. (Strictly sSadun, Lorenzo A. is the author of 'Applied Linear Algebra The Decoupling Principle', published 2000 under ISBN 9780130856456 and ISBN 0130856452 | 677.169 | 1 |
Paperback
Temporarily out of stock online.
Overview
Important mathematical concepts are presented in a moderately-paced manner so they may be easily learned. In fact, this is the only introductory engineering text that presents practical, worked-out examples for the engineering calculator. The presentation is also designed to motivate the reader to acquire more sophisticated skills in algebra, trigonometry, and geometry. Most importantly, this popular text helps readers to understand the broad spectrum of today's technologies, preparing them for future technical courses. | 677.169 | 1 |
This book provides a brief and accessible introduction to the theory of finite fields and to some of their many fascinating and practical applications. The first chapter is devoted to the theory of finite fields. After covering their construction and elementary properties, the authors discuss the trace and norm functions, bases for finite fields, and properties of polynomials over finite fields. Each of the remaining chapters details applications. Chapter 2 deals with combinatorial topics such as the construction of sets of orthogonal latin squares, affine and projective planes, block designs, and Hadamard matrices. Chapters 3 and 4 provide a number of constructions and basic properties of error-correcting codes and cryptographic systems using finite fields. Each chapter includes a set of exercises of varying levels of difficulty which help to further explain and motivate the material. Appendix A provides a brief review of the basic number theory and abstract algebra used in the text, as well as exercises related to this material. Appendix B provides hints and partial solutions for many of the exercises in each chapter. A list of 64 references to further reading and to additional topics related to the book's material is also included. Intended for advanced undergraduate students, it is suitable both for classroom use and for individual study. This book is co-published with Mathematics Advanced Study Semesters.
Editorial Reviews
Review
"This book gives a quick, clear introduction to finite fields and discusses applications in combinatorics, algebraic coding theory, and cryptography. ... The work contains more than 100 exercises, appendixes on number theoretic and algebraic background, exercise hints, and a list of 65 references." ---- CHOICE Magazine
"The book is very well written and most stimulating. In addition, it is well suited for self-study. The student having a course using this book or anyone studying it will be substantially rewarded. On the whole, the authors succeed in providing a nice introduction to an active field of research." ---- Mathematical Reviews
"Altogether, this undergraduate text is an extremely carefully written introduction to the subject for beginners, which stands out by its remarkable features: instructional mastery, lucidity, diversity, profundity, topicality, and user-friendly determination. Being a lovely invitation to this current topic of mathematical (and interdisciplinary) research, the book is perfectly suitable both for classroom use and for individual study." ---- Zentralblatt MATH
Top Customer Reviews
To transmit photos from Mars back to earth, to download files from Internet where the server resides a few thousand miles away, and to send email globally are fascinating. Thus to understand the mathematics which made "... possible for the person receiving a message to detect and correct errors that have arisen during the transmission process" is interesting. "The detection of errors is accomplished by noticing that the received sequence is not a codeword ... For some code, it is possible for the receiver to determine, with high probability, the intended message when the received sequence is not a codeword. Such codes are ... called error-correcting codes." Gary L. Mullen and Carl Mummert's "Finite Field and Applications" introduces the error-correcting codes (algebraic coding theory) and the related mathematics. The book has four chapters. They are: finite fields, combinatorics, algebraic coding theory, and cryptography. | 677.169 | 1 |
adventure in the application of mathematics to design, this unique resource helps students develop imagination, insight, and intuition--qualities that mark a stimulated and alert mind. Utilizing symmetries, designs, and patterns, students blend arithmetic and geometry to create an impressive array of geometric designs from Modular Arithmetic, Latin Squares, and Pascal's Triangle, to Fibonacci numbers, and more. Solutions are included. Grades K-8
Top Customer Reviews
I teach high school math, and this book was a great help to me! It provided a great set of extra activities for students in algebra to learn and understand Pascal's triangle and modular addition. In addition, they got a chance to use their creativity! A must-have as a tool for introducing art and design into the math classroom.
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I have used this book for about 30 years in teaching geometry classes. Often after we finish a chapter, I will spend a day or two doing projects from this book. I like hands-on projects. Students can learn slides,slips, and turns best by working with them. It is very rewarding to make beautiful designs. "Beauty is its own excuse for being." Students make designs from Latin squares, magic squares, Pascal's triangle, Fibonacci numbers, star polygons, and many other patterns. | 677.169 | 1 |
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ISBN: 1846282411
Publication Date: 2006
Publisher: Springer
AUTHOR
Odonnell, John, Hall, Cordelia
SUMMARY
'Discrete Mathematics Using A Computer' offers a new, 'hands-on' approach to teaching discrete mathematics. A simple functional language is used to allow students to experiment with mathematical notations which are traditionally difficult to pick up.Odonnell, John is the author of 'Discrete Mathematics Using a Computer ', published 2006 under ISBN 9781846282416 and ISBN 18462824 | 677.169 | 1 |
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Developing Mathematical Ideas
Geometry: Examining Features of Shapes
Participants examine aspects of 2D and 3D shapes, develop geometric vocabulary, and explore both definitions and properties of geometric objects. The seminar includes a study of angle, similarity, congruence, and the relationships between 3D objects and their 2D representations | 677.169 | 1 |
Find a Katy CalculusLinear Algebra is the study of matrices and their properties. The applications for linear algebra are far reaching whether you want to continue studying advanced algebra or computer science. I studied logic, as math major, both in the context of my courses requiring proofs, as well as a separat... | 677.169 | 1 |
06183488Mathematics for Elementary School Teachers, Student Solutions Manual
The first and only text of its kind, Mathematics for Elementary School Teachers, 2/e, offers a comprehensive, content approach designed to foster hands-on, discovery-based mathematics instruction.Students first encounter new concepts through hands-on activities in the Explorations Manual, enabling them to construct their own understanding of the material. The text then serves as a resource for further investigation, explanation, and clarification.
Unlike most texts, which present exercises very similar to examples, Bassarear demonstrates how real-life problems are generally complex and often filled with ambiguity. Students learn that there may be more than one way to find an answer and even more than one | 677.169 | 1 |
"I don't make culture, I sell it," Dick Clark once remarked. Indeed, the man who reigned as host of American Bandstand for nearly four decades may not have invented rock 'n' roll, but he sold it to the American public better than anyone before or since. Here is the first book to tell the full story of what happened in front of--and behind--the cameras... more...
CliffsQuickReview course guides cover the essentials of your toughest subjects. Get a firm grip on core concepts and key material, and test your newfound knowledge with review questions. Whether you're new to limits, derivatives, and integrals or just brushing up on your knowledge of the subject, CliffsQuickReview Calculus can help. This guide covers... more...
This unique resource provides 190 high-interest, ready-to-use activities to help students master basic math skills— including whole numbers, decimals, fractions, percentages, money concepts, geometry and measurement, charts and graphs, and pre-algebra— for use with students of varying ability levels. All activities are classroom-tested... more...
Math activities for Intermediate/Challenging level students.
The photocopy master activities are designed to consolidate comprehension skills as well as mathematical processes and problem solving strategies in a ?real life? context. For students aged 10+.
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Homework Help section of the Fear of Physics website provides solutions to a number of common physics problems for students of algebra-based introductory physics courses. A solution to each problem is provided in either text or video format. The solutions allow students to see examples of how to solve problems similar to those they may see on their homework.
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PowerMath CAS
Description
PowerMath is a computer algebra system designed for all science. It has the ability, but not the limitation, to complete advanced or simple science and math problems. It runs from DOS but programmers are needed to program a GUI for | 677.169 | 1 |
Synopses & Reviews
Publisher Comments
Besides being an important area of math for everyday use, algebra is a passport to studying subjects like calculus, trigonometry, number theory, and geometry, just to name a few. To understand algebra is to possess the power to grow your skills and knowledge so you can ace your courses and possibly pursue further study in math.
Algebra II For Dummies is the fun and easy way to get a handle on this subject and solve even the trickiest algebra problems. This friendly guide shows you how to get up to speed on exponential functions, laws of logarithms, conic sections, matrices, and other advanced algebra concepts. In no time you'll have the tools you need to:
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Synopsis
Packed with tips and ticks for cracking tough problems
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You've worked your way through Algebra I, and now you're ready to take algebr5a to the next level. This friendly guide shows you how to get up to speed on exponential functions, laws of logarithms, conic sections, matrices, and other advanced algebra concepts - without even breaking a sweat. It's just what you need for Algebra II success!
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Synopsis
You've worked your way through Algebra I, and now you're ready to take algebra to the next level. With a little boost from Algebra II for Dummies, you'll be up there in no time. This friendly guide brings you quickly up to speed on exponential functions, laws of logarithms, conic sections, matrices, and other advanced algebra concepts – and you won't even break a sweat.
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Packed with tips and tricks for cracking even the knottiest problems, Algebra II for Dummies is just what you need for Algebra II success
About the Author
Mary Jane Sterling has authored Algebra For Dummies, Trigonometry For Dummies, Algebra Workbook For Dummies, Trigonometry Workbook For Dummies, Algebra I CliffsStudySolver, and Algebra II CliffsStudySolver. She taught junior high and high school math for many years before beginning her current 25-year-and-counting career at Bradley University in Peoria, Illinois. Mary Jane enjoys working with her students both in the classroom and outside the classroom, where they do various community service projects. | 677.169 | 1 |
Succeed in math with FUNDAMENTALS OF MATHEMATICS! By offering a comprehensive review of all basic mathematics concepts, this mathematics text takes the intimidation out of arithmetic and makes learning accessible to everyone. Studying is made easy with tools found throughout the text such as objectives, vocabulary definitions, calculator examples, good advice for studying, concept reviews, and chapter tests. Through caution remarks that alert you to common pitfalls and how and why segments that explain and demonstrate concepts and skills in a step-by-step format, you will easily build confidence in your own skills.
Table of Contents
Each chapter begins with a Good Advice for Studying section and a chapter-specific application, then concludes with Key Concepts, Review Exercises, True-False Concept Review, Test, and a Group Project | 677.169 | 1 |
books.google.com - This... to Algebra
Introduction to Algebra
This confidence with abstractions, they are led to further developments in group and ring theory (simple groups and extensions, Noetherian rings, and outline of universal algebra, lattices andcategories) and to applications such as Galois theory and coding theory. There is also a chapter outlining the construction of the number systems from scratch and proving in three different ways that trascendental numbers exist. | 677.169 | 1 |
Precise Calculator has arbitrary precision and can calculate with complex numbers, fractions, vectors and matrices. Has more than 150 mathematical functions and statistical functions and is programmable (if, goto, print, return, for). | 677.169 | 1 |
Description:
This unit consists of two computer programs. The first teaches X,Y plotting; the second is a demonstration of coordinate transformations, matrices, vector equations of lines and perspective and will draw a picture of any geometric solid whose coordinates it has been given. These pictures will be stereograms and may be viewed in three dimensions. | 677.169 | 1 |
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Current Summer Graduate Schools
Tropical geometry uses a combination of techniques from algebraic geometry, combinatorics, and convex polyhedral geometry to study degenerations of algebraic varieties; the simplest tropical objects are tropical curves, which one can think of as "shadows" of algebraic curves. Linear equivalence of divisors on an abstract tropical curve is determined by a simple but rich combinatorial process called "chip firing", which was discovered independently in the discrete setting by physicists and graph theorists. From a pedagogical point of view, one can view tropical curves as a combinatorial model for the highly analogous but more abstract theory of algebraic curves, but there is in fact much more to the story than this: one can use tropical curves and chip firing to prove theorems in algebraic geometry and number theory. This field is relatively new, so participants will have the opportunity to start from scratch and still get a glimpse of the cutting edge in this active research area.
Updated on Feb 11, 2016 02:10 PM PST
MSRI receives major funding from the National Science Foundation Division of Mathematical Sciences
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Calculus: Problems and Solutions
Calculus: Problems and Solutions
Ideal for self-instruction as well as for classroom use, this text helps students improve their understanding and problem-solving skills in analysis, analytic geometry, and higher algebra. More than 1,200 problems appear in the text, with concise explanations of the basic notions and theorems to be used in their solution. Many are followed by complete answers; solutions for the others appear at the end of the book. Topics include sequences, functions of a single variable, limit of a function, differential calculus for functions of a single variable, fundamental theorems and applications of differential calculus, the differential, indefinite and definite integrals, applications of the definite integral, and infinite series. | 677.169 | 1 |
graduate text focuses on the study of semisimple Lie algebras, developing the necessary theory along the way. The material covered ranges from basic definitions of Lie groups to the classification of finite-dimensional representations of semisimple Lie algebras. Lie theory, in its own right, has become regarded as a classical branch of mathematics. Written in an informal style, this is a contemporary introduction to the subject which emphasizes the main concepts of the proofs and outlines the necessary technical details, allowing the material to be conveyed concisely. Based on a lecture course given by the author at the State University of New York at Stony Brook, the book includes numerous exercises and worked examples and is ideal for graduate courses on Lie groups and Lie algebras.
Editorial Reviews
Review
"The book is a very concise and nice introduction to Lie groups and Lie algebras. It seems to be well suited for a course on the subject. The exercises and examples will be useful in that case." Erik Koelink, Mathematical Reviews
"I strongly recommend this book as a possible selection for graduate course(s), as well as independent study, or individual reading." Mihaela Poplicher, MAA Reviews
Book Description
This graduate text focuses on the study of semisimple Lie algebras, developing the necessary theory along the way. Written in an informal style, this is a contemporary introduction to the subject. With numerous exercises and worked examples, it is ideal for graduate courses on Lie groups and Lie algebras.
Top Customer Reviews
I used this book as the primary text for an introductory course on Lie groups and Lie algebras. There are several aspects of the book which distinguish it from every other book on the same topic, making it an indespensable resource for the beginning student.
First, the book is, as its title indicates, an introduction, and a fairly brief one at that. It is not intended to be comprehensive in scope or in depth, rather to gently introduce some fairly complex ideas in the most basic way possible. This is the primary reason it is so useful to start with: The author knows just how much detail is necessary and skips cumbersome and unenlightening proofs. For example, he doesn't prove Serre's theorem or finish the proof of the PBW theorem, but rather refers to other books for these. In contrast to other books on the subject, the student doesn't have to sift the important points from the nitty-gritty details. Every section is important and worth reading. I particularly appreciate that the sections on Lie groups don't require that the reader is an expert in differential geometry and reviews all essential prerequisites.
Of particular value is the excellent collection of exercises. The majority of these are not particularly difficult, but most are enormously worthwhile. Having done lots of exercises from other books, including Knapp (Lie groups beyond an introduction), Hall, Humphreys, and others, I can safely say these are among the best, reaching both an optimal level of difficulty and a fair balance between computation and theory. (Note: Hall's book has great exercises too and are good for those who want more practice with computations).Read more ›
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this is a wickedly good book. it's concise (yeah!) and it's well written. it misses out on lots of stuff (spin representations, etc..). but once you read this book you will have the formalism down pat, and then everything else becomes easy.
if you put in the hours to read this book cover to cover -- like sitting down for 3 days straight 8 hours a day, then will learn the stuff. if you don't persevere and get overwhelmed with the stuff that is not clear at the beginning, then you will probably chuck it out the window.
lie groups and lie algebras in 200 pages done in an elegant way that doesn't look like lecture notes cobbled together is pretty impressive.
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It is trying too hard to be a rigorous introduction without mentioning the differential topology behind everything. It is clear that it assumes a background in differential topology but every time a proof involves stepping in that direction it is omitted.7 When you get to the exercises there are many that would be good, albeit very long, if there was more differential topology. It is fine to sweep many of the rigorous technical details under the rug, and needed to get to the juicy part of the subject, but craft the exercises to avoid such details as well. Once the book switches to Lie algebras and representations it does clean up its act a bit. There is also a glaring omission of maximal tori (or Borel subgroups) which are essential in any further study.
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0072907487Elementary Statistics: A Step by Step Approach
Elementary Statistics: A Brief Version is written for students in the beginning statistics course whose mathematical background is limited to basic algebra. The book uses a nontheoretical approach in which concepts are explained intuitively and supported by examples for your student. There are no formal proofs in the book. The applications are general in nature and the exercises include problems from agriculture, biology, business, economics, education, psychology, engineering, medicine, sociology, and computer science. The learning system found in Elementary Statistics: A Brief Version provides your student with a valuable framework in which to learn and apply concepts | 677.169 | 1 |
A Free Stanford Online Course to Develop Your Mathematical Thinking
Most math classes try to teach computation skills and not much more. This is true not only in school where students memorize mechanical procedures and how to plug numbers into formulas, but also in university courses targeting scientists and engineers. Computational fluency is important but it is only a stepping stone to mathematical maturity.
Higher level mathematics requires the ability to prove mathematical statements, which in turn, requires the ability to think logically and create convincing and rigorous arguments. If this sounds more like something taught in law school, it's because much of math education has been divorced from actual math. We've addressed the topic of mathematical thinking before, but now there is an online course that teaches the most foundational concepts in non-computational math.
That course, Introduction to Mathematical Thinking is offered by Stanford University via Coursera and is taught by Keith Devlin, who is a well known and charismatic math popularizer, educator, and researcher. The main purpose of this course is to serve as a transition between computation-fixated school math classes and undergraduate math major courses. In some ways, this is a traditional course that many university math departments require, but as a MOOC it is now accessible to anyone from high school students to math teachers. As Professor Devlin says in the introductory video (below), the course does not teach students new mathematics; instead it teaches them how to think mathematically and work with the standard mathematical language that involves notions like equivalence relations and logical quantifiers.
The course has been offered before with tens of thousands of students and has received excellent reviews. If you have never been exposed to anything beyond plug and chug math, this course is for you. Once you acquaint yourself with the basics of mathematical thinking, you will gain a deeper appreciation of some important topics that are usually left out of the regular school math curriculum. | 677.169 | 1 |
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This paper is concerned with the norm retrievable frames in $\mathbb{R}^n$. We present some equivalent conditions to the norm retrievable frames in $\mathbb{R}^n$. We will also show that the property of norm retrievability is stable under enough small perturbation of the frame set only for phase retrievable frames
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In this paper, we give a new characterization of the approximate solution given by hybridized mixed methods for second order self-adjoint elliptic problems. We apply this characterization to obtain an explicit formula for the entries of the matrix equation for the Lagrange multiplier unknowns resulting from hybridization. We also obtain necessary and sufficient conditions under which the multipliers of the Raviart–Thomas and the Brezzi–Douglas–Marini methods of similar order are identicalConsider the tangential trace of a vector polynomial on the surface of a tetrahedron. We construct an extension operator that extends such a trace function into a polynomial on the tetrahedron. This operator can be continuously extended to the trace space of H(curl ). Furthermore, it satisfies a commutativity property with an extension operator we constructed in Part I of this series. Such extensions are a fundamental ingredient of high order finite element analysis.
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In this paper, we introduce a new and efficient way to compute exactly divergence-free velocity approximations for the Stokes equations in two space dimensions. We begin by considering a mixed method that provides an exactly divergence-free approximation of the velocity and a continuous approximation of the vorticity. We then rewrite this method solely in terms of the tangential fluid velocity and the pressure on mesh edges by means of a new hybridization technique. This novel formulation bypasses the difficult task of constructing an exactly divergence-free basis for velocity approximations. Moreover, the discrete system resulting from our method has fewer degrees | 677.169 | 1 |
son is going to start with Saxon's algebra 1 this year. I' m trying to decide whether it would be better to buy the Saxon Teacher CD that is authorized by Saxon and has all of the problems in case the student gets stuck, or if I can spend less money and get the Dive CD. And is it just as good. Other than one being authorized by Saxon and one not, and male vs. female voice, price difference and one having the actual student problems, what's the difference? Which one is better to get and why? Can they both be used on a computer and a DVD player. I hope some of you more experienced moms can help me out. Thanks | 677.169 | 1 |
About this Series
Lessons: 6
Total Time: 0h 49m
Use: Watch Online & Download
Access Period: Unlimited
Created At: 09/30/2010
Last Updated At: 09/30/2010
This 6-video second-semester calculus series will teach you the basics of integration - what's an integral, how does notation work, how does integration work, how do integrals relate to derivatives, how do you solve basic integration problems, etc. | 677.169 | 1 |
This book presents three projects that demonstrate thefundamental problems of architectural design and urban composition ? the layoutdesign, evaluation and optimization. Part I describes the functional layoutdesign of a residential building, and an evaluation of the quality of a townsquare (plaza). The algorithm for the functional layout design is... more...
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?Progress in Expressive Image Synthesis?(MEIS2015), was held in Fukuoka, Japan, September 25?27, 2015. The aim of thesymposium was to provide a unique venue where various issues in computergraphics (CG) application fields could be discussed by mathematicians, CGresearchers, and practitioners. Through the previous symposiums MEIS2013 andMEIS2014, mathematicians... more...
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ICM 2010 proceedings comprises a four-volume set containing articles based on plenary lectures and invited section lectures, the Abel and Noether lectures, as well as contributions based on lectures delivered by the recipients of the Fields Medal, the Nevanlinna, and Chern Prizes. The first volume will also contain the speeches at the opening and closing... more...
This Ebook is designed for science and engineering students taking a course in numerical methods of differential equations. Most of the material in this Ebook has its origin based on lecture courses given to advanced and early postgraduate students. This Ebook covers linear difference equations, linear multistep methods, Runge Kutta methods and finite... more... | 677.169 | 1 |
...This algebra deals mostly with linear functions. Algebra 2 is a more advanced, more complex version of algebra 1. Here we get more involved with non-linear functions as well as imaginary and complex numbers | 677.169 | 1 |
Algerbra answers
$1. Lessons are written with the struggling student in mind. Product Description Saxon Algebra 1 Student Textbook, Professional editing rates
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offers New York teachers and other users of the Common Core State Standards free resources that simplify the integration of Regents exam questions into their | 677.169 | 1 |
Moderne Algebra
Learn about this topic in these articles:
history of algebra
...the idea of groups, offered a comprehensive means of investigating algebraic properties. Then in 1930 a textbook was published that presented a totally new image of the discipline. This was
Moderne Algebra, by the Dutch mathematician Bartel van der Waerden, who since 1924 had attended lectures in Germany by Emmy Noether at Göttingen and by Emil Artin at Hamburg. Van der | 677.169 | 1 |
Details: ***SIMPLY BRIT*** We have dispatched from our UK warehouse books of good condition to over 1 million satisfied customers worldwide. We are committed to providing you with a reliable and efficient service at all times.
Details: Complete with CD. Previously water damaged, slight waving to the page edges. Text fully legibleDetails: 2008. Paperback. Motivating readers by making maths easier to learn, this work includes complete past exam papers and student-friendly worked solutions which build up to practice questions, for all round exam preparation. It also includes a Live Text CDROM which features fully worked solutions examined step-by-step, and animations for key learning points. Series: Edexcel AS and A Level Modular Mathematics. Num Pages: 240 pages, full colour illustrations. BIC Classification: 4KL; PB. Category: (UA) A / AS level. Dimension: 265 x 196 x 11. Weight in Grams: 608......We ship daily from our Bookshop. | 677.169 | 1 |
3rd Grading Reviewer/Quick notes on Math VII * Algebra – is a branch of mathematics that focuses on demonstrating the properties and relationships of abstract things in symbolic form or in which symbols, usually letters of the alphabet, represent numbers or members of a specified set and are used to represent quantities and to express general relationships that hold for all members of the set. * Variable - is a symbol, usually a letter, such as x, y, or z, used to represent an unknown number. * Constant - an object or symbol that does not change its value in a particular problem or discussion e.g. pi. * Algebraic Expression - is a collection of numbers, variables, symbols for operations, and symbols for grouping. * Diophantus – a Greek mathematician in the 3rd century, was for some time called the "Father of Algebra". * Musa al-Khowarizimi – an Arabian mathematician in the 9th century was awarded the title the "Father of Algebra". The word algebra was taken from his treatise Al-jabr. * Polynomial - is an algebraic expression that represents a sum of one or more terms containing whole-number exponents on the variables. In a polynomial, a variable cannot appear (1) in a denominator, (2) as an exponent, or (3) within a radical sign. * A term is that part of an algebraic expression separated by + or – signs. Ex: 7x + 3 has two terms: 7x and 3. (7x is a variable term while 3 is a constant term). * The numerical part of a term is called its numerical coefficient. Ex: the numerical coefficient of 9m is 9, the numerical coefficient of –15x3y2 is –15, the numerical coefficient of x is 1. * Terms with exactly the same variables that have the same exponents are called like or similar terms. * Dissimilar terms are terms whose literal factors are not exactly alike. Ex. 3xy^2; 3x^2y; * Monomial – a polynomial with one term
* Binomial – a polynomial with two terms
* Trinomial – a polynomial with three terms...
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From zero to infinity, The Book of Numbers is a handy-sized volume which opens up a new realm of knowledge. Where else in one place could you find out how the illegal numbers racket worked, what makes some people see numbers as colours, why the standard US rail gauge exactly matches the axle width of an ancient Roman chariot, and the numerologic...... more...
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Multivariable calculus
In multivariable calculus, we progress from working with numbers on a line to points in space. It gives us the tools to break free from the constraints of one-dimension, using functions to describe space, and space to describe functions. | 677.169 | 1 |
1
A math text creates a path for students - one that should be easy to navigate, with clearly marked signposts, built-in footholds, and places to stop and assess progress along the way. Research-based and updated for today's classroom, Prentice Hall Mathematics is that well-constructed path. An outstanding author team and unmatched continuity of content combine with timesaving support to help teachers guide students along the road to success.
Provides the structure students need to take effective notes as you teach! Includes daily vocabulary, key concepts, examples, and Check Understanding excercises for every lessonThis handy workbook contains additional exercises for every lesson so students can practice their skills | 677.169 | 1 |
Algebraic Geometry -
Dave Rusin; The Mathematical Atlas
A short article designed to provide an introduction to algebraic geometry, which combines the algebraic with the geometric for the benefit of both. Thus the recent proof of "Fermat's Last Theorem" - ostensibly a statement in number theory - was proved with geometric tools. Conversely, the geometry of sets defined by equations is studied using quite sophisticated algebraic machinery. This is an enticing area but the important topics are quite deep. This area includes elliptic curves. Applications and related fields and subfields; textbooks, reference works, and tutorials; software and tables; other web sites with this focus.
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The Algebraic Geometry Notebooks - Aksel Sogstad
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Upcoming Webinars
Led by teachers for teachers, free upcoming T³™ webinars feature the most relevant topics in math and STEM education. These live webinars cover a variety of topics and TI technology. They are held on most Tuesdays during the school year, starting at 8:00 p.m. ET / 7:00 p.m. CT / 5:00 p.m. PT. They last about 60 minutes. A Certificate of Attendance will be provided at the end of each webinar.
Date
Title
Technology
7/26/2016
L'Hopitals Rule: Review and Preview with TI Technology
Speakers: Speakers: Vicki Carter, Steve Kokoska
Level: All
L'Hopitals Rule uses derivatives to help evaluate limits involving indeterminate forms, and is now included in the AP* Calculus Course and Exam Description for 2016-2017. To support your TI-84 and TI-Nspire™ Technology implementation, this session will focus on:*AP is a registered trademark of the College Entrance Examination Board. Visit
Did You Know Your TI-84 Plus Could Do This?
Get an insider's look at features of the TI-84 Plus graphing calculator family – designed to enhance teaching and learning - from a member of the product development team as well as a T³™ Instructor.
Whether you are new to the TI-84 Plus graphing calculator family or a veteran user looking to advance your TI technology skills, join us for this webinar to gain insight into TI-84 Plus features which support investigations and discoveries, as well as exam prep. This webinar will focus on:
Graph-Table Mode – Analytic, numeric and graph (plots) all in one view
Interactive Draw features – Drawing horizontal and vertical lines and much more
Inequality Graphing App – Enhancing Y= with relation graphing
Using the Boolean features to graph number lines and piecewise functions
How to set calculators for exams
Examples will be shown using the TI-84 Plus CE, as well as previous modelsFive New Features of TI-Nspire™ CX Technology for Back to School
Speakers: Speakers: Sean Bird, Elena Smirnova, Gosia Brothers
Level: All
Have you heard about the latest features for the TI-Nspire™ CX handheld and TI-Nspire™ CX Navigator™ system? Join us for this webinar to get an insider's look at TI-Nspire™ technology from a member of the development team as well as a T³™ Instructor.
This webinar will focus on the new features that provide a wow factor, as well as improve ease-of-use, including:
Relation Graphing – Easily graph vertical lines, conics in any form, and much more
Automatic Sliders in Graphs – Learn to graph f(x) = m*x+b or other general form equations and observe their behavior upon change of parameters and add these interactive features to manipulate graphs with just a few easy key strokes
Sliders in Notes – Explore analytic and numeric patterns, or with TI-Nspire™ CAS, even algebraic patterns, with an easy motion
Graphing from Notes and Calculator – Graph your favorite functions directly from where you define them
Plus, see what's new with the TI-Nspire™ CX Navigator™ system to help with Press-to-Test, configuring handheld settings, and other classroom management functionalityMath for the Win: Tying Math to Unexpected, Real World Scenarios
Speakers: Speakers: Jeff McCalla, Matt Owen
Level: All
Mathematics is a universal discipline that is applied in a multitude of careers, sports and hobbies. Yet, math is often viewed as a set of steps that should be memorized, rather than a language that is used to solve problems of all magnitudes. From finding the perfect angle at which to kick a soccer ball to make a goal, to calculating the rate at which a disease spreads through a population, math is essential to the decisions we make every day. As we are tasked with preparing students for college and careers, how might we inspire them to think differently about the math that can be used to explain their world and to solve problems?
This session will focus on empowering students to:
Tackle challenges using problem-solving skills and their TI technology
Evaluate, analyze and explain the math behind their favorite topics
Explore a variety of careers and the math that makes them possibleUsing TI-Nspire™ Technology to Promote Productive Struggle
One of the mathematics teaching practices from the National Council of Teachers of Mathematics' (NCTM's) Principles to Actions is to support productive struggle in learning mathematics.
This webinar will focus on:
Providing #SlowMath opportunities for all students to notice and question
Activities for visualization and conceptual development of fractions, ratios and equations
Exploring how TI technology can promote productive struggle Sports
Speakers: Speakers: Erick Archer, Curtis Brown
Level: All
Texas Instruments opens a new series of activities investigating the STEM Behind Sports. Each activity challenges your students to think critically about practical applications of science, technology, engineering and math.
This webinar will introduce two new activities:
First, tackle a graphical investigation of the success of a winning field goal and confirm the investigation algebraically. Consider the rush of the defense and the length of the kick while investigating activities appropriate for middle grades and algebra I up through and including algebra II and pre-calculus using the TI-84 CE.
Next, compete in a virtual triathlon, using TI-Nspire™ CX technology, in an effort to understand the concepts of work, energy and power at a physical science levelTips and Tricks for Taking Exams With TI Technology
Speakers: Speakers: Corey Boby, Beth Smith
Level: All
Looking for ways to better prepare your students for taking exams using TI technology? This webinar will focus on effective test taking strategies for your students using TI-84 Plus or TI-Nspire™ technology. In addition to general test taking strategies and hints, this session will highlight using TI technology to:
Save time and avoid mistakes
Understand the reasonableness of an answer
Explore answer choicesFlipping Your Classroom With TI Technology
Looking for more time with your students? The flipped classroom may help you find the time you need. This webinar will focus on:
Strategies for developing flipped lessons
Suggestions for in-class and out-of-class activities using TI-84 Plus and TI-Nspire™ technology
Identifying opportunities for rich tasks in and out of the math and science classroom Health: Breast Cancer Research
Speakers: Speakers: Jeff Lukens, Elizabeth McMillan
Level: All
As we approach Breast Cancer Awareness Month, STEM Behind Health will connect students to valuable classroom math and science content and allow them to explore current research and careers associated with seeking personal cures to breast cancer. This webinar will focus on:
The STEM Behind Health activity, "Breast Cancer: When Good DNA Goes Bad," and the genetics and exponential growth associated with cancer cells
Careers in medicine and research associated with the real world of cancer research
An introduction to the concepts of genetic mutations and precision medicine as it pertains to cancer diagnostics and treatmentData Collection With TI-SmartView™ CE
Speakers: Speakers: Jackie Bonneau, Judy Day
Level: All
If you've been thinking of using sensors in your TI-84 Plus math or science classroom, this webinar is a must-attend. Learn how to use TI-SmartView CE Emulator Software for the TI-84 Plus to create hands-on data collection experiences for your students. This session will focus on:
Collecting and analyzing data with sensors and TI-SmartView™ CE
Creating labs that incorporate screenshots and graphs of the data you collect
Using TI-SmartView™ CE as a teacher tool to create dynamic labs and worksheets10 Minutes of Code: STEM Projects
Speakers: Speakers: Fred Fotsch
Learn to create engaging and authentic STEM projects for your classes using 10 Minutes of Code for either TI-84 Plus CE or TI-Nspire™ CX technology. Example activities will be demonstrated ranging from getting started to more advancedSpooky Stories That STEMulate Student Thinking
Speakers: Speakers: Louise Chapman, Mike Cimino
Level: All
Join us for a SPOOKtakular webinar. We are going to take your favorite Science Nspired activities and infuse them with a Halloween twist that will have your students engaged and wanting more. We have lots of tricks up our sleeves, as this will be a treat for all attendeesMystery of the Ocelot and the Agouti" and Other Engaging TI-84 Plus Activities
Speakers: Speakers: Sherri Abel, Kathy Hale
Level: All
Catch your students' attention using engaging TI-84 Plus activities that bring real-world data to sometimes hard-to-grasp concepts for middle grades and high school mathematics students. This session will focus on the "Mystery of the Ocelot and the Agouti" and two additional TI-84 Plus activities that:
Use data to drive instruction and engage students
Integrate real data to teach students how to curve fit
Incorporate graphs to answer thought-provoking questionsExploring how TI-Nspire™ supports students in forming deep understandingCollecting Data With a CBR 2™ in Your Math Classroom
Speakers: Speakers: Audrey Cucci, Josh Mize
Level: All
Excite your students with the mathematics surrounding motion. Using the Calculator Based Ranger (CBR 2™) with either your TI-Nspire™ or TI-84 Plus technology allows your students to get out of their seats and experience the mathematics involving distance and time. In particular, kinesthetic learners will see the connections between the graphs and their movements. This session will focus on:
Increasing relevance using real world data to study piece wise function in algebra IScience Through Engineering Design With TI-Nspire™ CX
Speakers: Speakers: Todd Morstein, Cassie Whitecotton
Level: All
Are you interested in knowing more about the Engineering Design Process and employing it in the science classroom? Learn how you can take advantage of TI-Nspire™ CX technology to incorporate the Engineering Design Process into your science classroom. This webinar will highlight middle grades lessons that focus on topics such as photosynthesis, cellular respiration and uniform motion that will engage your students in new and exciting ways. Your students will get a chance to design and build models that interface with TI technology to explore these topics like never beforeBreaking the Code" With TI-Nspire™ Technology
Speakers: Speakers: Curtis Brown, Tom Reardon
Level: All
There is a buried treasure located somewhere in the nation's capital near a national monument. An algebra student finds a cryptic message that discusses how to locate the treasure using algebra, geometry and cryptology. The student uses TI-Nspire™ technology to:
Decipher the code using rotations and photos
Plot ordered pairs and write equations of lines
Use dynamic geometry and perpendicular bisectors to construct the circumcenter of the triangle created by the three lines
Discover how the student finds the treasure! | 677.169 | 1 |
Mathematical Thinking and Reasoning (TR): Posing questions characteristic of mathematics ("Is there….?, How do I find….?); ... Any locally certified enrichment programme which satisfies the criteria set by CXC For example, Citizenship Education and Community Service 3.
... that appeared on past CSEC CSEC The General Proficiency ... Biology Chemistry English A English B Human and Social Biology Information Technology Mathematics Office Administration Physics Principles of ... Principles of Business and Social Studies consist of Papers 1, 2 and 3/2 (an ...
Dr Mike Ta ylor Series Editor This new series of textbooks for Caribbean Examinations Council [CXC] General Proficiency examinations has been developed and written by teachers with many years' experience of preparing students for CXC ...
Additional MathematicsCXC " A very challenging subject for those who wish to be challenged in Maths." Additional Mathematics is useful for those persons who have intentions to study Further Mathematics,
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