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Overview This set of additional tests is perfect for siblings or co-ops! Accompanying Saxon Math's Algebra 1/2 Curriculum, these extra test forms will easily let extra students get the practice they need!I needed the test forms to complete my Algebra 1/2 set. I already had the text and the solution manual for the text and tests. Unfortunately, I have the second edition and the test forms I received are from the third edition. I can still use them, but I'll have to create my own test answer manual as the answers do not match up with the second edition. The description does not specify which edition the tests are for. I incorrectly assumed since they were selling the second edition text book, that the tests would also be second edition. Live and learn.
It is generally not needed - please read the warning Sep 17, 2009
WARNING - Saxon's Algebra is a great course but this is not 'it' - please read below.
All you get is the Test forms in a booklet that is ALWAYS included in the Home Study Kit.
The HOME STUDY KIT is made up of:
1 - The Algebra 1 student's textbook that includes the 123 lessons, the extra topics, index, glossary and an answer key to the odd-numbered problems. 2 - Test forms that include the 31 tests plus reproducible test answer sheets. 3 - Answers and tests solutions booklet that is just that. It only includes the answers not the ways to get there.
You only get #2 if you order this item, meaning that you will NOT be able to teach your students Algebra. Moreover, there is no value in purchasing this booklet as it is always included with the Home Study Kit so, to the extent that you get a textbook, you the tests booklet should be included. Also, keep in mind that the 32 pages can be easily copied and, in fact, the last 2 pages are test forms that are meant to be reproduced.
I love Saxon and I've been using Saxon courses for many years with my children but, unless you already have the other items, you should look for the Home Study Kit or any other offer that includes the 3 items I listed above | 677.169 | 1 |
Robert A. Carman & W. Royce Adams
ROBERT A. CARMAN, author of more than a dozen widely used math textbooks, was a professor at Santa Barbara Community College. MARILYN J. CARMAN was a teacher and administrative coordinator with the Santa Barbara High School District.
Robert A. Carman has published or released items in the following series...
This book is written for an individual that needs to find out how things work and how to get where they want to go. In other words, almost everybody. It really is a self-teaching guide. As you begin to read the book, you are instructed to go to a different section depending on your answers to questions. That way, you are finding out specifically what you need, and don't have to wade through lots of information that would apply to others, but not necessarily yourself. Not only does it cover 'making it in the system' it also goes over Improving General Reading Ability, Reading Textbooks, Writing Short Papers, Writing Term Papers, and taking Exams. Very | 677.169 | 1 |
Description
The 16 lessons in this group provide initial instruction or intervention for the introduction of rational and radical equations.
In the first four lessons rational expressions are introduced and simplified, operations are performed with rational expressions, and common factors are divided out to solve rational expressions.
The four lessons of this module are:
A15.1 Finding Restricted Values of a Rational Expression
A15.2 Simplifying Rational Expressions
A15.3 Multiplying and Dividing Rational Expressions
A15.4 Adding and Subtracting Rational Expressions
The next four lessons provide instruction on solving rational equations. Rational equations are used to solve work problems and uniform motion problems, and can be used to solve many other types of problems in areas such as physics.
A16.1 Solving Rational Equations
A16.2 Solving Problems Using Direct Variation
A16.3 Solving Problems Using Inverse Variation
A16.4 Solving Various Types of Problems Using Rational Equations
The next four lessons focus on simplifying radical expressions. Students learn to simplify radical expressions by removing any perfect square (cube) factors from the radical and combining like terms. They also remove radicals from the denominator of a fraction. Once radicals are simplified, students can add, subtract, multiply and divide the expressions.
A17.1 Simplifying Radicals
A17.2 Adding and Subtracting Radicals
A17.3 Multiplying Radicals
A17.4 Dividing Radicals
After simplifying radical expressions, students move to the next four lessons to solve radical expressions where they learn to solve both radical equations and problems that use distance and midpoint formulas. Understanding of these formulas is critical for success in analytic geometry and trigonometry.
A18.1 Solving One-Step Radical Equations
A18.2 Solving Multi-Step Radical Equations
A18.3 Solving Problems Using Radical Equations
A18.4 Solving Problems Using the Distance and Midpoint Formulas
Student print materials and Teachers' Notes are available for download at
Download the free Elevated Math app to view two complimentary lessons or buy the lessons individually | 677.169 | 1 |
Geometry Workbook
Overview them. Here's why: • Math is explained in simple language, in an easy-to-follow style • The workbooks allow students to learn at their own pace and master the subject • More than 20 lessons break down the material into the basics • Each lesson is fully devoted to a key math concept and includes many step-by-step examples • Paced instruction with drills and quizzes reinforces learning • The innovative "Math Flash" feature offers helpful tips and strategies in each lesson—including advice on common mistakes to avoid • Skill scorecard measures the student's progress and success • Every answer to every question, in every test, is explained in full detail • A final exam is included so students can test what they've learned When students apply the skills they've mastered in our workbooks, they can do better in class, raise their grades, and score higher on the all-important end-of-course, graduation, and exit exams. Some of the math topics covered in the Geometry Workbook include: • Basic Properties of Points, Rays, Lines, and Angles • Measuring Line Segments and Angles • Perimeter of Polygons • Triangles • Circles • Quadrilaterals and more! Whether used in a classroom, for home or self study, or with a tutor, this workbook gets students ready for important math tests and exams, set to take on new challenges, and helps them go forward in their studies!
Related Subjects
Read an Excerpt
About This Book
This book will help high school math students at all learning levels understand basic geometry. Students will develop the skills, confidence, and knowledge they need to succeed on high school math exams with emphasis on passing high school graduation exams.
More than 20 easy-to-follow lessons break down the material into the basics. In-depth, step-by-step examples and solutions reinforce student learning, while the "Math Flash" feature provides useful tips and strategies, including advice on common mistakes to avoid.
Students can take drills and quizzes to test themselves on the subject matter, then review any areas in which they need improvement or additional reinforcement. The book concludes with a final exam, designed to comprehensively test what students have learned.
The Ready, Set, Go! Geometry Workbook will help students master the basics of mathematics—and help them face their next math test—with confidence! | 677.169 | 1 |
ENGRL 1120 Iterative techniques for LAEs and non-square LAEs
1
ENGRL 1120: Iterative Methods for Solving Systems of Linear Algebraic Equations (LAEs)
We have been discussing strategies to solve the nn system of Linear Algebraic Equations (LAEs)
for the un
ENGRL 1120: Macroscopic energy balances
1
ENGRL 1120: Macroscopic energy balances and the rst law of thermodynamics.
Similar to mass, the total amount energy in the universe is constant. Energy is neither created or
destroyed, instead, energy is converted | 677.169 | 1 |
Symmetry is an immensely important concept in mathematics and throughout the sciences. In this Very Short Introduction, Ian Stewart highlights the deep implications of symmetry and its important scientific applications across the entire subject. more...
The first part of a two-volume text providing a readable and lively presentation of large parts of geometry in the classical sense, this book appeals systematically to the reader's intuition and vision, and illustrates the mathematical text with many figures. more...
Generalized Polygons is the first book to cover, in a coherent manner, the theory of polygons from scratch. In particular, it fills elementary gaps in the literature and gives an up-to-date account of current research in this area, including most proofs, which are often unified and streamlined in comparison to the versions generally known. Generalized... more...
Aims Theaimofthisbookistoprovideaguidetoarichandfascinatings- ject: algebraic curves, and how they vary in families. The revolution that the ?eld of algebraic geometry has undergone with the introd- tion of schemes, together with new ideas, techniques and viewpoints introduced by Mumford and others, have made it possible for us to understandthebehaviorofcurvesinwaysthatsimplywerenotpos-... more...
This book is an introduction to manifolds at the beginning graduate level. It contains the essential topological ideas that are needed for the further study of manifolds, particularly in the context of di?erential geometry, algebraic topology, and related ?elds. Its guiding philosophy is to develop these ideas rigorously but economically, with minimal... more...
How do you convey to your students, colleagues and friends some of the beauty of the kind of mathematics you are obsessed with? If you are a mathematician interested in finite or topological geometry and combinatorial designs, you could start by showing them some of the (400+) pictures in the "picture book". Pictures are what this book is all about;... more...
There are three new appendices, one by Stefan Theisen on the role of Calabi? Yau manifolds in string theory and one by Otto Forster on the use of elliptic curves in computing theory and coding theory. In the third appendix we discuss the role of elliptic curves in homotopy theory. In these three introductions the reader can get a clue to the far-reaching... more... | 677.169 | 1 |
Please note that all prices are correct at time of going to press but are subject to change without notice.
Basic Maths Practice Problems For
Dummies
Colin Beveridge
Whether you're returning to school, studying for
an adult numeracy test, helping the kids with
homework or seeking the confidence that a firm
math foundation provides in everyday encounters,
Basic Maths Practice Problems For Dummies
provides you with the practice you need to commit mathematics
techniques to memory. This UK edition contains 2,000 practice
problems and solutions and can be use either alone or in conjunction
with Basic Maths For Dummies.
ISBN 111835162-2
336 pages • July 2012
Print: 978-1-118-35162-8 • PB • US$22.95
9 781118 351628
Dispersion Decay and Scattering Theory
Alexander Komech, Elena Kopylova
Thoroughly classroom tested, this book applies
scattering theory methods to modern problems
within a variety of areas in advanced mathematics,
quantum physics, and mathematical physics. It
features the application of functional analysis,
complex analysis, and theory of distributions for
a comprehensive treatment of problems in mathematical physics
and covers the Agmon-Jensen-Kato theory, along with eigenvalues,
Klein-Gordon, and wave equations. Filled with exercises, hints, and
explanatory figures throughout, this book will prove invaluable for
students and professionals looking to expand their knowledge of
scattering theory and partial differential equations.
224 pages • August 2012
Print: 978-1-118-34182-7 • CL • US$94.95
Online: 978-1-118-38286-8
applied Mathematics
A Classical Introduction to Galois Theory
Stephen C. Newman, Univ. of Alberta, Canada
This book provides an introduction to Galois theory and focuses
on one central theme – the solvability of polynomials by radicals.
Both classical and modern approaches to the subject are described,
with the theme of the book providing a platform for exploring both
classical and modern concepts. This book examines a number of
problems arising in the area of classical mathematics. The classical
material within the book includes theorems on polynomials, fields,
and groups due to such luminaries as Gauss, Kronecker, Lagrange,
Ruffini and, of course, Galois.
Over the course of the book, three versions of the Impossibility
Theorem are presented: the first relies entirely on polynomials and
fields, the second incorporates a limited amount of group theory,
and the third takes full advantage of modern Galois theory. This
progression through methods that involve more and more group
theory characterizes the first part of the book. The latter part of the
book is devoted to topics that illustrate the power of Galois theory as
a computational tool, but once again in the context of solvability of
polynomial equations by radicals.
296 pages • July 2012
Print: 978-1-118-09139-5 • CL • US$74.95
Online: 978-1-118-33681-6
applied Mathematics in science
ISBN 111809139-6
9 781118 091395
MatheMatIcs
General & introductory Mathematics
ISBN 111834182-1
9 781118 341827
applied Mathematics in engineering
Bayesian Estimation and Tracking
A Practical Guide
Anton J. Haug
This book presents a practical approach to
estimation methods that are designed to provide a
clear path to programming all algorithms. Readers
are provided with a firm understanding of Bayesian
estimation methods and their interrelatedness.
Starting with fundamental principles of Bayesian theory, the book
shows how each tracking filter is derived from a slight modification
to a previous filter. Such a development gives readers a broader
understanding of the hierarchy of Bayesian estimation and tracking.
This book also presents the development and application of track
performance metrics, including how to generate error ellipses when
implementing in real-world applications, how to calculate RMS errors
in simulation environments, and how to calculate Cramer-Rao lower
bounds for the RMS errors. These are also illustrated in the case study
presentations.
400 pages • June 2012
Print: 978-0-470-62170-7 • CL • US$124.95
Online: 978-1-118-28779-8
ISBN 047062170-2
9 780470 621707
visit our website at
1
MatheMatIcs / statIstIcs
statistics – text & Reference
introductory calculus
2ND EDITION
Common Errors in Statistics
(and How to Avoid Them)
Yang Kuang, Elleyne Kase
4TH EDITION
Pre-Calculus For Dummies®
Pre-Calculus For Dummies is an un-intimidating,
hands-on guide that walks you through all the
essential topics, from absolute value and quadratic
equations to logarithms and exponential functions
to trig identities and matrix operations.
With this guide's help, you'll get a handle on all of the concepts – not just
the number crunching and also understand how to perform all pre-calc
tasks, from graphing to tackling proofs. You'll also get a new appreciation
for how these concepts are used in the real world.
ISBN 111816888-7
384 pages • June 2012
Print: 978-1-118-16888-2 • PB • US$19.95
9 781118 168882
Quantitative and Statistical
Research Methods
Textbook
From Hypothesis to Results
William E. Martin, Krista D. Bridgmon
Quantitative and Statistical Research Methods
offers a guide for psychology, counseling and
education students in the use of statistics and
research designs, combined with guidance on using SPSS in the course
of their research. Each chapter covers a research problem, taking the
student through identifying research questions and hypotheses. This
book shows students how to plan research and conduct statistical
analyses using several different procedures.
ISBN 047063182-1
Emphasizing key theoretical concepts while incorporating real-world
applications, Probability and Stochastic Processes presents a rigorous
approach to probability and uses it as a foundation for understanding
concepts and examples related to stochastic processes. Extensively
classroom-tested, the material is both motivating and comprehensive.
Engaging real-world applications and novel examples from business,
mathematical finance, and engineering shed light on the topic's
relevance in modern research. Topics of coverage are mathematically
rigorous but also incorporate traditional applied concepts so that
upper-undergraduate and graduate students can gain an intuitive sense
of theory and its practice.
512 pages • August 2012
Print: 978-0-470-62455-5 • CL • US$125.00
The Blind Side of Statistics
Herbert I. Weisberg
This provocative new book posits how the tacit principle of
"willful ignorance" has led to a deep divide between qualitative
and quantitative modes of research that will increasingly constrain
scientific progress unless bridged by a broadened conception of
statistical methodology. It presents numerous examples, both
hypothetical and real, to illustrate and support the main premise;
offers a non-technical, historical survey of core statistical concepts;
and speculates about the future evolution of statistics. Clinicians will
find this book to be of particular interest, as will others interested in
observational statistics.
M. Panik, Univ. of Hartford
Statistical Inference: A Short Course is general in nature
and is appropriate for undergraduates and first-year
graduates majoring in the natural sciences, the social
sciences, or in business. While most beginning statistics
books discuss the concepts of simple random sampling
and normality, this book takes such discussions a bit further. Other unique
topics found in the book include determining a confidence interval for
a population median, ratio estimation (a technique akin to estimating a
population proportion), general discussions of randomness and causality,
and nonparametric methods that serve as an alternative to parametric
routines when the latter are not strictly applicable.
440 pages • June 2012
Print: 978-1-118-22940-8 • CL • US$114.95
Online: 978-1-118-30977-3
ISBN 111822940-1
9 781118 229408
applied Probability & statistics
Classic Problems of
Probability
Previously
Announced
Prakash Gorroochurn
Detailing the history of probability, this book
examines the classic problems of probability, many
of which have shaped the field, and emphasizes
problems that are counter-intuitive by nature.
Classic Problems of Probability is rich in the history of probability
while keeping the explanations and discussions as accessible as
possible. Each of 33 presented problems contains listing of the latest
relevant publications on the topic, and the author provides detailed and
rigorous mathematical proofs as needed.
For example, in the discussion of the Buffon needle problem, readers
will find much more than the conventional discussion found in other
books on the topic. The author discusses alternative proofs by Barbier
that lead to much more profound and general results. The choice of
random variables for which a uniform distribution is possible is also
presented, which then naturally leads to a discussion on invariance.
Other topics discusses include Cardano, the Chevalier de Mere
paradoxes, Jacob Bernoulli and the law of large numbers, the discovery
of the normal curve, the lady tasting tea, the Monty-Hall problem, and
many more.
330 pages • June 2012
Print: 978-1-118-06325-5 • PB • US$59.95
Online: 978-1-118-31434-0
Introduction to Probability
and Stochastic Processes
with Applications
Textbook
Liliana Blanco Castaned, Viswanathan
Arunachalam, S. Dharmaraja
This text book is designed for a one-year course
in probability and stochastic processes with
applications, especially for students who wish
to specialize in probabilistic modeling. This book bridges the gap
between elementary texts and advanced texts in probability and is
easily accessible for students with diverse backgrounds and majoring
in engineering, applied sciences, business and finance, statistics,
mathematics, and operations research. The text contains many
examples and exercises which have been tested in classrooms and
are chosen from diverse areas such as queuing models, reliability and
finance. Chapter coverage includes: basic concepts; random variables
and their distributions; discrete distributions; continuous distributions;
random vectors; multivariate normal distributions; conditional
expectation; limit theorems; stochastic processes; queuing models;
stochastic calculus; and mathematical finance.
640 pages • May 2012
Print: 978-1-118-29440-6 • CL • US$124.95
Online: 978-1-118-34497-2
statIstIcs
Statistical Inference
ISBN 111829440-8
9 781118 294406
Modelling Under Risk and Uncertainty
An Introduction to Statistical, Phenomenological and
Computational Methods
Etienne de Rocquigny, Electricite De France
Modelling Under Risk and Uncertainty goes beyond
the 'black-box' view that some risk analysts or
statisticians develop the underlying phenomenology
of the environmental or industrial processes,
without valuing enough their physical properties and inner modelling
potential; conversely it is also to attract environmental or engineering
modellers to more elaborate statistical and risk analysis material
beyond for example, elementary variance analysis, taking advantage
of advanced scientific computing, to face new regulations departing
from deterministic design or decision-making. Researchers in applied
statistics, scientific computing, reliability, advanced mechanics,
physics or environmental science will benefit from this book.
488 pages • May 2012
Print: 978-0-470-69514-2 • CL • US$115.00
Online: 978-1-119-96949-5
ISBN 047069514-5
9 780470 695142
ISBN 111806325-2
9 781118 063255
visit our website at
3
statIstIcs
engineering statistics
statistics for finance, business & economics
Statistics for Scientists and Engineers
A Modern Theory of Random Variation
R. Chattamvelli
With Applications in Stochastic Calculus, Financial
Mathematics, and Feynman Integration
One of the key uses of descriptive statistics is to summarize data,
rather than use the data to learn about the population that the data
represents. This important reference offers scientists and engineers
an introduction to descriptive statistics with an emphasis on
scientific and engineering applications. The book provides a thorough
introduction to data scales, data transformations and popular data
discretization techniques as well as sampling distributions, joint and
conditional distributions, statistical quality control, reliability analysis
and failure modes. The book includes data sets, illustrative examples,
end-of-chapter exercises as well as new algorithms and equations not
available anywhere else.
ISBN 111822896-0
Survival analysis concerns sequential occurrences
of events governed by probabilistic laws. Recent
decades have witnessed many applications of
survival analysis in various disciplines. This book
introduces both classic survival models and theories along with newly
developed techniques. Readers will learn how to perform analysis of
survival data by following numerous empirical illustrations in SAS.
The content is written as to be understandable to readers with minimal
knowledge of SAS whilst enabling more experienced users to learn
new techniques of data input and manipulation. Numerous examples
of SAS code illustrate each of the methods, along with step-by-step
instructions to perform each technique. Each technique is also
accessed on its strengths and limitations.
Covering a wide scope of survival techniques and methods, from
the introductory to the advanced, this book can be used as a useful
reference book for planners, researchers, and professors who are
working in settings involving various lifetime events. Scientists
interested in survival analysis should find it a useful guidebook for the
incorporation of survival data and methods into their projects.
472 pages • July 2012
Print: 978-0-470-97715-6 • CL • US$99.95
Online: 978-1-118-30765-6
4
ISBN 047097715-9
9 780470 977156
visit our website at
Patrick Muldowney, Univ. of Ulster
This book presents a self-contained study of
the Riemann approach to the theory of random
variation and assumes only some familiarity with
probability or statistical analysis, basic Riemann integration, and
mathematical proofs. The author focuses on non-absolute convergence
in conjunction with random variation. The central and recurring theme
throughout this book is that, provided the use a non-absolute method
of summation, every finitely additive, function of disjoint intervals is
integrable.
In contrast, more traditional methods in probability theory exclude
significant classes of such functions whose integrability cannot be
established whenever only absolute convergence is considered. An
example such as the Feynman "measure-which-is-not-a-measure" – the
so-called probability amplitudes used in the Feynman path integrals
of quantum mechanics. This book presents a framework in which the
Feynman path integrals are actual integrals, and they are utilized to
express Feynman diagrams as convergent series of integrals.
512 pages • July 2012
Print: 978-1-118-16640-6 • CL • US$114.95
Online: 978-1-118-34595-5
ISBN 111816640-X
9 781118 166406
Handbook of Exchange Rates
Lucio Sarno, City University London; Jessica James,
Citibank, United Kingdom; Ian Marsh, City University London
Handbook of Exchange Rates is an impressive
compilation of research from more than thirty-five
leading researchers and experts on the topic. The
book is clearly organized into five succinct sections
that explore the foreign exchange (FX) market,
from its background and economic foundation to current practices,
obstacles, and policies in the modern foreign exchange market.
Each chapter follows the same easy-to-follow format. Following an
introduction, a description of theory is presented along with key
formulae. Next, the discussed theory is applied to a real data set
and accompanied with illustrative descriptions. Exercises and realworld examples from the finance industry are spread throughout each
chapter, and a summary provides a brief overview of main points
and concepts.
ISBN 047076883-5
832 pages • May 2012
Print: 978-0-470-76883-9 • CL • US$149.95
9 780470 768839
Bart L. Weathington, University of Tennessee at Chattanooga;
Christopher J. L. Cunningham, University of Tennessee at
Chattanooga; David J. Pittenger, Marietta College
Designed to assist readers in the fields of business,
finance, and management science, this book
provides step-by-step coverage of the research
process including research design, models for
design, statistical considerations, and guidance on
writing and presenting results. Filled with simple explanations, realworld examples, and numerous illustrations to help readers understand
complex and abstract concepts, this is an ideal book for MBA-level
students, as well as researchers and practitioners who want understand
and utilize qualitative and quantitative research methods in applied
scenarios.
524 pages • August 2012
Print: 978-1-118-13426-9 • CL • US$129.95
Online: 978-1-118-34297-8
ISBN 111813426-5
9 781118 134269
econometric & statistical Methods
Financial Statistics and Mathematical Finance
Methods, Models and Applications
A. Steland
Financial Statistics and Mathematical Finance: Methods, Models and
Applications introduces the financial methodology and the relevant
mathematical tools in a style that is mathematically rigorous and
yet accessible to advanced level practitioners and mathematicians
alike. The book focuses on elementary financial calculus, statistical
models for financial data and option pricing. Elements of the statistical
analysis of financial time series, central limit theorems for time series
as well as Itô calculus and stochastic integration are discussed.
400 pages • July 2012
Print: 978-0-470-71058-6 • CL • US$90.00
Online: 978-1-118-31644-3
ISBN 047071058-6
9 780470 710586
statistics for social sciences
Agent-Based Computational Sociology
Flaminio Squazzoni, University of Brescia
Most of the intriguing social phenomena of our time,
such as international terrorism, social inequality,
and urban ethnic segregation, are consequences of
complex forms of agent interaction that are difficult
to observe methodically and experimentally. This
book looks at a new research stream that makes
use of advanced computer simulation modelling techniques to
spotlight agent interaction that allows us to explain the emergence
of social patterns. It presents a method to pursue analytical sociology
investigations that look at relevant social mechanisms in various
empirical situations, such as markets, urban cities, and organisations.
statIstIcs
Understanding Business Research
Agent-Based Computational Sociology is written in a common
sociological language and features examples of models that look at
all the traditional explanatory challenges of sociology. Researchers
and graduate students involved in the field of agent-based modelling
and computer simulation in areas such as social sciences, cognitive
sciences and computer sciences will benefit from this book.
248 pages • May 2012
Print: 978-0-470-71174-3 • CL • US$89.95
Online: 978-1-119-95420-0
ISBN 047071174-4
9 780470 711743
The Visualisation of Spatial Social Structure
2ND EDITION
Danny Dorling
The Visualization of Spatial Social Structure introduces the reader
to new ways of thinking about how to look at social statistics,
particularly those about people in places. The author presents a unique
combination of statistical focus and understanding of social structures
and innovations in visualization, describing the rationale for, and
development of, a new way of visualizing information in geographical
research. These methods are illustrated through extensive full colour
graphics; revealing mistakes, techniques and discoveries which present
a picture of a changing political and social geography. More complex
aspects on the surface of social landscapes are revealed with sculptured
symbols allowing us to see the relationships between the wood and the
trees of social structure. Today's software can be so flexible that these
techniques can now be emulated without coding.
368 pages • August 2012
Print: 978-1-119-96293-9 • CL • US$45.00
Online: 978-1-118-35392-9
ISBN 111996293-5
9 781119 962939
visit our website at
5
statIstIcs
biostatistics
clinical trials
Bayesian Biostatistics
Previously
Announced
Emmanuel Lesaffre, Catholic University of
Leuven; Andrew B. Lawson, University of South Carolina
The growth of biostatistics has been phenomenal in
recent years and has been marked by considerable
technical innovation in both methodology and
computational practicality. The growing use of
Bayesian methodology has taken place partly due to
an increasing number of practitioners valuing the Bayesian paradigm
as matching that of scientific discovery. In addition, computational
advances have allowed for more complex models to be fitted routinely
to realistic data sets.
Through examples, exercises and a combination of introductory
and more advanced chapters, this book provides an invaluable
understanding of the complex world of biomedical statistics
illustrated via a diverse range of applications taken from epidemiology,
exploratory clinical studies, health promotion studies, image analysis
and clinical trials.
536 pages • July 2012
Print: 978-0-470-01823-1 • CL • US$75.00
Online: 978-1-119-94241-2
9 780470 018231
Alexander J. Sutton, University of Leicester, UK;
Keith R. Abrams, A.E Ades,
Nicola J. Cooper, Nicky J. Welton
Evidence Syntesis for Decision Making intends to
provide a practical guide to the appropriate methods
for synthesizing evidence for use in analytical
decision models. More specifically, it proposes a comprehensive
evidence synthesis framework, which models all the available data
appropriately and efficiently in a format that can be incorporated
directly into a decision model.
Key Features:
• Presents evidence synthesis methods compatible with NICE (and
equivalent international) guidelines
• Introduces the subject with an overview of Bayesian methods
• Emphasizes the importance of model critique and checking for evidence
consistency, as well as appropriate diagnostics and their application
• Adopts a probabilistic approach throughout, via the use of MCMC
simulation
• Each chapter contains worked examples, exercises and solutions
drawn from a variety of medical disciplines
• Based upon a tried and tested course presented by the authors
• Provides adaptable WinBUGS code via a Web site allowing readers to
apply to their own analyses
6
Gerald van Belle, Univ. of Washington;
Kathleen F. Kerr, Univ. of Washington
This volume provides technical professionals
and students with three uniquely integrative
enhancements to the study of predictive modeling
not typically found in data-mining books: an applied
approach, immediate practice using Microsoft Excel, and easy-to-use
access to multiple online model-building tools. Since actual datasets
are employed, users deal with real-life modeling issues and situations
such as handling missing values, applying variable transformations,
and addressing outliers, among others. An easy-to-learn Microsoft
Excel add-in (Predictive MinerXL) and all applicable datasets are
available for free on an associated Web site.
248 pages • June 2012
Print: 978-0-470-12727-8 • CL • US$59.95
Online: 978-1-118-27972-4
This book offers a practical, concise introduction
to regression analysis for upper-level undergraduate
students of diverse disciplines including, but not
limited to statistics, the social and behavioral
sciences, MBA, and vocational studies. The book's
overall approach is strongly based on an abundant use of illustrations,
examples, case studies, and graphics. It emphasizes major statistical
software packages, including SPSS®, Minitab®, SAS®, R, and R/SPLUS®. Detailed instructions for use of these packages, as well as
for Microsoft Office Excel®, are provided on a specially prepared and
maintained author web site. Select software output appears throughout
the text. To help readers understand, analyze, and interpret data and
make informed decisions in uncertain settings, many of the examples
and problems use real-life situations and settings.
The book introduces modeling extensions that illustrate more
advanced regression techniques, including logistic regression, Poisson
regression, discrete choice models, multilevel models, Bayesian
modeling, and time series and forecasting. New to this edition are
more exercises, simplification of tedious topics (such as checking
regression assumptions and model building), elimination of repetition,
and inclusion of additional topics (such as variable selection methods,
further regression diagnostic tests, and autocorrelation tests).
384 pages • July 2012
Print: 978-1-118-09728-1 • CL • US$129.95
Online: 978-1-118-34505-4
ISBN 111809728-9
9 781118 097281
Quality, Productivity & Reliability
Methods of Multivariate Analysis
Using the Weibull Distribution
Lead
Title
Reliability, Modeling and Inference
3rD EDITION
Alvin C. Rencher, Brigham Young Univ.,
Textbook
Provo, Utah
This new edition, now with a co-author, offers a
complete and up-to-date examination of the field.
The authors have streamlined previously tedious
topics, such as multivariate regression and MANOVA
techniques, to add newer, more timely content. Each chapter contains
exercises, providing readers with the opportunity to test and extend
their understanding. The new edition also presents several expanded
topics in Kronecker product; prediction errors; maximum likelihood
estimation; and selective key, but accessible proofs. This resource
meets the needs of both statistics majors and those of students and
professionals in other fields.
ISBN 047017896-5
800 pages • August 2012
Print: 978-0-470-17896-6 • CL • US$119.95
9 780470 178966
categorical Data analysis
Textbook
John I. McCool
This book presents the theory, statistical
background and specialized software including R,
for understanding newly-evolving applications of the
Weibull distribution across various fields of study.
Unlike other engineer-written publications, which
omit the probabilistic and statistical basis for the methodology, this
text provides all the necessary information so that readers can gain a
deeper understanding of statistical methods and become knowledgeable
practitioners, rather than number-crunchers. It is ideal for courses
on applied statistics and reliability engineering or as a reference
for statisticians, engineers, business analysts, operation research
professionals, and risk analysts.
368 pages • August 2012
Print: 978-1-118-21798-6 • CL • US$119.95
Online: 978-1-118-35199-4
statIstIcs
Multivariate analysis
ISBN 111821798-5
9 781118 217986
time series
Log-Linear Modeling
Causality
Alexander von Eye, Eun-Young Mun
Over the past ten years, there have been many important advances
in log-linear modeling, including the specification of new models, in
particular non-standard models, and their relationships to methods
such as Rasch modeling. While most literature on the topic is
contained in volumes aimed at advanced statisticians, Applied
Log-Linear Modeling presents the topic in an accessible style that
is customized for applied researchers who utilize log-linear modeling in
the social sciences.
The book begins by providing readers with a foundation on the basics
of log-linear modeling, introducing decomposing effects in crosstabulations and goodness-of-fit tests. Popular hierarchical log-linear
models are illustrated using empirical data examples, and odds ratio
analysis is discussed as an interesting method of analysis of crosstabulations. Next, readers are introduced to the design matrix approach
to log-linear modeling, presenting various forms of coding (effects
coding, dummy coding, Helmert contrasts etc.) and the characteristics
of design matrices. Additional topics of coverage include models of
marginal homogeneity, rater agreement, methods to test hypotheses
about differences in associations across subgroup, the relationship
between log-linear modeling to logistic regression, and reduced designs.
Throughout the book, Computer Applications chapters feature
SYSTAT, Lem, and R illustrations of the previous chapter's material,
utilizing empirical data examples to demonstrate the relevance of the
topics in modern research.
Statistical Perspectives and Applications
Previously
Announced
Carlo Berzuini, Centre for Mathematical Sciences;
Philip Dawid, Professor of Statistics, Cambridge;
Luisa Bernardinell, Institute of Public Health
Causality: Statistical Perspectives and Applications
presents a wide-ranging collection of seminal
contributions by renowned experts in the field,
providing a thorough treatment of all aspects of statistical causality.
It covers the various formalisms in current use, methods for applying
them to specific problems, and the special requirements of a range
of examples from medicine, biology and economics to political
science. This book will be of great interest to postgraduate students,
professional statisticians and researchers in academia and industry.
Key features:
• Provides a clear account and comparison of formal languages,
concepts and models for statistical causality
• Addresses examples from medicine, biology, economics and political
science to aid the reader's understanding
• Is authored by leading experts in their field
• Is written in an accessible style
384 pages • May 2012
Print: 978-0-470-66556-5 • CL • US$90.00
Online: 978-1-119-94571-0
This book uniquely focuses on graph mining and
classification techniques and introduces novel graph
classes appropriate for countless applications across
many disciplines. It explores the relationship of
novel graph classes among each other; existing and classical methods
to analyze networks; similarity and classification techniques based
on machine learning methods; and applications of classification and
mining. With emphasis on computational aspects such as machining
learning, data mining, and information theory techniques, this book
will benefit both professionals and graduate-level students in the field.
352 pages • July 2012
Print: 978-0-470-19515-4 • CL • US$114.95
Online: 978-1-118-34699-0
ISBN 047019515-0
9 780470 195154
bayesian analysis
Bayesian Analysis of
Stochastic Process Models
Previously
Announced
David Insua, Universidad Rey Juan Carlos, Spain;
Fabrizio Ruggeri, CNR IMATI, Italy;
Mike Wiper, Universidad Carlos III de Madrid, Spain
This book provides analysis of stochastic processes
from a Bayesian perspective with coverage of the
main classes of stochastic processing, including
modeling, computational, inference, prediction, decision-making and
important applied models based on stochastic processes. It offers an
introduction of MCMC and other statistical computing machinery that
have pushed forward advances in Bayesian methodology. Addressing
the growing interest for Bayesian analysis of more complex models,
based on stochastic processes, this book aims to unite scattered
information into one comprehensive and reliable volume.
320 pages • April 2012
Print: 978-0-470-74453-6 • CL • US$99.95
Online: 978-0-470-97591-6
ISBN 047074453-7
9 780470 744536
Textbook
This easy accessible textbook provides an overview
of solar to electric energy conversion, followed by a
detailed look at one aspect, namely photovoltaics,
including the underlying principles and fabrication
methods. The author, an experienced author and
teacher, reviews such green technologies as solar-heated-steam power,
hydrogen, and thermoelectric generation, as well as nuclear fusion.
Throughout the book, carefully chosen, up-to-date examples are used to
illustrate important concepts and research tools.
The book is self-contained so as to be suitable for students with
introductory calculus-based courses in physics, chemistry or
engineering. It introduces concepts in quantum mechanics, atomic and
molecular physics, plus the solid state and semiconductor junction
physics needed to attain a quantitative understanding of the current
status of this field. With its comments on economic aspects, it is also a
useful tool for those readers interested in a career in alternative energy.
300 pages • August 2012
ISBN 352741052-X
Print: 978-3-527-41052-1 • CL • US$140.00
Print: 978-3-527-41046-0 • PB • US$65.00
Online: 978-3-527-64628-9
9 783527 410521
The book gives an overview on the current
knowledge of the Active Galactic Nuclei
phenomenon. The spectral energy distribution will
be discussed, pointing out what can be observed in
different wavebands and with different physical models. Furthermore,
the authors discuss the AGN with respect to its environment, host
galaxy, feedback in galaxy clusters, etc. and finally the cosmological
evolution of the AGN phenomenon.
This AGN textbook includes phenomena based on new results
in the X-Ray domain from new telescopes such as Chandra and
XMM Newton not mentioned in any other book. Furthermore,
it considers also the Fermi Gamma Ray Space Telescope with its
revolutionary advances of unprecedented sensitivity, field of view and
all-sky monitoring. Those and other new developments as well as
simulations of AGN merging events and formations, enabled through
latest super-computing capabilities. This book fills the gap of a wide
scope textbook which is appropriate for first year graduate/final year
undergraduate students on active galaxies.
ISBN 352741091-0
350 pages • August 2012
Print: 978-3-527-41091-0 • CL • US$170.00
Print: 978-3-527-41078-1 • PB • US$80.00 9 783527 410910
8
visit our website at
ISBN 352741078-3
9 783527 410781
physIcs
Optical Imaging and Metrology
optics & Photonics
Advanced Technologies
Computational Colour Science
Using MATLAB
Wolfgang Osten, Nadya Reingand, CeLight Inc.,
Textbook
2ND EDITION
Stephen Westland, University of Leeds; Caterina Ripamonti, University
College London; Vien Cheung, University of Leeds
Computational Colour Science Using MATLAB 2nd Edition offers a
practical, problem-based approach to colour physics. The book focuses
on the key issues encountered in modern colour engineering, including
efficient representation of colour information, Fourier analysis of
reflectance spectra and advanced colorimetric computation. Emphasis
is placed on the practical applications rather than the techniques
themselves, with material structured around key topics. These
topics include colour calibration of visual displays, computer recipe
prediction and models for colour-appearance prediction.
Each topic is carefully introduced at three levels to aid student
understanding. First, theoretical ideas and background information
are discussed, then explanations of mathematical solutions follow and
finally practical solutions are presented using MATLAB.
Silver Spring, USA
A comprehensive review of the state of the art and
advances in the field, while also outlining the future
potential and development trends of optical imaging
and optical metrology, an area of fast growth with
numerous applications in nanotechnology and nanophysics. Written
by the world's leading experts in the field, it fills the gap in the current
literature by bridging the fields of optical imaging and metrology, and is
the only up-to-date resource in terms of fundamental knowledge, basic
concepts, methodologies, applications, and development trends.
458 pages • August 2012
Print: 978-3-527-41064-4 • CL • US$140.00
Online: 978-3-527-64844-3
ISBN 352741064-3
9 783527 410644
laser
Fiber Lasers
Review from First Edition:
Oleg G. Okhotnikov
"...highly recommended as a concise introduction to the practicalities
of colour science..." (Color Technology, 2004)
This monograph provides a comprehensive account
of the latest developments and applications in this
rapidly developing field, covering a wide range
of topics such as power scaling and short pulse
generation, dispersion management and modeling,
broadband supercontinuum generation and
wavelength tailoring.
This book is based on tried and tested courses
taught by the author, George Stegeman, who is one
of the experimental pioneers in nonlinear optics.
Presenting complex nonlinear optics problems and
applications in a clear and comprehensible way, the book starts with
second order phenomena, goes on to explain the derivation of nonlinear
susceptibilities, and finishes with a thorough discussion of third order
nonlinear effects.
The book brings together contributions from the world's leading experts at
major collaborative research centers throughout Europe, Australia, Russia
and the USA. Each chapter presents a tutorial style introduction to the
selected topic suitable for scientists, researchers and experts, as well as
graduate and postgraduate students with a basic background in optics.
300 pages • July 2012
Print: 978-3-527-41114-6 • CL • US$140.00
Online: 978-3-527-64864-1
ISBN 352741114-3
9 783527 411146
The content includes in-depth discussion of nonlinear materials,
especially third order and second order nonlinear devices and their
optimization which are dealt with in detail. A simple "electron on a
spring" model is used to help readers begin their journey through the
field of nonlinear optics. The book comes complete with exercises and
a helpful Solutions Manual.
456 pages • July 2012
Print: 978-1-118-07272-1 • CL • US$99.95
Online: 978-1-118-27146-9
ISBN 111807272-3
9 781118 072721
visit our website at
9
physIcs
solid state Physics
Plasma Physics
Discovering Superconductivity
Textbook
An Investigative Approach
2ND EDITION
Weston M. Stacey, Georgia Institute of Technology
Gren Ireson, Nottingham Trent University
The highly-illustrated text will serve as excellent introduction to
superconductivity for students with and without a physics background.
With a strong practical and experimental emphasis, it will provide
readers with an overview of the topic preparing them for more advanced
texts used in more advanced undergraduate and post-graduate courses.
The book will feature a series of practical, investigative, activities
which can be used as tutor demonstrations or, more usefully, as
student lab' exercises. All of the activities have been designed and
tested by the author and colleagues and are designed to be as low cost
as possible in order to promote whole class use.
216 pages • August 2012
Print: 978-1-119-99141-0 • CL • US$120.00
Print: 978-1-119-99140-3 • PB • US$49.95
Online: 978-1-118-34318-0
ISBN 111999141-2
ISBN 111999140-4
9 781119 991410
9 781119 991403
Dynamics at Solid State
Surfaces and Interfaces
2 Volume Set
Uwe Bovensiepen, University of Duisburg-Essen;
Hrvoje Petek, University of Pittsburgh;
Martin Wolf, Free University Berlin
This 2-volume set presents the most up-to-date
coverage of ultrafast/femtosecond dynamics of
elementary processes at solid surfaces and interfaces: from techniques
and methods, to the most recent advances and results in the field.
Volume 1 of this two-volume-set covers the methods, techniques, and
advances that are currently being used/made in the field. Volume 2
covers the methods, techniques, advances and identifies fields of future
developments. Both volumes are of vital interest to surface physicists,
surface chemists, solid-state physicists, solid-state chemists, materials
scientists, materials institutes, PhD students, photochemists, and
spectroscopists.
ISBN 352740938-6
900 pages • May 2012
Print: 978-3-527-40938-9 • CL • US$225.00
10
Fusion Plasma Physics
9 783527 409389
visit our website at
With the investment of the international
community in major new projects such as ITER,
the area of fusion and plasma physics is at a rapidly
changing state in its development. Authored by
Professor Weston M. Stacey, an active researcher
and teacher in plasma physics for more than 35 years who received
the Wigner Reactor Physics Award, this book is a very comprehensive
introduction to fusion plasma physics. It spans the subject from the
classical textbook knowledge to more recent developments. The new
edition keeps the original book up to date by adding information on
alternate concepts. It introduces basic theory of plasmas and relates
it to contemporary applications to controlled nuclear fusion. It is a
unique contribution to the genre, in that it discusses the practical
achievement of controlled fusion.
ISBN 352741134-8
550 pages • July 2012
Print: 978-3-527-41134-4 • CL • US$143.95
9 783527 411344
Author
ISBN13
Page
Reference
Prices
(US$)
Bind
A Classical Introduction to Galois Theory
Newman
9781118091395
1
74.95
CL
A Modern Theory of Random Variation: With Applications
in Stochastic Calculus, Financial Mathematics, and Feynman
Integration
Hardback • 2350 pages • May 2012 • ISBN: 978-3-527-40728-6
The go-to reference for the scientific basics in Biophotonics and the latest
applications in life science.
volumes break new ground by providing readers with the physics basics
as well as the biological and medical background, together with detailed
reports on recent technical advances.
The Handbook of Biophotonics: 3 Volume Set:
• Covers latest research trends with review style contributions
• Covers the physics basics as well as the biological and medical background
• Adopts an application related approach in a fully coloured handbook
Visit for full details and to order
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/physicsbywileyblackwell
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This classic on the general history of functions was written by one of the twentieth century's best-known mathematicians. Hermann Weyl, who worked with Einstein at Princeton, combined function theory and geometry in this high-level landmark work, forming a new branch of mathematics and the basis of the modern approach to analysis, geometry, and topology. The author intended this book not only to develop the basic ideas of Riemann's theory of algebraic functions and their integrals but also to examine the related ideas and theorems with an unprecedented degree of rigor. Weyl's two-part treatment begins by defining the concept and topology of Riemann surfaces and concludes with an exploration of functions of Riemann surfaces. His teachings illustrate the role of Riemann surfaces as not only devices for visualizing the values of analytic functions but also as indispensable components of the theory. | 677.169 | 1 |
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Product Description
This book is intended to help students with reading mathematics correctly. As a Calculus teacher, I have discovered that many of my upper level math students still struggle with reading mathematical expressions and equations correctly. When a student lacks mathematical literacy, they are prevented from being successful in all levels of math. This book allows students who are visual learners to use graphic organizers to help them read and solve mathematical equations correctly. This book focuses on one-and-two step equations. My hope is for teachers to use what I have created and modify it to benefit their own students. Asking students to organize their mathematical thoughts is a perfect way to transition them into Common Core. | 677.169 | 1 |
Linear programming, or LP, is one of the most powerful tools of management science. It is a mathematical technique used to allocate limited resources among competing demands in an optimal way. LP is a mathematical optimization technique.
Linear programming problems must have limited resources, workers, equipment, finances or material. They must also have an explicit objective such as to maximize profit or minimize cost. There must be linearity and homogeneity. Another constraint is divisibility. Normal linear programming assumes that products and resources can be subdivided into fractions. If this subdivision is not possible, a modification of linear programming called integer programming is used.
The steps in the graphical linear programming optimizing process are to formulate the problem in mathematical terms, plot the constraint equations, determine the area of feasibility, plot the objective function, and finally find the optimal point.
Spreadsheets can be used to solve linear programming problems and most spreadsheets have built-in optimization routines that are very easy to use and understand. For example, Microsoft Excel has an optimization tool called Solver. The CD that accompanies your textbook has a new product from Frontline Systems, called Premium Solver, that will work with Microsoft's Solver and provide additional features including those for doing genetic searches for non-linear problems. | 677.169 | 1 |
Mathematics 08
This math course will take you sequentially through these topics: Squares, Square Roots, Pythagorean Theorem, Multiplication, Division and order of Operations of Fractions, Ratio, Rates and Percent, Surface Area and Volume of Rectangular and Triangular Prisms and Cylinders, Multiplication, Division and Order of Operations of Integers, Tesselations, Drawing and Reading 2-D views of Cubic Structures, Data Analysis, Probability, and Linear Equations. There will be 27 assignments, 8 tests and 1 final exam for marks. There will also be Practice Questions for each lesson. These must be done in order to get access to the Weekly Assignment. There will also be 4 cumulative reviews to help students in their studying. Students should review their understanding of addition and subtraction of fractions and integers prior to the course.
Optional supplies are: a set of interlocking cubes, isometric dot paper and graph paper. | 677.169 | 1 |
Purchase all 8 Chapters of Algebra Quizzes for each section - total of 80 assessments! Learn, Practice, Assess, Track
Algebra Series By Chapter (Ch 1 - 8)
Item# AlgCh1-8
$299.00
Media Type:
Product Description
Receive all the materials (Chapters 1 - 8) to learn a complete course of first year algebra! These 8 chapters also represent the first semester of an Algebra 2 course. (Functionality includes automatic scoring and recording of assessments as well as "bookmarking" for the student to return to the last screen they worked on in their previous session.)Assessments Every section of every program has a corresponding quiz to test the student's achievement. Scores are calculated and automatically sent to the included database. Use the quizzes for placement or accountability. Once the students have proven proficiency, they will take the Practice Test which contains questions from all the learning sections in the chapter and includes HINTs and solutions. After the Practice Test, students will take the concluding Chapter Test which will be scored and recorded.Curriculum Note: Continuing Advanced Algebra and Pre-Calculus students should also consider purchasing the Advanced Math Series:
Chapter 9: Conic Sections
Chapter 10: Exponents & Logs
Chapter 11: Sequences & Series
Chapter 12: Matrices & Determinants
Trigonometry - A Complete Course Calculus BasicsDelivery Methods: Downloadable for Windows, CD for Windows, Online Subscription for Windows | 677.169 | 1 |
...GCSE Mathematics What is GCSE Mathematics? The course is run primarily as a resit of the course you followed at school. We highly recommend anyone... Learn about: GCSE Mathematics, Basic Mathematics, Data analysis...
...Level: This is a Level 2 course. What is the Mathematics GCSE course? This maths GCSE course is a one year modular course where you will be with other adult students who need a more up-to-date qualification to the one they have already. No previous experience is required and this course goes right... Learn about: Basic Mathematics...
...Mathematics is an essential subject for all learners, IGCSE Mathematics encourages the development of mathematical knowledge as a key life skill. This course has been designed to help meet the needs of those who want an introduction to this challenging subject either with a view to further study... Learn about: Sequences functions, Numerical Problems, numbering system...
GCSE
Online
12 Months
Flexible
Requirements
This course is for residents in England, Scotland, Wales, Northern Ireland
...The course is for you if you want to improve your mathematical skills for work, further study or personal development and need or would like to gain the GCSE qualification. It follows the AQA syllabus and the qualification is gained by taking exams at the end of the course in June... Learn about: Skills and Training, GCSE Mathematics...
... is not just fundamental to life but can lead on to fascinating subjects like engineering, computing and business. A good grounding in mathematics is critical to most pathways in further education as well as always valuable to your career. This course covers everything from algebra and geometry to handling... Learn about: GCSE Mathematics...
I wanted to access a PGCE course a couple of years ago. That is why, I decided to resit my Maths GCSE and Science GCSE to improve my marks. I enjoyed both courses: the material received was easy to follow. Now I enrolled on the Psychology GCSE course just for fun!
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I always regretted not getting my Maths GCSE at school and had found it detailed as a requirement for a lot of jobs that I was interested in. As I work part-time and have 2 young boys, distance learning seemed like the best option for me. The course materials were easy to follow and my tutor was a great help with marking my work quickly. I have now passed my exams with a B Grade and I am even thinking of doing my A Level.... Scary!
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Distance Learning Centre
...If you need to prove your understanding of numbers and mathematics, this distance learning course on Mathematics will help you obtain your dream job. GCSE Mathematics is a key requirement for many job roles as it demonstrates an understanding of numbers and an ability to work with... Learn about: Skills and Training, GCSE Mathematics...
...A/AS Mathematics What is Mathematics? It's the last 5 minutes of the World Cup qualifier and David Beckham is about to take a free... Learn about: Mathematics Series, Basic Mathematics, Engineering Mathematics...
...Two thirds of the course is Core Mathematics. It is the doing part of the subject. Areas covered include coordinate geometry, calculus and trigonometry. The work begins where GCSE finishes. On enrolment you will be given a foundation exercise which covers algebraic topics covered at GCSE which... Learn about: Basic Mathematics...
... from the following three units: Using mathematics – academic subjects Using mathematics – professional and vocational contexts Using mathematics – public and personal... Learn about: Teacher Education, GCSE Mathematics... | 677.169 | 1 |
Math 112 Bailey - Mathematics 112 Spring 2010 Textbook...
Mathematics 112 Spring, 2010 Textbook : Stewart, James; Single Variable Calculus , 6th ed., for reference Instructor : Dr. Evelyn C. Bailey, Office in Pierce 122 Office Hours : Will be posted weekly on the class conference. Email: [email protected]or type Evelyn Bailey on Learn Link Conference : There is a conference, Math 112 spring 2010. Announcements, daily notes, scheduled SI sessions, questions related to problems, information can be posted at any time. Each student should place this conference on his/her desktop and check daily. Content : Mathematics 112 is the second semester of calculus and is designed specifically for students who have completed a semester of college calculus (Math 111, Math 110B, AP Calculus). Course content includes methods of integration, improper integrals, polar coordinates, sequences and infinite series, power series, and introduction to differential equations. Specific topics by class day are attached. Goals : (1) Students should have a basic understanding of derivative, of anti-derivative, and of limit. (2) Students should be able to use the rules of differentiation as they apply to algebraic and transcendental functions. (3) Students should be able to evaluate a variety of limits. (4) Students should be able to sketch graphs of transcendental functions by building on concepts from Calculus I. (5) Students should be able to demonstrate appropriately the methods of integration (substitution, parts, trigonometric substitution, partial fractions) and use these methods with typical indefinite, definite, and improper integrals. (6) Students should be able to graph and to find area using simple polar coordinate expressions. (7) Students should be able to determine convergence of appropriate infinite series by giving logical arguments. (8) Students should have a basic understanding of power series and be able to determine the domain of appropriate power series. (9) Students should be able to derive a power series expression for specified transcendental expressions using a geometric series or Taylor's Theorem. (10) Students should be able to solve simple first-order differential equations (separable, exact, linear). Major Tests/Final Exam : Four major tests will be given in Seney 209 and 215, at 2:15 - 4:00 p.m. on the following Friday afternoons: February 5, February 19, March 26, and April 16. The final exam will be comprehensive and will be given according to the final exam schedule. Each student is expected to take tests at the scheduled times. Any conflicts or problems will be handled on an individual basis. If your instructor considers the excuse legitimate, arrangements will be made to take a test on the afternoon prior to the testing time. No make up tests are given after the test date and time . Emergencies will be handled on an individual basis. Any student requiring special testing arrangements must provide documentation and give sufficient time for appropriate arrangements to be made.
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Video Organizer for Intermediate Algebra
The Video Organizer encourages students to take notes and work practice exercises while watching Elayn Martin-Gay's lecture series (available in MyMathLab® and on DVD). All content in the Video Organizer is presented in the same order as it is presented in the videos, making it easy for students to create a course notebook and build good study habits! The Video Organizer provides ample space for students to write down key definitions and rules throughout the lectures, and "Play" and "Pause" button icons prompt students to follow along with Elayn for some exercises while they try others on their own. | 677.169 | 1 |
Pages
9.05.2010
SBG: Back to the Drawing Board
I'm an Illinois girl which means we have vague state standards, we're assessed on ACT College Readiness Standards, we recently agreed to Common Core Standards,and basically have to decipher this on our own.
I have decided to go by ACT Standards until Illinois gets smart enough to write their own state test, which will take at least 3-4 years from now. So using a very helpful ACT resource book, I've listed all the topics addressed in the ACT.
Now I just need help deciding which topics fit specifically into Algebra 1 as opposed to Algebra II.
Once I get those nailed down, should I list the prerequisite skills needed? How specific should I get?
How do I assign grades on topics instead of skills?
Or maybe I could break down the topics a bit more specifically and use them as shorter skill list?
3 comments:
From my experience, these would be Algebra 2: -I would just touch on quadratic formula -radical expressions - all I would worry about here is being able to simplify radicals (need for Geom) -rational expressions -most sequence stuff -systems of equations (but you could fit those in if you have time) -logarithms -roots of polynomials -complex numbers
I can't help you with dividing the skills into separate courses. That seems to depend on how far you can go with the Algebra 1 class. You know your kids and your pacing skills better than we do. If the state standards don't tell you where the kids should be after each course, then they aren't much use.
Actually, you are lucky to have external standards like the ACT rather then state-written tests. Most states have been doing an absolutely terrible job of writing standardized tests. | 677.169 | 1 |
Most Recent Episode
Episode 9: We Like Problems, Yes We Do
1 day ago
·
60 minutes
Wes Carroll- MIT grad, podcast host, math tutor, educational coach/consultant, and Ted-proclaimed "Master Problem Solver"- joins the show for Episode 9. Wes explains how learning to problem solve is more than simply learning how to answer test questions correctly; in fact, this type of thinking is a skill that will serve students through college and beyond. Ted and Wes discuss the difference between classroom exercises and true problem solving and weigh in on which is more important to a student's success. (0:50)
Check out Wes' website to learn more about his tutoring and how he can help high school students become better thinkers. Also, take a listen to his podcast for more fun problem solving practice!
Ted and Wes chat about mathematician George Polya's 1947 book How To Solve It, a guide to the principles of problem solving. Both are big fans of Polya's strategy of making a plan and breaking a challenging problem into parts to find the answer. (9:00)
Ted asks Wes if good thinking and problem solving are things a tutor can teach and a student can develop. They discuss the self-evaluative process of a successful problem solver. Ted humbly admits he has a tendency to always immediately pick the best method to solve a problem but Wes believes there's more to it than chance. (17:00)
After discussing the theories, quandaries, and philosophies of problem solving, Ted and Wes finally get to solving some problems! Wes has brought some interesting examples from his puzzler podcast so he and Ted work on a few to close out the show. (33:06)
Have a question or comment for Ted and the show? Email and check out his website! | 677.169 | 1 |
Powers, Roots, and Radicals - REVIEW
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This worksheet is a good review at the end of the powers, roots, and radicals unit, to help students prepare for the test. Concepts include:
- nth root and rational exponents
- properties of rational exponents
- power functions and function operations
- solving radical equations
There are a total of 36 practice questions. All answers are on the last page. | 677.169 | 1 |
Mathematics
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Mathematics
Mathematics is the study and application of arithmetic, algebra, geometry, and analysis. Mathematical methods and tools, such as MATLAB® and Mathematica®, are used to model, analyze, and solve diverse problems in a range of fields, including biology, computer science, engineering, finance, medicine, physics, and the social sciences. Important subareas of mathematics include combinatorics, differential equations, game theory, operations research, probability, and set theory and logic.
Fads are as common in mathematics as in any other human activity, and it is always difficult to separate the enduring from the ephemeral in the achievements of one's own time. An unfortunate effect of the predominance of fads is that if a student doesn't learn about such worthwhile topics as the...
A Course in Differential Equations with Boundary Value Problems, 2nd Edition adds additional content to the author's successful A Course on Ordinary Differential Equations, 2nd Edition. This text addresses the need when the course is expanded.
The focus of the text is on applications and methods...
Written as a textbook, A First Course in Functional Analysis is an introduction to basic functional analysis and operator theory, with an emphasis on Hilbert space methods. The aim of this book is to introduce the basic notions of functional analysis and operator theory without requiring the student als...
Based on the many approaches available for dealing with large-scale systems (LSS), Decentralized Control and Filtering in Interconnected Dynamical Systems supplies a rigorous framework for studying the analysis, stability, and control problems of LSS. Providing an overall assessment of LSS theories...
The application of quantum mechanics to many-particle systems has been an active area of research in recent years as researchers have looked for ways to tackle difficult problems in this area. The quantum trajectory method provides an efficient computational technique for solving both stationary...
Explores the Origin of the Recent Banking Crisis and how to Preclude Future Crises
Shedding new light on the recent worldwide banking debacle, The Banking Crisis Handbook presents possible remedies as to what should have been done prior, during, and after the crisis. With contributions from...
The H∞ control has been one of the important robust control approaches since the 1980s. This book extends the area to nonlinear stochastic H2/H∞ control, and studies more complex and practically useful mixed H2/H∞ controller synthesis rather than the pure H∞ control. Different from the commonly...
A properly structured financial model can provide decision makers with a powerful planning tool that helps them identify the consequences of their decisions before they are put into practice. Introduction to Financial Models for Management and Planning, Second Edition enables professionals and...
Designed as a self-contained account of a number of key algorithmic problems and their solutions for linear algebraic groups, this book combines in one single text both an introduction to the basic theory of linear algebraic groups and a substantial collection of useful algorithms.
Computation with...
Strange Functions in Real Analysis, Third Edition differs from the previous editions in that it includes five new chapters as well as two appendices. More importantly, the entire text has been revised and contains more detailed explanations of the presented material. In doing so, the book explores...
This book is intended for a one-semester course in discrete mathematics. Such a course is typically taken by mathematics, mathematics education, and computer science majors, usually in their sophomore year. Calculus is not a prerequisite to use this book. Part one focuses on how to write proofs,... "...
What Makes Variables Random: Probability for the Applied Researcher provides an introduction to the foundations of probability that underlie the statistical analyses used in applied research. By explaining probability in terms of measure theory, it gives the applied researchers a conceptual | 677.169 | 1 |
ISBN 13: 9780071001830
Complex Analysis
Theory of functions is one of the most elegant branches of mathematics with many important results and a variety of applications which can be covered even in an introductory course. I had the occasion to teach the subject several times and, over the years, the lecture notes prepared for the course progressively took a more formal form. The cyclostyled notes were found useful by the local students and it was felt that they would prove useful to others also. This book is largely based on these notes. The contents of the book are described in detail in the introduction which follows this preface. Although there are no formal prerequisites for this work, it is expected that a student will already have taken one or two courses in calculus of real variables and will probably be studying some abstract algebra and real analysis along with this course. The book contains several diagrams, and a large number of illustrative examples and exercises. As such, it is hoped that it can also be used for self-study. The subject matter is so arranged that each of the chapters depends upon the previous ones. There is probably sufficient material in the book for a one year introductory course on complex analysis. The first part can also be covered at the undergraduate level. (From the Preface by the author.) | 677.169 | 1 |
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These notes accompany my RULES for systems of equations powerpoint ( and contain more examples to go through with students and to have them practice.
These notes outline each step of the RULES process to illustrate how important they are. My students really enjoyed the RULES method because it gave them a "to do" list when it came to solving word problems. It is in the same format as my RULES for equations and inequalities notes. Using them together over the course of first semester will set us up for success in solving systems of equations the same way | 677.169 | 1 |
The General Theory of Relativity: A Mathematical Exposition will serve readers as a modern mathematical introduction to the general theory of relativity. Throughout the book, examples, worked-out problems, and exercises (with hints and solutions) are furnished. Topics in this book include, but are not limited to:
tensor analysis the special theory of relativity the general theory of relativity and Einstein's field equations spherically symmetric solutions and experimental confirmations static and stationary space-time domains black holes cosmological models algebraic classifications and the Newman-Penrose equations the coupled Einstein-Maxwell-Klein-Gordon equations appendices covering mathematical supplements and special topics Mathematical rigor, yet very clear presentation of the topics make this book a unique text for both university students and research scholars.
Anadijiban Das has taught courses on Relativity Theory at The University College of Dublin, Ireland, Jadavpur University, India, Carnegie-Mellon University, USA, and Simon Fraser University, Canada. His major areas of research include, among diverse topics, the mathematical aspects of general relativity theory.
Andrew DeBenedictis has taught courses in Theoretical Physics at Simon Fraser University, Canada, and is also a member of The Pacific Institute for the Mathematical Sciences. His research interests include quantum gravity, classical gravity, and semi-classical gravity.
"The book under review grew out of courses that were taught by the senior author (Das) over the years … . it does serve as a useful reference work for researchers who would prefer a direct exposition of the topics that they are involved with." (David H. Delphenich, Mathematical Reviews, May, 2013)
"This advanced work is only for those who are familiar with high-level tensor mathematics or who specialize in gravitation theory. … The authors provide detailed proofs to theorems with elaborate discussion. … Every chapter contains exercises with hints to solve them or complete solutions. An exhaustive list of references and a good index support the text. Summing Up: Recommended. Graduate students, researchers/faculty, and professionals." (N. Sadanand, Choice, Vol. 50 (5), January, 2013)
"The General Theory of Relativity: A Mathematical Exposition is … written in a very clear style and the mathematics is done carefully and in detail. There are also a lot of 'examples, worked-out problems, and exercises (with hints and solutions),' … so it is certainly a pedagogically sound enterprise well worth the price of admission. I am happy to be able to recommend it." (Michael Berg, The Mathematical Association of America, August, 2012) | 677.169 | 1 |
Working with Algebraic Expressions
Learners understand that algebra is a branch of mathematics that uses symbols or letters to represent unknown numbers in problems. They also understand the definition for an algebraic expression. Make sure to click on the Download the Activity bear so that you can access a top-notch task document that walks learners through the evaluation of algebraic expressions. It can be used as part of your instructional activity or sent home as reinforcement. | 677.169 | 1 |
Description
Math Formulae is an advanced application which provides you with not only the description of the formulae but also the interaction i.e, you can enter the values for a formula and look at calculation.
For example, if you want to do multiplication of two matrices, you have to enter values of the matrices in the given text fields and then click the calculate button and the resultant matrix will be shown along with the description of the formula.
This application provides you with a wide range of mathematics formulae which includes the following categories, | 677.169 | 1 |
MOTGAGES
Foreclosures
Introduction
I choose the topic mortgages because it is one of the
most accepted financing when one needs to own a home. It
is the cheapest way to own a home.
Question: Review the data on home prices in a city of your
choosing. How
Measurement
Y520
Strategies for Educational Inquiry
Robert S Michael
Measurement-1
How are variables measured?
First, variables are defined by conceptual definitions
(constructs) that explain the concept the variable is
attempting to capture.
Second, vari
Week 1 Signature Assignment
Payam Maveddat
February 26, 2017
Samantha Mills
I choice the topic, how has the average home value changed in the past 15 years? Predict what
the average home value will be in 2035. In Tampa, Florida the housing market has made
Critical Thinking Worksheet
MAT/219 Version 1
University of Phoenix Material
Critical Thinking Worksheet
In the following exercises, determine whether each statement makes sense or does not make
sense and explain your reasoning in 50 to 100 words for each
153. Explain the negative exponent rule and give an example.
The negative exponent rule is if B is any real number and is not 0. If N is also a natural number
then you will have this:
An example would be:
154. How do you know if an exponential expression
Introduction to College Algebra Advice
Showing 1 to 2 of 2
great course, good Teacher. breaks down the information into easily understandable material
Course highlights:
MATH Math math, great course a Lot of information.
Hours per week:
6-8 hours
Advice for students:
It is best to complete all of the study guide materials, they really help with the tests and quizzes
Course Term:Fall 2016
Professor:Rolfe Nast
Course Required?Yes
Feb 12, 2016
| Would highly recommend.
Not too easy. Not too difficult.
Course Overview:
This course introduces algebraic concepts providing a solid foundation for college algebra. Topics range from properties of real numbers, the order of operations, and algebraic expressions to solving equations and inequalities. Additional topics include polynomials, factoring methods, rational and radical expressions as well as graphing and functions.
Course highlights:
I got to learn things like
Mathematical Models, Equations, and Inequalities.
•Perform arithmetic operations on algebraic expressions.
•Identify properties of real numbers.
•Evaluate numerical expressions using order of operations.
•Solve linear equations.
•Solve linear inequalities.
•Solve a formula for a given variable.
•Use mathematical models to solve application problems.
•Solve compound inequalities.
•Solve absolute value equations.
Linear Equations in Two Variables, Polynomials, and Introduction to Factoring Polynomials
•Graph linear equations in two variables.
•Determine the slope of a linear equation.
•Perform arithmetic operations on polynomials.
•Factor polynomials using the greatest common factor.
•Factor polynomials by grouping.
•Factor trinomials when the leading coefficient is one.
Hours per week:
6-8 hours
Advice for students:
If you already know some algebra this course will be very easy for you, if you don't you will have some work to do because I never took it in high school so it was all new to me. | 677.169 | 1 |
ILastPublishedMy daughter has never been a fan of Math, so as we were approaching higher Math levels in our homeschooling, I was getting apprehensive about which curriculum she would use. She used Singapore Math during her primary school years but we both wanted to shift to something that she could do on her own and at the same time, make Math easier to understand . We've tried several other Math programs before but they didn't work for her so we wanted to try Teaching Textbooks.
Teaching Textbooks is a Math curriculum which comes in CD format and/or physical book format. It starts from 3rd grade Math until Pre-Calculus. There is a placement test your child can take so you can be sure which level suits child.
We purchased a pre-loved copy of the Teaching Textbooks Math 7 from another homeschooling family. We got the complete set which includes the self-grading CDs, Textbook and Answer Keys.
The Math 7 curriculum has 4 CDs and each CD contains around 25 lessons each. There is a total of 117 lessons. Each lesson starts with a lecture with someone explaining the concepts. The narrator is able to explain the problem in an interesting and non-boring manner. The lecture is interactive allowing the student to answer the questions by typing. After lecture, there are 5 practice sets followed by around 25 problems that she needs to solve . The answers are checked on the spot and if student gets it wrong, there is an option where the student can watch the correct solution. There is also a gradebook which records all the quizzes and grades of the student.
My daughter thinks Teaching Textbooks is an awesome way to study math! Hearing that from her is enough to confirm that this product is worth buying.
I personally like Teaching Textbooks because it fosters independence in my daughter. It also dispels my fear of being unable to teach higher Math to my homeschoolers. I also like it that the technical support responds promptly to my inquiries through email.
After several years of homeschooling my children, I am glad to have finally found the Math curriculum that fits my daughter's needs. | 677.169 | 1 |
Unlike most programming tools, Bootstrap uses algebra as the vehicle for
creating images and animations. That means that concepts students encounter in Bootstrap
behave the exact same way that they do in math class. | 677.169 | 1 |
The development of dynamics theory began with the work of Isaac Newton. In his theory the most basic law of classical mechanics is f = ma, which describes the motion n in IR. of a point of mass m under the action of a force f by giving the acceleration a.
This book is at once an analytical study of one of the most important mathematical texts of antiquity, the Mathematical Collection of the fourth-century AD mathematician Pappus of Alexandria, and also an examination of the work's wider cultural setting.
This text presents the mathematical concepts of Grassmann variables and the method of supersymmetry to a broad audience of physicists interested in applying these tools to disordered and critical systems, as well as related topics in statistical physics. Based on many courses and seminars held by the author, one of the pioneers in this field, the reader is given a systematic and tutorial introduction to the subject matter. | 677.169 | 1 |
Maths
A-level Maths
A level Mathematics provides a thorough grounding in the mathematical tools and techniques often needed in the workplace. It provides a foundation for further studies in a variety of subjects including Science, Engineering and Economics subjects.
The logic and reasoning skills developed by studying A level Mathematics makes sure the qualification is widely respected even in non-mathematical areas.
Entry Requirements A good base in Mathematics is essential – preferably an A or B at GCSE level – along with an enthusiastic interest in the subject and a determination to work hard. To progress onto A2 Mathematics, it is important that you have satisfactorily completed the AS course.
Course Content
The Pure Core makes up two thirds of the AS and A level qualification and provides the techniques in algebra, geometry, trigonometry and calculus that form the fundamental building blocks of the subject.
Mathematical applications make up the remaining third of the qualification and there are various options to suit the particular needs of individual students.
A level Further Mathematics is designed to broaden and deepen the mathematical knowledge and skills of the mathematician. It is studied in the second year and provides a stimulating experience for those who really enjoy the subject. Topics such as matrices and complex numbers are introduced, whilst others already studied are taken to greater depth. It follows the same format as for Mathematics, but three units are from the Further Pure Core, and three are a choice from the available applications.
AS and A level Statistics are offered when required to complement the needs of other subject areas. It is particularly helpful alongside A levels in Sociology, Psychology, Biology or Geography which have a large statistical element.
Assessment
A level Mathematics is made up of six units – three at AS and three at A2. Graphics calculators are allowed in all units except Pure Core 1. Each module examination is a written paper for 1 hour and 30 minutes.
A level Further Mathematics has a similar structure, but is usually studied only in the second year. | 677.169 | 1 |
Description of the book "A Level Mathematics for Edexcel: Mechanics M1":
Oxford A Level Mathematics for Edexcel covers the latest 2008 curriculum changes and also takes a completely fresh look at presenting the challenges of A Level. It specifically targets average students, with tactics designed to offer real chance of success to more students, as well as providing more stretch and challenge material. This Mechanics 1 book includes a background knowledge chapter to help bridge the gap between GCSE and A Level study.
Reviews of the A Level Mathematics for Edexcel: Mechanics M1
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Exponent Rules Exploration - Guided Notes
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247 KB|11 pages
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5 pages of guided notes (including key) about exponent rules, set up in Cornell Note form for students to write questions and/or main ideas in the left-hand margin. This is set up in a way that emphasizes important rules and vocabulary. | 677.169 | 1 |
Who Can Do My Math Homework: Online Solutions
Math is one of the subjects that not everyone understands it clearly. However, just like it is the case with all subjects, professors give math homework as part of their subject. Some mathematical problems are easy, but others are not and you should better ask for help, than struggling at your own. There are lot of web sites that offer help with school subjects, some of them are free some of them require payment. However they offer three types of help:
Tutoring through online chat rooms
Question & answer forums
Pages with free sources
Tutoring through online chat rooms
Lots of web sites offer help with this principle. You choose the subject with which you need help, in this case Math, and the page offers you an available tutor to start with your classes. However, you must follow the instructions of the tutor in order to have effective class. Some web pages are free, but other pages require particular fee.
Question & answer web sites
There are two types of these pages:
Pages that give immediate answer
Pages that you can contact with tutors through e-mail
However, problems such as mathematical calculations can be dealt on pages that give immediate answer, due to their nature. You write what you need and the page automatically calculates and gives you the answer.
Other types of mathematical problems, such as textual assignments that need thinking and deeper analysis, they have to be addressed to pages where you can contact with tutors through e-mail, or in online chat rooms.
Pages with free sources for help
These pages have sources, such as books, encyclopedias and helpful documents that deal with the particular math problem. You have to go through these books in order to find the answer of your problem. Just like the other web sites, some of these are free, others are not.
Helpful web sites for your math homework
discoveryeducation.com – This is a page where you have various sources for all your subjects. It offers sources for help and online help through e-mail.
math.com – This is math specializes page and you can find online help within all the fields of math. You can ask for help online and you can set the assignment and get immediate answer automatically. This page offers its services for free.
hotmath.com – Chatting online with tutor, explanation of answers, learning activities and math videos, you can find everything here, but it requires fee for every online class.
If you think you can't cope with your assignments, just go to the some of the above pages and you will deal with the problem very quickly. | 677.169 | 1 |
Thursday, March 1, 2007
ARE YOU TIRED OF SNOW?
Sometimes while teaching or learning, you may be hit by a surprise storm. It may be a difficult subject that you are studying or teaching, and you find that you are stuck. You don't understand it, or you really don't like it, and you are just not making any progress. You may choose to dig in, and plow your way through. Or, maybe you decide to take a break, rest your mind, or at least, pursue a different subject for awhile, then return to the difficult subject with a fresh perspective. Or you may decide that the subject is not what you want to pursue at all, and try a new route. There is nothing wrong with any of the above methods, but sometimes, it is possible to become so blinded by the snowstorm of difficulty, that you may not see the alternative approaches.
In other words, if algebra is proving to be so challenging that you are stuck, sometimes you force yourself to keep trying (digging) to understand it for so long, that you begin to hate the word algebra. But, if you stop for a minute and relax, you may realize that a short break can bring a fresh perspective, before you put up such a high mental wall of frustration. After a week or two, you may want to try something that might shed some new light on the subject. Maybe books that take a unique approach, such as "Algebra Unplugged," by Kenn Amdahl, or "The Algebra Survival Guide," by Josh Rappaport. Or, you may even decide to postpone the study of algebra, and try geometry, business math, economics or some other branch of mathematics. Maybe you decide to take a long break from all math, and dig into literature for a while, or plow through an interesting period of history. A break won't hurt you, and will probably be helpful in the long run.
I hope that the snowstorm provided you with time to rest and rejuvenate your body and mind, and filled you with a blizzard of ideas and inspiration. It's all in your attitude, you know | 677.169 | 1 |
DataFitting is a powerful statistical analysis program that performs linear and nonlinear regression analysis (i.e. curve fitting). DataFitting determines the values of parameters for an equation, whose form you specify, that cause the equation to best fit a set of data values. DataFitting can handle linear, polynomial, exponential, and general nonlinear functions.
DataFitting performs true nonlinear regression analysis, it does not transform the function into a linear form. As a result, it can handle functions that are impossible to linearize such as:
y = (a - c) * exp(-b * x) + c
Quickly Find the Best Equations that Describe Your Data:
DataFitting gives students, teachers, engineers, researchers and other professionals the power to find the ideal model for even the most complex data, by putting a large number of equations at their fingertips. It has built-in library that includes a wide array of linear and nonlinear models from simple linear equations to high order polynomials.
Graphically Review Curve Fit Results:
Once your data have been fit, DataFitting automatically sorts and plots the fitted equations by the statistical criteria of Standard Error. You can preview your graph and output publication-quality graphs in several different configurations. A residual graph as well as parameter output is generated for the selected fitted equation. Data, statistical and numeric summaries are also available from within the report-panel.
DataFitting has the following capabilities:
* A 38-digit precision math emulator for properly fitting high order polynomials and rationals.
* A robust fitting capability for nonlinear fitting that effectively copes with outliers and a wide dynamic Y data range34)
Geometric Constructions 1.03
The program includes 142 animations. Each animation provides detailed, step-by-step description of a straightedge and compass geometric construction studied in geometry courses. The program is designed for high school students and teachers.
5)
A1 Multiplicatron 3.0
A1 Multiplicatron is a fun Award Winning Graphical method of learning the Multiplication Table for Kids that integrates the concept of multiplication chart and multiplication worksheets into a highly effective multimedia program.
7)
Machinist MathGuru 1.0.90
Solve common trade maths problems in a whiz with Machinist's Math Guru software.
This inexpensive, easy to use utility is designed primarily for students, machinists, toolmakers and CNC programmers.
8)Math Mechanixs 1.1.0.1
Math Mechanixs is a general purpose math program with a Math Editor for solving mathematical problems and taking notes, extensive Function Library and Function Solver, 2D & 3D Graphing, and a Calculus Utility with integration and differentiation.
9)
Math Logic 4.0
Math Logic 4.0 is a Math Lesson Plans software for children that is an integral component of math lesson plans. Math Logic is a fun and straightforward computerized method of learning and solving math problems for students.
5)
FindGraph 2.39
FindGraph is a graphing, curve-fitting, and digitizing tool for engineers, scientists and business. Discover the model that best describes your data.
6)
BREAKTRU QUICK CONVERSION 9.7
Lite version converts several units of length. Plus version converts length, weight and capacity measures. By typing a number into box provided will instantly display the results without the user having to search through a confusing menu of choices.
10)
Calculator Prompter 2.7
Calculator Prompter is a math expression calculator. Calculator Prompter has a built-in error recognition system that helps you get correct results. With Calculator Prompter you can enter the whole expression, including brackets, and operators. | 677.169 | 1 |
Algebra 1 Assessments or Quizzes ENTIRE YEAR 100% EDITABLE
Math Assessments or Quizzes for Algebra 1. These Algebra 1 Math Quizzes are aligned with the common core math standards. These Algebra 1 Math assessments can also be used as quick checks, spiral math review, and progress monitoring. Covers the Entire Year of Algebra 1 and includes pacing chart and answer keys.
Help Finding Zeros on the Graphing Calculator
Every year my Algebra 2 students get really confused by the whole "Left Bound / Right Bound" thing on the graphing calculator. This is especially true when finding zeros. I've found it really wordy trying to explain that Left and Right Bound will sometimes be above the x-axis and will sometimes be below it, so this year I am going to hand out the above reference sheet to help. If you think this sheet would help your students, you can find the file called "left bound right bound" | 677.169 | 1 |
Elementary Algebra - PDF
Elementary Algebra is a work text that covers the traditional topics studied in a modern elementary algebra course. It is intended for students who (1) have no exposure to elementary algebra, (2) have previously had an unpleasant experience with elementary algebra, or (3) need to review algebraic concepts and techniques.
This is only PDF preview of first few pages of Elementary Algebra by Wade Ellis and Denny Burzynski | 677.169 | 1 |
Life as a UBC PHYL Student with a PHYS Minor
Friday, 4 January 2013
Differential equations provide a framework for describing processes in terms of rates of change of their components. For example, a bathtub filled with a concentrated solution of dye X can be drained at the same rate as pure water is introduced. Over time, the concentration of dye X goes down. This we can determine intuitively, but we can also model the process with a differential equation, an equation with derivatives as variables, i.e. with dy/dt dependent on y. Here, "solving" the differential equation gives us y in terms of t in simple terms. In the example I have given, it turns out y decays exponentially with time. The neat thing about differential equations is that we can model a system entirely in terms of rates (which are intuitive to grasp). Using calculus, we can then deduce mechanically from those rates the exact behavior of the system.
Unfortunately, solving a differential equation is not always easy. This is where MATH 215 comes into the picture. As a first course in differential equations, MATH 215 introduces the key concepts and methods necessary to solve first and second order linear ordinary differential equations (ODEs). As my instructor put it, this actually constitutes 0% of all possible differential equations, but it gives us excellent tools for modeling systems nonetheless.
Systems often depend upon more than one variable. For example, an extension of the simple "Newton's Cooling" ODE presented in first year mathematics courses is the Heat Equation. Here, heat dissipates along a direction, x, and its "diffusion" is observed through time, t. This requires a description that have rates depending independently on two different variables, requiring partial derivatives. Such a differential equation is called a partial differential equation (PDE). This is the subject of MATH 316.
An Outline of MATH 215 I divide MATH 215 into five distinct sections
Introduction, first order ODEs, and their applications (Midterm 1).
Second order ODEs and their applications (Midterm 2).
The Laplace transform (Midterm 3).
Systems of First Order Linear ODEs (Midterm 3).
Qualitative description of nonlinear systems of ODEs.
The first two sections deal with solving ODEs directly using calculus-based methods. First order ODEs are solved using separation of variables, integrating factors, and other such techniques. Second order ODEs come in many varieties, and are treated in a series of different cases and using specially derived theorems.
The third section covers a different method altogether for solving ODEs, in which the ODE is transformed into different variables that allow it to be solved using purely algebraic methods (including LOTS of partial fractions) and then transformed back.
In the fourth section, linear algebra is used in order to solve systems of first order ODEs. This was tricky at first, because it had been over a year since MATH 223, but I soon found it very clear. Linear algebra knowledge is a must for this, but rarely does the course go beyond 2x2 matrices. The last section worked on the ability to draw vector fields formed by ODEs freehand without solving the ODE in detail. It turns out that many ODEs that cannot be solved in this course can nonetheless be drawn. I found this section to be the most fascinating of the course.
Assignments were given out weekly and typically took me around 10 hours to complete on average. There were 3 midterms. Between the 3 midterm grades and 9 of 11 assignment grades, one of the four is discarded in the calculation of a final grade. The three remaining average out to give 45% of the final grade. Students must pass the assignment component in order to pass the course. The remaining 55% came from the final exam. Please note that these details are relevant to the course as I took it, and may change depending on the instructor. They serve only to give a general idea of structure and grading.
An outline of Math 316
Math 316 starts with a review of ODEs. It then introduces one more powerful method for solving ODEs: the series solution. Using this method, ODEs and their solutions are represented in terms of power series. This is a much more general approach for solving ODEs and can solve many problems that previously introduced methods cannot.
After this, there was a brief section on solving partial differential equations using computational/numerical methods. To keep things simple, these methods were carried out on Microsoft Excel. This involves representation of PDE solutions in discrete rather than continuous form. The excel templates for this section were provided online, and we only needed to modify them to solve the desired problems.
The remaining two thirds of the course dealt in analytic solutions to PDEs. This was first performed mechanically using algebraic and calculus-based methods, and then generalized into eigenvalue/eigenfunction problems under the heading of "Sturm-Liouville" theory, which I found to be the most challenging part of the course.
My experience with these courses
The theory behind differential equations is dense and detailed. However, for physicists and engineers, they serve their purposes mainly by their applications. As a result, these courses can be taught in two ways: (1) by exploring the detailed mathematical theory, or (2) by focusing on the applications and problem-solving aspects of the course. For me, MATH 215 was taught in the former way, and MATH 316 was taught in the latter. Naturally, MATH 215 turned out to be very difficult, and MATH 316 very easy for me. Both were taught excellently and clearly, however, which may account for my fascination with differential equations and consequently the ridiculous length of this post.
MATH 215 was very time-consuming, in particular its assignments. To be honest, the course terrified me. I did not understand much in the first section because it was densely theoretical and I hadn't taken a math course since my first term of first year. I choked on my first midterm exam and received 50%. Luckily, my grade was salvageable, but this would mean I had to complete assignments diligently and ace both remaining midterm exams. This was the first term I seriously considered a minor in physics, and it all hinged on my performance in MATH 215.
I found that the course got clearer and more interesting as it progressed. It became a balancing act between understanding theory and carrying out procedural calculations. As my interest piqued, so did my motivation, and this ended up being my second highest mark of first and second year. With this encouragement, I happily declared a minor in Physics.
Studying for MATH 215 required a thorough review of all concepts of the course, as well as assignments, textbook problems, and past final exams. I also found the course on ODEs from MIT Open Courseware extremely helpful. It is a great course to look at for a preview on MATH 215 or as a supplement. Link: The professor, Dr. George Bluman, was a key asset. His treatment of the content promoted thorough and conceptual understanding. In addition, his devotion to student learning was clear, and his end of term review sessions were incredibly helpful. If he is your instructor, I strongly recommend seeing him during office hours or review sessions as he is genial, patient, and clear when helping to explain difficult problems and concepts.
The following term, I took MATH 316, and found it thoroughly easy in comparison to MATH 215. The course was less theory driven and focused more on calculations. Assignments took about half the time as in MATH 215. Again, this was likely a product of teaching style rather than course content.
Closing statement
In the term I took MATH 215, I also took CHEM 205, CHEM 211, and BIOL 201. An understanding of differential equations gave me a deeper understanding of chemical kinetics in these courses, not to mention a facility with mathematical reasoning. Differential equations constitute a critically important field in mathematics. Their theory brings concepts of calculus into clear focus and their applications to fields throughout science and engineering are countless. It is difficult to get by in the study of physics without a thorough understanding.
Sunday, 13 May 2012
The topic of electives came up recently among discussions with my fellow third year Physiology students. One of the hot topics was PHIL 433. This post is dedicated to the two ethics electives I've takenOn my second day of classes at UBC, I stumbled into IKB 182 for Introduction to Moral Theory. The previous year, I had found a video of a Harvard lecture, containing material that would eventually find its way into Michael Sandel's series entitled "Justice: What's the Right Thing To Do?" ( It seems that Ethical Philosophy is a cornerstone course in Harvard, and one of the best attended. In my first term at UBC, I was taking a break from the study of basic science, seeking instead to gain understanding of general scholarly thought, whether in mathematics, literature, or philosophy.
I quickly realized that I had no clue what Philosophy really was. I quickly did some research to put Ethics in context. I found that Philosophy is broken into four key loosely related fields:
Epistemology: The construction of a framework for the understanding of truth and reason.
Metaphysics: The study of the reality, existence, time, and purpose. Questions such as "who are we, really?" and "why do we exist?"
Ethics: The study of morality and its applications to human acts and behaviour. Questions such as "what do we ought to do?"
Aesthetics: The study of the origins and perception of beauty.
In the first half of the course, we studied a number of less widely accepted moral theories (egoism, divine command, cultural relativism, natural law theory). In the second half of the course, we studied more established theories (consequentialism, deontology, virtue ethics, and pluralism). The discussion of each topic entailed an introduction of the theory and its applications, and evaluation of the theory, including arguments for and against it. The course thus enables students to think critically about the limits of ethical theories.
I have found that, even years later, I still consider what I learned in this class when I discuss ethical issues. It did more than provide a grab bag of frameworks, it was my introduction to philosophy, argument, and logical thought. I came to understand that philosophy is not just about "thinking about thinking" or "thinking about things very hard," but rather was a complex framework of logical reason, an analytical system for the evaluation of abstract ideas. I found PHIL 230 fascinating, and through interesting discussions made contacts in the class from many different backgrounds.
Moral problems arising in the health sciences, especially in medicine but also in biology, psychology, and social work. Topics include abortion, death and euthanasia, genetic engineering, behaviour modification, compulsory treatment, experimentation with human beings and animals, and the relationship between professionals and their patients, subjects or clients. No philosophical background is requiredAs with many courses in an applied field of arts theory, this course is divided into two clear parts. We first discussed the ethical theory, principally covering topics in consequentialism, deontology, virtue ethics, pluralism, and social contract theory. We then applied such theory to the following topics: (1) euthanasia and physician directed suicide, (2) right to refuse treatment, (3) abortion, (4) 2-tier healthcare system, (5) allocation of scarce resources, (6) alternative medicine, and (7) neuroethics and cognitive enhancers.
Having taken PHIL 230A, I found the first half of the course a relaxing refresher. It allowed me to ease my way back into moral philosophy by considering these theories a second time. A previous course in either ethics or logic/critical thinking/argument is useful, but not a necessity. The introduction is certainly sufficient either way. The second half is where we got to sink our teeth into key issues, topics which Physiology students (and students studying medical science in general) will be fascinated by.
My instructor was Dr. Rana Ahmad, a post-doctoral fellow and an expert in the field of biomedical and scientific ethics. Although she used lecture slides through the whole course, she taught in an open, and discussion-oriented way, and made the course enjoyable, fascinating, and exciting. We used a textbook called "Debating Health Care Ethics" by Smolkin, Bourgeois, and Findler. This book is largely written in a conversational, scripted-debate format, which made it much easier to read than most textbooks, and conveyed its methods very clearly.
Exams were mostly essay-based, with the emphasis on a student's ability to form and communicate sound philosophical arguments grounded in concepts of moral theory. The result was the course was not incredibly time-consuming, but was highly rewarding and informative nonetheless.
In summary
Whether it's metaphysics or morals, philosophy is a topic in which we as humans as intrinsically fascinated. It deals in questions which we consider, consciously or not, every day of our lives. Philosophy as a field is also a teacher, one which guides us by principles of logical thought. It is the construction of a framework by which to comprehend the fabric of our being. Courses such as PHIL 230A and 433A are a fascinating look at classically constructed frameworks for understanding our own moral convictions. Some believe that these moral questions have no true answer. But, whether we go on to become scientists, doctors, politicians, or industrial/corporate workers, it is the consideration of such concerns which is our great teacher.
Of course, this means that students will, from now on, have to take ANAT 390 and PHYL 421. As a result, students will have 12 elective credits in year three, and 3 elective credits in year 4. The Physiology student's courseload has, in effect, been made heavier, and future physiology students will come out of the program with a better understanding of cellular physiology.
For current third year students, the faculty have agreed to waive the ANAT 390 prerequisite. This will not be so for future students. The course sounds exceedingly interesting, but it threatens to make me ridiculously busy next year. What does this mean for students taking a Physics minor?
It should still be possible to do, but this will definitely make it more difficult. I could have fit ANAT 390 into my third year courses. I would have been forced to drop MATH 300 (which I did very well in, so thankfully this was not the case), which was one of my elective courses. Fourth year is tight as it is, and this change means you may want to save no more than one minor requirement for fourth year (unless you're planning to take an extra term).
As for me, I'm on the fence. The course sounds fascinating and useful, and fourth year physiology courses tend to have relatively high class averages. I have a feeling, however, that I will opt out of this one. I am already forced to take PHYS 301 in conflict with PHYL 430, and I don't think it would be wise to take another 36 credit year.
Friday, 20 April 2012
This is one of those courses that students from a variety of backgrounds still have to take. Math 200 is a course in multivariable calculus. Your standard first year calculus courses cover topics in differential and integral calculus, analytical methods for understanding rate relations and modeling systems. What you may have noticed, however, is that these courses are largely one-dimensional. You deal with some quantity as it varies with respect to some dependent variable. Math 200 is the extension of first year calculus to multiple dimensions.
By multiple, I mean 2 or 3. Concepts remain well within the realm of visualizability, though they are more brain-bending than anything in Math thus far. You will begin with concepts in vector analysis such as using dot products and cross products, lead into partial derivatives (which are just derivatives with respect to one variable, treating all the other variables as constants) and their applications, and then into multiple integration (integrating over a 2 dimensional or 3 dimensional space instead of just along one axis).
Math 200 sets the groundwork for future courses in Mathematics, such as Math 215 and Math 317, and is therefore a cornerstone for prerequisites in Physics. Physical systems are often modelled in one or two dimensions to start with, but most concepts are extended to three or more dimensions when applied to real-world situations. It is key to develop a solid understanding of the analytic approach to such problems, which can be difficult to interpret by visualization alone.
Math 317 (vector calculus) starts where 200 left off. It can be divided into three main parts: (1) analysis of space curves, (2) scalar and vector fields in two dimensions, (3) parametric surfaces and scalar and vector fields in three dimensions. This course builds on the concept of parametrization, showing that if components of a curve's parametrization are known, then one can prove some powerful theorems to analyse such systems. Here, you will study Green's theorem, Stokes theorem, and divergence/Gauss' theorem. Between these three, there lies a considerable amount of abstract conceptual imagery.
Math 200 and 317 are key for the study of topics in classical mechanics and electrodynamics, and are useful for modelling systems in all practically all fields of Physics. I would consider Math 200 a survey course in mathematical concepts, akin to first year Math courses (at least the way I was taught it anyhow), while Math 317 delves deeper into analytical theory. Much of Math 317 follows a "theorem-proof" format, more suggestive of higher level Math courses.
Both courses are central to the study of Physics, and I found them both interesting. I found Math 200 a little dry, but that may have been because I took it as a first year student and was finding my way around the University. Math 317, on the other hand, is the last Math course I've taken (and the last one I'll probably ever take). It has been an absolute pleasure to be in Dr. Ed Perkins's class. His manner of teaching is crystal clear and well-paced, and this course not only was enjoyable and relaxing, but helped to elucidate concepts which I had long grappled with.
Saturday, 31 December 2011
This post is in response to a question posed by a student I am in discussion with for SPAC (Science Peer Academic Coaching). For more information, please visit
I'll try not to tout either program, but I'll give a rundown of the details of each and let you come up with your own ideas.
Physiology
Physiology only offers an honours program, make sure that you're capable and willing to commit to an honours program prior to signing up (i.e. if you bomb a course and are forced to drop honours status, you have to switch specializations).
The class size is usually between 10 and 20 students (at least in recent years). You can expect to cover a broad foundation in Physiological sciences, particularly the structure and function of human body processes.
Discussion of required 3rd year courses:
PHYL 301 - General overview of physiology. This is like British Columbia's BIOL 12 human body sections on steroids. The sections of the course are listed in great detail on the Cellphys website above. This is a very popular course, and gives a foundation of understanding in many human body systems. Different sections are covered to different depths, and there is, in my opinion, about a 50/50 balance between conceptual understanding and rote memorization. Different sections are taught by different instructors, which has so far been not a jarring experience. There are two exams, one at the end of term one and one at the end of term two. This course requires a fair bit of studying, but is fascinating and rewarding.
PHYL 303 - Our wonderful physiology lab. Whereas PHYL 301 has some 500 students, PHYL 303 is restricted to members of Hons. PHYL and Hons. PCTH (by option). Preparation for labs and the process of labs are generally fairly low-stress, though lab reports can be significantly more troubling. Weeks alternate between formal lab reports (fit to submit to a scientific journal), and brief results reports/worksheets. Marking is not easy, and this course serves as an excellent introduction to scientific writing. First term labs include a blood lab, a number of electrophysiology (nerve conduction velocity, electromyogram, etc.) labs, an electrocardiogram lab, and a neuroanatomy lab in which we examine and handle donated human brains. A highlight in second term is sessions on surgical techniques. This lab course is excellent for solidifying a number of concepts from PHYL 301, and offers insight on techniques most undergraduate students do not have the privilege to explore.
BIOC 301 - In this course we go through a series of experiments which mimic techniques many students would do in a biochemical research job such as PCR, gel electrophoresis, PCR cleanup, restriction enzyme digestion, DNA ligation etc. We aim to ligate a protein coding segment into a plasmid containing Ampicillin resistance. We then introduce this plasmid into competent bacteria, and plate the bacteria on Ampicillin-containing LB media. In the next term, we will attempt to use the bacteria to grow the protein of interest.
I did not take BIOC 302 this term, and I do not need to take STAT 200 or BIOL 300 this year.
Physics courses
I don't have too much insight on specializing in Physics, aside from doing a minor. In order to complete a minor, one must take 18 upper level PHYS/ASTR credits. I took 2 physics courses and 2 math courses this term, and I found it extremely rewarding. These courses have given me deeper insight on the way science is done and the nature of the universe.
In addition, I find the social atmosphere around the Physics program very warm and supportive. The UBC Physics society is located in the room HENN 307, and is a great place for physics students to hang out, have lunch, and discuss homework.
Taking physics courses had a semi-expected consequence. Weekly assignments. This term, I had to complete assignments weekly for 3 courses, which took me anywhere from 4 to 12 hours each. I had biweekly assignments for a fourth course. This is on top of weekly lab reports for PHYL 303, and occasional lab reports for BIOC 301. Needless to say, these courses keep students busy, but it does also keep you on track, and prevent last-minute cramming.
MATH 316 - We start with brief review of MATH 215, followed by a discussion of series solutions to ODE's. We then discussed a number of methods to algebraically and numerically come up with solutions to partial differential equations. Weekly assignments took 4-8 hours to do. Very important for applied mathematics and physics courses (particularly PHYS 304, I found).
MATH 300 - I didn't actually have to take this course for any reasons whatsoever. It just sounded interesting. This is a course in complex variables, an introduction to analysis in the complex plane. This was incredibly eye-opening for a number of mathematical concepts which I had not had a complete understanding of before. The course is broken in to 5 sections: (1) Algebraic and geometric representations of complex numbers, (2) functions of a complex variable, (3) integration of functions in the complex plane, (4) sequences and series, and (5) residue calculus. Assignments were weekly and took about 6-10 hours each.
PHYS 304 - Quantum mechanics. I found the quantum mechanics in PHYS 200 somewhat disjointed and hard to follow. This course hammered the mathematical models of quantum mechanics into place. From first principles, we work our way into 3-dimensional radial quantum mechanics, and the mathematical construction of the Hydrogen molecule. Using an algebraic approach, we also develop the trends inherent in the periodic table of elements. This course is also an introduction to a number of mathematical concepts such as linear algebra topics, operator approaches to linear algebra, bra-ket vector notation, a number of well established polynomial sequences, partial differential equations, and more. This course had weekly assignments taking about 8-12 hours each.
PHYS 404 - Medical physics. I found this course fascinating but disjointed. It is divided into six 4-lecture topics, each taught by a different instructor. They are (1) introduction to imaging, (2) MRI, (3) nuclear medicine, (4) radiological imaging, (5) biomedical optics, and (6) radiation therapy. This was not an easy course, and not an exciting one either. It did serve as a good introduction to imaging concepts and radiation physics, as well as a great chance to tie together medical and physical concepts. There were bi-weekly assignments taking 1-3 hours each. There is a term paper (2000-3000 words) on a medical physics topic of your choice.
I will likely do a similar post at the end of next term. Cheers, and take care!
My first task on the Wednesday which marked the start of my classes was to locate the small and aged mathematics building on campus under a blanket of rainy weather. This is nothing new to a seasoned UBC student, but I was heading to my very first class.
This is my personal addition to the reams that have been said regarding the intimidation of approaching University classes for the first time. I had a sneaking suspicion that MATH 223 was not going to be easy, and I was not sure I had even close to enough background to handle it.
I'll get some details out of the way. I attended University Hill Secondary School from 2004 to 2009, a school known by many as one of the consistently highest academically rated public schools in the lower mainland. My study focus was primarily in the sciences.
I took an accelerated science course in 8th grade, which covered 9th grade material as well, and took Science 10 in my 9th grade. I then took our school's AP version of Chemistry, Physics, and Biology courses, as well as Math 12 in the following years, and AP Calculus BC at Kerrisdale academy. I received 4's and 5's on AP exams in Chemistry, Physics B and C (mechanics), Biology, and Calculus BC. In my 12th grade, I shifted my focus slightly and took many humanities courses.
I had no idea how MATH 223 would pan out, considering I had not taken any University courses in mathematics. The day before the first class, I read through some basic matrix algebra concepts including matrix addition, subtraction, multiplication, and determinants. As a result, the first class was relatively comfortable, touching on these concepts and using them to determine the properties of inverse matrices.
Professor Anstee was extremely welcoming and friendly, displaying expertise, wit, and warmness. I was surprised that he gave us his home phone number rather than his office number. My impression was that he was quite dedicated to his role as a teacher and cared about his students' learning.
He said something on the first class which both unnerved and excited me. He looked through the class list, checking on the specialities of the students. There were students from mathematics, physics, and computer science, all expected. He appeared slightly confused, however, at the number of students from other fields, and advised that most of us were likely in the wrong course. I smiled, driven by the same academically masochistic drive that led me to take AP courses in high school.
But MATH 223 was like no course I had thus far experienced. The proofs started at the second class. I realized quickly that this would be a course in which I desperately copy down chalk markings during lectures in hopes that I might have time later on to actually understand them.
Weekly assignments were difficult. I cannot put it any way but frankly. The majority of questions were proofs, which took anywhere between 10 minutes and one hour each. Assignments were made up of anywhere between 8 and 12 questions, some with multiple parts. I would often get stuck on questions for long stretches of time. I would take breaks, ask friends for their thoughts, and come back to the questions after some time, burning up pages and pages of blank paper scribbling every approach I could come up with.
Exams were similarly difficult, but definitely fair. Professor Anstee was careful to give us exams which were hard to ace, but easy to pass. 60% of every exam would be based on basic algorithmic concepts, and usually required little cleverness (gaussian elimination, the determinant, inverse matrices, Gram-Schmidt process etc.). The remaining 40% would usually be made up of four 10% questions of increasing difficulty. These would usually be based on smaller concepts derived in proofs in class, and some were proofs themselves. Often only 1 or 2 students in the class would get any points at all on the final question of the exam.
My conclusion, the hardest part of this course was the homework. I spent from 5-10 hours on homework weekly for this course alone (why this course is worth only 3 credits, I have no clue). Exams were not easy, but doable, and if I had studied more, I imagine I could actually have done well. Unless you're very confident in your abilities, don't expect to get over 90% in this class, but I think with solid effort devoted to studying for exams, the 80's are attainable.
One last point up for debate is whether this course is better as a base in linear algebra than 221. After the fact, I barely remembered any but the most key concepts in this course. I feel that material raced by so rapidly that I had no time to grasp much of the conceptual framework with confidence. However, where this course was useful was that it challenged me to think at a level of mathematical abstraction that I had never even knew I could. The homework beat into me a new-found confidence for approaching problems of a mathematical nature, and this, I feel, has helped me immensely in subsequent courses.
Tuesday, 30 August 2011
This is not a metaphysical quest, merely an introduction. My name is Eric, and I am an undergraduate student at the University of British Columbia, located in Vancouver, British Columbia, Canada. I am currently working on an Honours degree in Physiology.
The number of Physiology students at UBC is already quite limited, but what makes me academically different is that I am completing a minor in Physics and Astronomy. The spectrum of courses I am taking therefore encompasses life science courses as well as math, physics, and astronomy.
I am here (on blogger) because I believe that a personal perspective on one's experience in University can often be an incredibly helpful and thought-provoking tool for others considering their academic paths. Of course, this is just a thought. If such a result ensues, I would be deeply thankful that my experiences might help guide or encourage others towards their proper callings.
My specific motivation, however, lies in the fact that, although my choice in program may be unusual, it is likely far from unique. The next physiology student (or any life science student for that matter) who decides to minor in physics (or any mathematical, analytical, or physical science field) might find use in these pages.
The structure of this blog will be fairly loose, but posts are likely to fall into a number of categories.
Course evaluations
I tend to give detailed online evaluations at the ends of terms. However, it is disheartening to think that those words cannot reach students deciding which courses to take. In these posts, I will share my experience with specific courses. It is likely that these will make up the bulk of my early posts, as I blast through all my first and second year courses.
Note that I will not be making specific reference to my impressions of instructors. This is partly because I will make little effort to conceal my identity, but mainly because this information is likely to grow out of date as courses inevitably switch instructors over time. I will, however, make very brief reference to instructors if their teaching style or specific teaching methods significantly impacted my experience in a course, or if I believe that the instructor warrants an extremely high level of applause for their ability. For all other evaluations of instructors, ratemyprofs is generally decent as long as one keeps in mind its inherent biases.
Courses for which evaluations can be expected are...
Relatively soon:
MATH 200, 223, 215.
ENGL 112, 120.
PHIL 230a, 433.
PHYS 108, 200.
BIOL 112, 200, 201, 205, 234.
CHEM 123, 205, 211, 233, 235.
STAT 241.
MICB 202.
MUSC 167.
After term 1 this year:
PHYS 404.
ASTR 303.
MATH 300, 317.
After term 2 this year:
PHYL 301, 303.
BIOC 301, 302.
PHYS 304.
MATH 316.
ANTH 227.
After next year (tentative):
PHYL 422, 423, 424, 426, 430, 449.
PHYS 301 (or 305, but not both).
One other physics or astronomy course (likely ASTR 403).
Study tips
I am continually discovering more effective study methods. At some point, I'll share insight on my most significant study strategies from the last couple of years. Additionally, I'll post study tips as they come to mind, or as I learn them from others.
Due to my interest in study methods, this year I am taking part in SPAC (Science Peer Academic Coaches), and my first big study tip is to check out their website I highly recommend checking it out.
General experience/day to day life
This is, after all, a blog. Plenty of exciting stuff happens at UBC all the time, and when something really exciting happens, I'll write about it here if I have time.
In general, I'll try to remember to declare what "type" of post each post is.
Well, I essentially just wrote up a course outline. Perhaps just writing in this blog would have made all of this self explanatory, but whatever. In a way, this was as much for me as for you, which hopefully means my subsequent posts will be all the more organized and helpful to whoever might wish to read.
For now, cheers, and adios fellow UBCites. Enjoy the last week of summer before Imagine day! | 677.169 | 1 |
Algebra provides a systematic way to represent mathematical relationships and analyze change. Students need to understand the concepts and symbols of algebra, the structures that govern the manipulation of the symbols, and how the symbols can be used to record ideas and events. Students will explorepatterns that are linear and quadratic in the first year of algebra, and should develop the notion of families of functions. A solid conceptual foundation in algebra should be developed before students engage extensively in symbolic manipulation.
Some of the topics we will cover include Real Number Systems, Expressions, Equations, Inequalities, Problem Solving, Probability, Statistics, Angles, Transformations, 2D/3D Figures, and Patterns, Functions, Algebra.
Class Materials: (1) 100 Page Composition Notebook
Pencils
Class Donation (we use a lot of these throughout the year):
Glue Sticks
Dry Erase Markers
Tissues
News and Notes: WATER PARK PROJECT Due Friday, May 26th
Will not be accepted late
Copy in POST SOL folder below
All students should sign-up for a ReCap account. Go to letsrecap.com/student students should enter the pin: vgtjxrh
Upcoming Events | 677.169 | 1 |
16
MULTIPLE INTEGRALS
MULTIPLE
MULTIPLE INTEGRALS
16.3
Double Integrals
over General Regions
In this section, we will learn:
How to use double integrals to find
the areas of regions of different shapes.
SINGLE INTEGRALS
For single integrals, the region
ov
Math 4A Castroconde
Exam 3BExam 3AQuiz 1BMath 4A Castroconde
Quiz 1AMATH 4A Advice
Showing 1 to 2 of 2
Honestly this course is very difficult. Taking Gupta is essential to success. He will show you what you need to do to get an A in his course. If you re-read his notes, do the homework, read the book (skim it is fine), do some of the previous tests (he gives for practice) you are guaranteed an A.
Course highlights:
You will learn double and triple integrals. A lot of awesome stuff about the third dimension (space, x-y-z plane). You will also learn about a few cool theorems like Stoke's Theorem and Green's Theorem. rdrdthetadz polar coordinates are the best thing for me!!
Hours per week:
9-11 hours
Advice for students:
DO THE HOMEWORK! Also please pay attention to the advice Gupta gives, they are key to success in class. Sometimes he gives life advice so please listen to that. His advice has helped me so much. | 677.169 | 1 |
The mathematical theory of population, of its character and fluctuations, and of the factors which influence them, being an examination of the general scheme of statistical representation, with deductions of necessary formulae; the whole being applie! for Sharing!
You submitted the following rating and review. We'll publish them on our site once we've reviewed them. | 677.169 | 1 |
I highly recommend this course due to a few reasons. Although higher math may not seem very useful in non-STEM majors, it has to be one of the most significant skills that applies to almost anywhere in life. Calculus is an adventure through a very high level of problem solving. This level of logical reasoning is a rarity, and it is in very high demand in the market place. Solving math problems apply to solving almost any problem in life whether it's finding and predicting the rate of change in profits for business, or even calculating how many calories we will burn. I entered calculus fearing the difficulty, but I have found studying the language of the universe has to be among the most intriguing topics I've ever explored.
Course highlights:
The first two weeks of calculus was a bit intimidating to me. This had been from the brisk review of many algebraic and trigonometric concepts that I had simply forgotten over summer break. The first month and a half was increasingly difficult for me due to having to use the longest method to derive equations and then simplifying the complex result. This is why it's important to be proficient in basic mathematical skills, because simplifying is the half of it. After the first month and a half, the calculus had become much easier to understand, and we learned much faster, easier, and more proficient methods to derive equations. Although the problems become more complex, the calculus really does become easier, but forgetting the basic concepts causes most mistakes.
Hours per week:
9-11 hours
Advice for students:
I would suggest to future students that they familiarize themselves with many of the mathematical concepts they've learned prior to this course. The worst culprit of most mistakes seems to be forgetting algebraic and trigonometric concepts. Even though most students may be somewhat experienced with these concepts, proficiency in these become very important when simplifying very complex results. Also I would suggest for many of those who find themselves struggling or even questioning their capabilities in the first month or two of this course, simply keep pushing on. Even though it all doesn't make sense right away, it really does get better and become easier. Make sure there's at least a few hours or so every other day to focus on mastering practice problems in this course, because there is not a whole lot of time to be slow on the exams. I started this course a bit afraid of not knowing if I have it in me, but after much effort I truly enjoy learning calculus.
Course Term:Fall 2016
Professor:Loge, Erik
Course Required?Yes
May 23, 2016
| Would highly recommend.
Not too easy. Not too difficult.
Course Overview:
The homework left me extremely well prepared for the tests. Also the teacher offers a lot of extra credit so it's easy to get an A even if you bomb one test.
Course highlights:
This course is a thorough grounding in the details and techniques for differentiation as well as the basics of integration. | 677.169 | 1 |
Featured Products 4+. 242 pagesHelp students make the transition from arithmetic to algebra and geometry! Perfect for use as full units of study, as supplements to the curriculum, or as tutorial resources at home. Each book includes step-by-step instructions with examples, practice problems using the concepts, real-life applications, a list of symbols and terms, tips, and answer keys. 128 pages.
Help students to practice the strategies and acquire the skills needed to successfully perform on Common Core State Standards assessments. Each book includes test-taking tips, instructional resources, practice assessments using literature, informational text, and paired passages. 64 pages. | 677.169 | 1 |
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Parabola is an online magazine published by the School of Mathematics and Statistics, UNSW Australia, with editorial support from the Australian Mathematics Trust (AMT). AMT is a not-for-profit organisation that administers mathematical enrichment activities for Australian and international students and publishes materials on mathematical enrichment. Its trustee is the University of Canberra and its Board comprises representation from the Australian Association of Mathematics Teachers, as well as representatives from academia and the corporate sector. For a brief history of Parabola, see our History page.
Our web presence began in 2009 with the Parabola Online Project, initiated by a grant from the U-Committee of UNSW. In 2014 the redesigned website was launched.
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Parabola publishes articles that can contribute to the teaching and learning of mathematics at the senior secondary school level or equivalent, in the areas of applied mathematics, mathematical modelling, pure mathematics, statistics and the history of mathematics. The journal's readership consists of mathematics students, teachers and researchers with interests in promoting excellence in senior secondary school mathematics education - as well as anyone else who is interested in mathematics. Contributions to the journal are invited as scholarly articles, surveys and problems subject to approval by the editors and editorial board. Parabola is keen to encourage a wide range of contributions, particularly from but not limited to, educators and students at secondary and tertiary levels.
For information on how to submit an article, please visit our Contact page. | 677.169 | 1 |
Linear Algebra in Physics
(Summer Semester, 2006)
1 Introduction
The mathematical idea of a vector plays an important role in many areas of physics.
• Thinking about a particle traveling through space, we imagine that its speed
and direction of travel can be represented by a vector v in 3-dimensional
Euclidean space R
3
. Its path in time t might be given by a continuously
varying line — perhaps with self-intersections — at each point of which we
have the velocity vector v(t).
• A static structure such as a bridge has loads which must be calculated at
various points. These are also vectors, giving the direction and magnitude of
the force at those isolated points.
• In the theory of electromagnetism, Maxwell's equations deal with vector fields
in 3-dimensional space which can change with time. Thus at each point of space
and time, two vectors are specified, giving the electrical and the magnetic fields
at that point.
• Given two different frames of reference in the theory of relativity, the trans-
formation of the distances and times from one to the other is given by a linear
mapping of vector spaces.
• In quantum mechanics, a given experiment is characterized by an abstract
space of complex functions. Each function is thought of as being itself a kind
of vector. So we have a vector space of functions, and the methods of linear
algebra are used to analyze the experiment.
Looking at these five examples where linear algebra comes up in physics, we
see that for the first three, involving "classical physics", we have vectors placed at
different points in space and time. On the other hand, the fifth example is a vector
space where the vectors are not to be thought of as being simple arrows in the
normal, classical space of everyday life. In any case, it is clear that the theory of
linear algebra is very basic to any study of physics.
But rather than thinking in terms of vectors as representing physical processes, it
is best to begin these lectures by looking at things in a more mathematical, abstract
way. Once we have gotten a feeling for the techniques involved, then we can apply
them to the simple picture of vectors as being arrows located at different points of
the classical 3-dimensional space.
2 Basic Definitions
Definition. Let X and Y be sets. The Cartesian product X Y , of X with Y is
the set of all possible pairs (x, y) such that x ∈ X and y ∈ Y .
1
Definition. A group is a non-empty set G, together with an operation
1
, which is a
mapping ' ' : GG → G, such that the following conditions are satisfied.
1. For all a, b, c ∈ G, we have (a b) c = a (b c),
2. There exists a particular element (the "neutral" element), often called e in
group theory, such that e g = g e = g, for all g ∈ G.
3. For each g ∈ G, there exists an inverse element g
−1
∈ G such that g g
−1
=
g
−1
g = e.
If, in addition, we have a b = b a for all a, b ∈ G, then G is called an "Abelian"
group.
Definition. A field is a non-empty set F, having two arithmetical operations, de-
noted by '+' and '', that is, addition and multiplication
2
. Under addition, F is an
Abelian group with a neutral element denoted by '0'. Furthermore, there is another
element, denoted by '1', with 1 ,= 0, such that F ¸ ¦0¦ (that is, the set F, with
the single element 0 removed) is an Abelian group, with neutral element 1, under
multiplication. In addition, the distributive property holds:
a (b + c) = a b + a c and (a + b) c = a c + b c,
for all a, b, c ∈ F.
The simplest example of a field is the set consisting of just two elements ¦0, 1¦
with the obvious multiplication. This is the field Z/2Z. Also, as we have seen in
the analysis lectures, for any prime number p ∈ N, the set Z/pZ of residues modulo
p is a field.
The following theorem, which should be familiar from the analysis lectures, gives
some elementary general properties of fields.
Theorem 1. Let F be a field. Then for all a, b ∈ F, we have:
1. a 0 = 0 a = 0,
2. a (−b) = −(a b) = (−a) b,
3. −(−a) = a,
4. (a
−1
)
−1
= a, if a ,= 0,
5. (−1) a = −a,
6. (−a) (−b) = a b,
7. a b = 0 ⇒ a = 0 or b = 0.
1
The operation is usually called "multiplication" in abstract group theory, but the sets we will
deal with are also groups under "addition".
2
Of course, when writing a multiplication, it is usual to simply leave the '' out, so that the
expression a b is simplified to ab.
2
Proof. An exercise (dealt with in the analysis lectures).
So the theory of abstract vector spaces starts with the idea of a field as the
underlying arithmetical system. But in physics, and in most of mathematics (at
least the analysis part of it), we do not get carried away with such generalities.
Instead we will usually be confining our attention to one of two very particular fields,
namely either the field of real numbers R, or else the field of complex numbers C.
Despite this, let us adopt the usual generality in the definition of a vector space.
Definition. A vector space V over a field F is an Abelian group — with vector
addition denoted by v + w, for vectors v, w ∈ V. The neutral element is the "zero
vector" 0. Furthermore, there is a scalar multiplication F V → V satisfying (for
arbitrary a, b ∈ F and v, w ∈ V):
1. a (v +w) = a v + a w,
2. (a + b) v = a v + b v,
3. (a b) v = a (b v), and
4. 1 v = v for all v ∈ V.
Examples
• Given any field F, then we can say that F is a vector space over itself. The
vectors are just the elements of F. Vector addition is the addition in the field.
Scalar multiplication is multiplication in the field.
• Let R
n
be the set of n-tuples, for some n ∈ N. That is, the set of ordered lists
of n real numbers. One can also say that this is
R
n
= R R R
. ¸¸ .
n times
,
the Cartesian product, defined recursively. Given two elements
(x
1
, , x
n
) and (y
1
, . . . , y
n
)
in R
n
, then the vector sum is simply the new vector
(x
1
+ y
1
, , x
n
+ y
n
).
Scalar multiplication is
a (x
1
, , x
n
) = (a x
1
, , a x
n
).
It is a trivial matter to verify that R
n
, with these operations, is a vector space
over R.
3
• Let C
0
([0, 1], R) be the set of all continuous functions f : [0, 1] →R. This is a
vector space with vector addition
(f + g)(x) = f(x) + g(x),
for all x ∈ [0, 1], defining the new function (f + g) ∈ C
0
([0, 1], R), for all
f, g ∈ C
0
([0, 1], R). Scalar multiplication is given by
(a f)(x) = a f(x)
for all x ∈ [0, 1].
3 Subspaces
Let V be a vector space over a field F and let W ⊂ V be some subset. If W is itself
a vector space over F, considered using the addition and scalar multiplication in V,
then we say that W is a subspace of V. Analogously, a subset H of a group G, which
is itself a group using the multiplication operation from G, is called a subgroup of
G. Subfields are similarly defined.
Theorem 2. Let W ⊂ V be a subset of a vector space over the field F. Then
W is a subspace of V ⇔ a v + b w ∈ W,
for all v, w ∈ W and a, b ∈ F.
Proof. The direction '⇒' is trivial.
For '⇐', begin by observing that 1 v +1 w = v +w ∈ W, and a v +0 w =
a v ∈ W, for all v, w ∈ W and a ∈ F. Thus W is closed under vector addition
and scalar multiplication.
Is W a group with respect to vector addition? We have 0 v = 0 ∈ W, for
v ∈ W; therefore the neutral element 0 is contained in W. For an arbitrary v ∈ W
we have
v + (−1) v = 1 v + (−1) v
= (1 + (−1)) v
= 0 v
= 0.
Therefore (−1) v is the inverse element to v under addition, and so we can simply
write (−1) v = −v.
The other axioms for a vector space can be easily checked.
The method of this proof also shows that we have similar conditions for subsets
of groups or fields to be subgroups, or subfields, respectively.
Theorem 3. Let H ⊂ G be a (non-empty) subset of the group G. Then H is a
subgroup of G ⇔ ab
−1
∈ H, for all a, b ∈ H.
4
Proof. The direction '⇒' is trivial. As for '⇐', let a ∈ H. Then aa
−1
= e ∈ H. Thus
the neutral element of the group multiplication is contained in H. Also ea
−1
= a
−1
∈
H. Furthermore, for all a, b ∈ H, we have a(b
−1
)
−1
= ab ∈ H. Thus H is closed
under multiplication. The fact that the multiplication is associative (a(bc) = (ab)c,
for all a, b and c ∈ H) follows since G itself is a group; thus the multiplication
throughout G is associative.
Theorem 4. Let U, W ⊂ V be subspaces of the vector space V over the field F.
Then U∩ W is also a subspace.
Proof. Let v, w ∈ U∩ W be arbitrary vectors in the intersection, and let a, b ∈ F
be arbitrary elements of the field F. Then, since U is a subspace of V, we have
a v +b w ∈ U. This follows from theorem 2. Similarly a v +b w ∈ W. Thus it
is in the intersection, and so theorem 2 shows that U∩ W is a subspace.
4 Linear Independence and Dimension
Definition. Let v
1
, . . . , v
n
∈ V be finitely many vectors in the vector space V over
the field F. We say that the vectors are linearly dependent if there exists an equation
of the form
a
1
v
1
+ + a
n
v
n
= 0,
such that not all a
i
∈ F are simply zero. If no such non-trivial equation exists, then
the set ¦v
1
, . . . , v
n
¦ ⊂ V is said to be linearly independent.
This definition is undoubtedly the most important idea that there is in the theory
of linear algebra!
Examples
• In R
2
let v
1
= (1, 0), v
2
= (0, 1) and v
3
= (1, 1). Then the set ¦v
1
, v
2
, v
3
¦ is
linearly dependent, since we have
v
1
+v
2
−v
1
= 0.
On the other hand, the set ¦v
1
, v
2
¦ is linearly independent.
• In C
0
([0, 1], R), let f
1
: [0, 1] → R be given by f
1
(x) = 1 for all x ∈ [0, 1].
Similarly, let f
2
be given by f
2
(x) = x, and f
3
is f
3
(x) = 1 − x. Then the set
¦f
1
, f
2
, f
3
¦ is linearly dependent.
Now take some vector space V over a field F, and let S ⊂ V be some subset
of V. (The set S can be finite or infinite here, although we will usually be dealing
with finite sets.) Let v
1
, . . . , v
n
⊂ S be some finite collection of vectors in S, and let
a
1
, . . . , a
n
∈ F be some arbitrary collection of elements of the field. Then the sum
a
1
v
1
+ + a
n
v
n
is a linear combination of the vectors v
1
, . . . , v
n
in S. The set of all possible linear
combinations of vectors in S is denoted by span(S), and it is called the linear span
5
of S. One also writes [S]. S is the generating set of [S]. Therefore if [S] = V, then
we say that S is a generating set for V . If S is finite, and it generates V, then we
say that the vector space V is finitely generated.
Theorem 5. Given S ⊂ V, then [S] is a subspace of V.
Proof. A simple consequence of theorem 2.
Examples
• For any n ∈ N, let
e
1
= (1, 0, . . . , 0)
e
2
= (0, 1, . . . , 0)
.
.
.
e
n
= (0, 0, . . . , 1)
Then S = ¦e
1
, e
2
, . . . , e
n
¦ is a generating set for R
n
.
• On the other hand, the vector space C
0
([0, 1], R) is clearly not finitely gener-
ated.
3
So let S = ¦v
1
, . . . , v
n
¦ ⊂ V be a finite set. From now on in these discussions,
we will assume that such sets are finite unless stated otherwise.
Theorem 6. Let w = a
1
v
1
+ a
n
v
n
be some vector in [S] ⊂ V, where a
1
, . . . , a
n
are arbitrarily given elements of the field F. We will say that this representation of
w is unique if, given some other linear combination, w = b
1
v
1
+ b
n
v
n
, then we
must have b
i
= a
i
for all i = 1, . . . , n. Given this, then we have that the set S is
linearly independent ⇔ the representation of all vectors in the span of S as linear
combinations of vectors in S is unique.
Proof. '⇐' We certainly have 0 v
1
+ 0 v
n
= 0. Since this representation of the
zero vector is unique, it follows S is linearly independent.
'⇒' Can it be that S is linearly independent, and yet there exists a vector in the
span of S which is not uniquely represented as a linear combination of the vectors in
S? Assume that there exist elements a
1
, . . . , a
n
and b
1
, . . . , b
n
of the field F, where
a
j
,= b
j
, for at least one j between 1 and n, such that
a
1
v
1
+ + a
n
v
n
= b
1
v
1
+ + b
n
v
n
.
But then
(a
1
−b
1
)v
1
+ + (a
j
−b
j
)
. ¸¸ .
=0
v
i
+ + (a
n
−b
n
)v
n
= 0
shows that S cannot be a linearly independent set.
3
In general such function spaces — which play a big role in quantum field theory, and which
are studied using the mathematical theory of functional analysis — are not finitely generated.
However in this lecture, we will mostly be concerned with finitely generated vector spaces.
6
Definition. Assume that S ⊂ V is a finite, linearly independent subset with [S] =
V. Then S is called a basis for V.
Lemma. Assume that S = ¦v
1
, . . . , v
n
¦ ⊂ V is linearly dependent. Then there
exists some j ∈ ¦1, . . . , n¦, and elements a
i
∈ F, for i ,= j, such that
v
j
=
i=j
a
i
v
i
.
Proof. Since S is linearly dependent, there exists some non-trivial linear combination
of the elements of S, summing to the zero vector,
n
i=1
b
i
v
i
= 0,
such that b
j
,= 0, for at least one of the j. Take such a one. Then
b
j
v
j
= −
i=j
_
b
i
−
b
j
a
i
a
j
_
v
i
This shows that [S
′
] = V.
In order to show that S
′
is linearly independent, assume that we have
0 = cw +
i=j
c
i
v
i
= c
_
n
i=1
a
i
v
i
_
+
i=j
c
i
v
i
=
n
i=1
(ca
i
+ c
i
)v
i
(with c
j
= 0),
for some c, and c
i
∈ F. Since the original set S was assumed to be linearly inde-
pendent, we must have ca
i
+ c
i
= 0, for all i. In particular, since c
j
= 0, we have
ca
j
= 0. But the assumption was that a
j
,= 0. Therefore we must conclude that
c = 0. It follows that also c
i
= 0, for all i ,= j. Therefore, S
′
must be linearly
independent.
Theorem 9 (Steinitz Exchange Theorem). Let S = ¦v
1
, . . . , v
n
¦ be a basis of V
and let T = ¦w
1
, . . . , w
m
¦ ⊂ V be some linearly independent set of vectors in V.
Then we have m ≤ n. By possibly re-ordering the elements of S, we may arrange
things so that the set
U = ¦w
1
, . . . , w
m
, v
m+1
, . . . , v
n
¦
is a basis for V.
Proof. Use induction over the number m. If m = 0 then U = S and there is nothing
to prove. Therefore assume m ≥ 1 and furthermore, the theorem is true for the
case m − 1. So consider the linearly independent set T
′
= ¦w
1
, . . . , w
m−1
¦. After
an appropriate re-ordering of S, we have U
′
= ¦w
1
, . . . , w
m−1
, v
m
, . . . , v
n
¦ being a
basis for V. Note that if we were to have n < m, then T
′
would itself be a basis for
V. Thus we could express w
m
as a linear combination of the vectors in T
′
. That
would imply that T was not linearly independent, contradicting our assumption.
Therefore, m ≤ n.
Now since U
′
is a basis for V, we can express w
m
as a linear combination
w
m
= a
1
w
1
+ + a
m−1
w
m−1
+ a
m
v
m
+ + a
n
v
n
.
8
If we had all the coefficients of the vectors from S being zero, namely
a
m
= a
m+1
= = a
n
= 0,
then we would have w
m
being expressed as a linear combination of the other vectors
in T. Therefore T would be linearly dependent, which is not true. Thus one of the
a
j
,= 0, for j ≥ m. Using theorem 8, we may exchange w
m
for the vector v
j
in U
′
,
thus giving us the basis U.
Theorem 10 (Extension Theorem). Assume that the vector space V is finitely
generated and that we have a linearly independent subset S ⊂ V. Then there exists
a basis B of V with S ⊂ B.
Proof. If [S] = V then we simply take B = S. Otherwise, start with some given
basis A ⊂ V and apply theorem 9 successively.
Theorem 11. Let U be a subspace of the (finitely generated) vector space V. Then
U is also finitely generated, and each possible basis for U has no more elements than
any basis for V.
Proof. Assume there is a basis B of V containing n vectors. Then, according to the-
orem 9, there cannot exist more than n linearly independent vectors in U. Therefore
U must be finitely generated, such that any basis for U has at most n elements.
Theorem 12. Assume the vector space V has a basis consisting of n elements.
Then every basis of V also has precisely n elements.
Proof. This follows directly from theorem 11, since any basis generates V, which is
a subspace of itself.
Definition. The number of vectors in a basis of the vector space V is called the
dimension of V, written dim(V).
Definition. Let V be a vector space with subspaces X, Y ⊂ V. The subspace X+
Y = [X ∪ Y ] is called the sum of X and Y. If X∩ Y = ¦0¦, then it is the direct
sum, written X⊕Y.
Theorem 13 (A Dimension Formula). Let V be a finite dimensional vector space
with subspaces X, Y ⊂ V. Then we have
dim(X+Y) = dim(X) + dim(Y) −dim(X∩ Y).
Corollary. dim(X⊕Y) = dim(X) + dim(Y).
Proof of Theorem 13. Let S = ¦v
1
, . . . , v
n
¦ be a basis of X ∩ Y. According to
theorem 10, there exist extensions T = ¦x
1
, . . . , x
m
¦ and U = ¦y
1
, . . . , y
r
¦, such
that S ∪ T is a basis for X and S ∪ U is a basis for Y. We will now show that, in
fact, S ∪ T ∪ U is a basis for X+Y.
9
To begin with, it is clear that X+Y = [S ∪T ∪U]. Is the set S ∪T ∪U linearly
independent? Let
0 =
n
i=1
a
i
v
i
+
m
j=1
b
j
x
j
+
r
k=1
c
k
y
k
= v +x +y, say.
Then we have y = −v−x. Thus y ∈ X. But clearly we also have, y ∈ Y. Therefore
y ∈ X∩ Y. Thus y can be expressed as a linear combination of vectors in S alone,
and since S ∪ U is is a basis for Y , we must have c
k
= 0 for k = 1, . . . , r. Similarly,
looking at the vector x and applying the same argument, we conclude that all the b
j
are zero. But then all the a
i
must also be zero since the set S is linearly independent.
Putting this all together, we see that the dim(X) = n +m, dim(Y) = n +r and
dim(X∩ Y) = n. This gives the dimension formula.
Theorem 14. Let V be a finite dimensional vector space, and let X ⊂ V be a
subspace. Then there exists another subspace Y ⊂ V, such that V = X⊕Y.
Proof. Take a basis S of X. If [S] = V then we are finished. Otherwise, use
the extension theorem (theorem 10) to find a basis B of V, with S ⊂ B. Then
4
Y = [B ¸ S] satisfies the condition of the theorem.
5 Linear Mappings
Definition. Let V and W be vector spaces, both over the field F. Let f : V → W
be a mapping from the vector space V to the vector space W. The mapping f is
called a linear mapping if
f(au + bv) = af(u) + bf(v)
for all a, b ∈ F and all u, v ∈ V.
By choosing a and b to be either 0 or 1, we immediately see that a linear mapping
always has both f(av) = af(v) and f(u + v) = f(u) + f(v), for all a ∈ F and for
all u and v ∈ V. Also it is obvious that f(0) = 0 always.
Definition. Let f : V → W be a linear mapping. The kernel of the mapping,
denoted by ker(f), is the set of vectors in V which are mapped by f into the zero
vector in W.
Theorem 15. If ker(f) = ¦0¦, that is, if the zero vector in V is the only vec-
tor which is mapped into the zero vector in W under f, then f is an injection
(monomorphism). The converse is of course trivial.
Proof. That is, we must show that if u and v are two vectors in V with the property
that f(u) = f(v), then we must have u = v. But
f(u) = f(v) ⇒ 0 = f(u) −f(v) = f(u −v).
4
The notation B ¸ S denotes the set of elements of B which are not in S
10
Thus the vector u −v is mapped by f to the zero vector. Therefore we must have
u −v = 0, or u = v.
Conversely, since f(0) = 0 always holds, and since f is an injection, we must
have ker(f) = ¦0¦.
Theorem 16. Let f : V → W be a linear mapping and let A = ¦w
1
, . . . , w
m
¦ ⊂ W
be linearly independent. Assume that m vectors are given in V, so that they form a
set B = ¦v
1
, . . . , v
m
¦ ⊂ V with f(v
i
) = w
i
, for all i. Then the set B is also linearly
independent.
Proof. Let a
1
, . . . , a
m
∈ F be given such that a
1
v
1
+ + a
m
v
m
= 0. But then
0 = f(0) = f(a
1
v
1
+ +a
m
v
m
) = a
1
f(v
1
) + +a
m
f(v
m
) = a
1
w
1
+ +a
m
w
m
.
Since A is linearly independent, it follows that all the a
i
's must be zero. But that
implies that the set B is linearly independent.
Remark. If B = ¦v
1
, . . . , v
m
¦ ⊂ V is linearly independent, and f : V → W is
linear, still, it does not necessarily follow that ¦f(v
1
), . . . , f(v
m
)¦ is linearly inde-
pendent in W. On the other hand, if f is an injection, then ¦f(v
1
), . . . , f(v
m
)¦ is
linearly independent. This follows since, if a
1
f(v
1
) + + a
m
f(v
m
) = 0, then we
have
0 = a
1
f(v
1
) + + a
m
f(v
m
) = f(a
1
v
1
+ + a
m
v
m
) = f(0).
But since f is an injection, we must have a
1
v
1
+ + a
m
v
m
= 0. Thus a
i
= 0 for
all i.
On the other hand, what is the condition for f : V → W to be a surjection
(epimorphism)? That is, f(V) = W. Or put another way, for every w ∈ W, can
we find some vector v ∈ V with f(v) = w? One way to think of this is to consider
a basis B ⊂ W. For each w ∈ B, we take
f
−1
(w) = ¦v ∈ V : f(v) = w¦.
Then f is a surjection if f
−1
(w) ,= ∅, for all w ∈ B.
Definition. A linear mapping which is a bijection (that is, an injection and a sur-
jection) is called an isomorphism. Often one writes V
∼
= W to say that there exists
an isomorphism from V to W.
Theorem 17. Let f : V → W be an isomorphism. Then the inverse mapping
f
−1
: W → V is also a linear mapping.
Proof. To see this, let a, b ∈ F and x, y ∈ W be arbitrary. Let f
−1
(x) = u ∈ V
and f
−1
(y) = v ∈ V, say. Then
f(au + bv) = (f(af
−1
(x) + bf
−1
(v)) = af(f
−1
(x)) +bf(f
−1
(v)) = ax + by.
Therefore, since f is a bijection, we must have
f
−1
(ax + by) = au + bv = af
−1
(x) + bf
−1
(y).
11
Theorem 18. Let V and W be finite dimensional vector spaces over a field F, and
let f : V → W be a linear mapping. Let B = ¦v
1
, . . . , v
n
¦ be a basis for V. Then
f is uniquely determined by the n vectors ¦f(v
1
), . . . , f(v
n
)¦ in W.
Proof. Let v ∈ V be an arbitrary vector in V. Since B is a basis for V, we can
uniquely write
v = a
1
v
1
+ a
n
v
n
,
with a
i
∈ F, for each i. Then, since the mapping f is linear, we have
f(v) = f(a
1
v
1
+ a
n
v
n
)
= f(a
1
v
1
) + + f(a
n
v
n
)
= a
1
f(v
1
) + + a
n
f(v
n
).
Therefore we see that if the values of f(v
1
),. . . , f(v
n
) are given, then the value of
f(v) is uniquely determined, for each v ∈ V.
On the other hand, let A = ¦u
1
, . . . , u
n
¦ be a set of n arbitrarily given vectors
in W. Then let a mapping f : V → W be defined by the rule
f(v) = a
1
u
1
+ a
n
u
n
,
for each arbitrarily given vector v ∈ V, where v = a
1
v
1
+ a
n
v
n
. Clearly the map-
ping is uniquely determined, since v is uniquely determined as a linear combination
of the basis vectors B. It is a trivial matter to verify that the mapping which is so
defined is also linear. We have f(v
i
) = u
i
for all the basis vectors v
i
∈ B.
Theorem 19. Let V and W be two finite dimensional vector spaces over a field F.
Then we have V
∼
= W ⇔ dim(V) = dim(W).
Proof. "⇒" Let f : V → W be an isomorphism, and let B = ¦v
1
, . . . , v
n
¦ ⊂ V be a
basis for V. Then, as shown in our Remark above, we have A = ¦f(v
1
), . . . , f(v
n
)¦ ⊂
W being linearly independent. Furthermore, since B is a basis of V, we have
[B] = V. Thus [A] = W also. Therefore A is a basis of W, and it contains precisely
n elements; thus dim(V) = dim(W).
"⇐" Take B = ¦v
1
, . . . , v
n
¦ ⊂ V to again be a basis of V and let A =
¦w
1
, . . . , w
n
¦ ⊂ W be some basis of W (with n elements). Now define the mapping
f : V → W by the rule f(v
i
) = w
i
, for all i. By theorem 18 we see that a linear
mapping f is thus uniquely determined. Since A and B are both bases, it follows
that f must be a bijection.
This immediately gives us a complete classification of all finite-dimensional vector
spaces. For let V be a vector space of dimension n over the field F. Then clearly
F
n
is also a vector space of dimension n over F. The canonical basis is the set of
vectors ¦e
1
, . . . , e
n
¦, where
e
i
= (0, , 0, 1
.¸¸.
i-th Position
, 0, . . . , 0¦
for each i. Therefore, when thinking about V, we can think that it is "really" just
F
n
. On the other hand, the central idea in the theory of linear algebra is that
12
we can look at things using different possible bases (or "frames of reference" in
physics). The space F
n
seems to have a preferred, fixed frame of reference, namely
the canonical basis. Thus it is better to think about an abstract V, with various
possible bases.
Examples
For these examples, we will consider the 2-dimensional real vector space R
2
, together
with its canonical basis B = ¦e
1
, e
2
¦ = ¦(1, 0), (0, 1)¦.
• f
1
: R
2
→ R
2
with f
1
(e
1
) = (−1, 0) and f
1
(e
2
) = (0, 1). This is a reflection of
the 2-dimensional plane into itself, with the axis of reflection being the second
coordinate axis; that is the set of points (x
1
, x
2
) ∈ R
2
with x
1
= 0.
• f
2
: R
2
→ R
2
with f
2
(e
1
) = e
2
and f
1
(e
2
) = e
1
. This is a reflection of the
2-dimensional plane into itself, with the axis of reflection being the diagonal
axis x
1
= x
2
.
• f
3
: R
2
→R
2
with f
3
(e
1
) = (cos φ, sin φ) and f
1
(e
2
) = (−sin φ, cos φ), for some
real number φ ∈ R. This is a rotation of the plane about its middle point,
through an angle of φ.
5
For let v = (x
1
, x
2
) be some arbitrary point of the
plane R
2
. Then we have
f
3
(v) = x
1
f
3
(e
1
) + x
2
f(e
2
)
= x
1
(cos φ, sin φ) + x
2
(−sin φ, cos φ)
= (x
1
cos φ −x
2
sin φ, x
1
sin φ + x
2
cos φ).
Looking at this from the point of view of geometry, the question is, what
happens to the vector v when it is rotated through the angle φ while preserving
its length? Perhaps the best way to look at this is to think about v in polar
coordinates. That is, given any two real numbers x
1
and x
2
then, assuming that
they are not both zero, we find two unique real numbers r ≥ 0 and θ ∈ [0, 2π),
such that
x
1
= r cos θ and x
2
= r sin θ,
where r =
_
x
2
1
+ x
2
2
. Then v = (r cos θ, r sin θ). So a rotation of v through
the angle φ must bring it to the new vector (r cos(φ + θ), r sin(φ + θ)) which,
if we remember the formulas for cosines and sines of sums, turns out to be
(r(cos(θ) cos(φ) −sin(θ) sin(φ)), r(sin(θ) cos(φ) −cos(θ) sin(φ)).
But then, remembering that x
1
= r cos θ and x
2
= r sin θ, we see that the
rotation brings the vector v into the new vector
(x
1
cos φ −x
2
sin φ, x
1
sin φ + x
2
cos φ),
5
In analysis, we learn about the formulas of trigonometry. In particular we have
cos(θ + φ) = cos(θ) cos(φ) −sin(θ) sin(φ),
sin(θ + φ) = sin(θ) cos(φ) −cos(θ) sin(φ).
Taking θ = π/2, we note that cos(φ + π/2) = −sin(φ) and sin(φ + π/2) = cos(φ).
13
which was precisely the specification for f
3
(v).
6 Linear Mappings and Matrices
This last example of a linear mapping of R
2
into itself — which should have been
simple to describe — has brought with it long lines of lists of coordinates which are
difficult to think about. In three and more dimensions, things become even worse!
Thus it is obvious that we need a more sensible system for describing these linear
mappings. The usual system is to use matrices.
Now, the most obvious problem with our previous notation for vectors was that
the lists of the coordinates (x
1
, , x
n
) run over the page, leaving hardly any room
left over to describe symbolically what we want to do with the vector. The solution
to this problem is to write vectors not as horizontal lists, but rather as vertical lists.
We say that the horizontal lists are row vectors, and the vertical lists are column
vectors. This is a great improvement! So whereas before, we wrote
v = (x
1
, , x
n
),
now we will write
v =
_
_
_
x
1
.
.
.
x
n
_
_
_
.
It is true that we use up lots of vertical space on the page in this way, but since
the rest of the writing is horizontal, we can afford to waste this vertical space. In
addition, we have a very nice system for writing down the coordinates of the vectors
after they have been mapped by a linear mapping.
To illustrate this system, consider the rotation of the plane through the angle φ,
which was described in the last section. In terms of row vectors, we have (x
1
, x
2
)
being rotated into the new vector (x
1
cos φ −x
2
sin φ, x
1
sin φ + x
2
cos φ). But if we
change into the column vector notation, we have
v =
_
x
1
x
2
_
being rotated to
_
x
1
cos φ −x
2
sin φ
x
1
sin φ + x
2
cos φ
_
.
But then, remembering how we multiplied matrices, we see that this is just
_
cos φ −sin φ
sin φ cos φ
__
x
1
x
2
_
=
_
x
1
cos φ −x
2
sin φ
x
1
sin φ + x
2
cos φ
_
.
So we can say that the 2 2 matrix A =
_
cos φ −sin φ
sin φ cos φ
_
represents the mapping
f
3
: R
2
→R
2
, and the 2 1 matrix
_
x
1
x
2
_
represents the vector v. Thus we have
A v = f(v).
That is, matrix multiplication gives the result of the linear mapping.
14
Expressing f : V → W in terms of bases for both V and W
The example we have been thinking about up till now (a rotation of R
2
) is a linear
mapping of R
2
into itself. More generally, we have linear mappings from a vector
space V to a different vector space W (although, of course, both V and W are
vector spaces over the same field F).
So let ¦v
1
, . . . , v
n
¦ be a basis for V and let ¦w
1
, . . . , w
m
¦ be a basis for W.
Finally, let f : V → W be a linear mapping. An arbitrary vector v ∈ V can be
expressed in terms of the basis for V as
v = a
1
v
1
+ + a
n
v
n
=
n
j=1
a
j
v
j
.
The question is now, what is f(v)? As we have seen, f(v) can be expressed in terms
of the images f(v
j
) of the basis vectors of V. Namely
f(v) =
n
j=1
a
j
f(v
j
).
But then, each of these vectors f(v
j
) in W can be expressed in terms of the basis
vectors in W, say
f(v
j
) =
m
k=1
a
j
c
ij
d
ki
x
k
.
There are so many summations here! How can we keep track of everything? The
answer is to use the matrix notation. The composition of linear mappings is then
simply represented by matrix multiplication. That is, if
v =
_
_
_
a
1
.
.
.
a
n
_
_
_
,
then we have
f ◦ g(v) = g(f(v)) =
_
_
_
d
11
d
1m
.
.
.
.
.
.
.
.
.
d
r1
d
rm
_
_
_
_
_
_
c
11
c
1n
.
.
.
.
.
.
.
.
.
c
m1
c
mn
_
_
_
_
_
_
a
1
.
.
.
a
n
_
_
_
= BAv.
So this is the reason we have defined matrix multiplication in this way.
6
7 Matrix Transformations
Matrices are used to describe linear mappings f : V → W with respect to particular
bases of V and W. But clearly, if we choose different bases than the ones we had
been thinking about before, then we will have a different matrix for describing the
same linear mapping. Later on in these lectures we will see how changing the bases
changes the matrix, but for now, it is time to think about various systematic ways
of changing matrices — in a purely abstract way.
6
Recall that if A =
_
_
_
c
11
c
1n
.
.
.
.
.
.
.
.
.
c
m1
c
mn
_
_
_ is an m n matrix and B =
_
_
_
d
11
d
1m
.
.
.
.
.
.
.
.
.
d
r1
d
rm
_
_
_ is an
r m matrix, then the product BA is an r n matrix whose kj-th element is
m
i=1
d
ki
c
ij
.
17
Elementary Column Operations
We begin with the elementary column operations. Let us denote the set of all nm
matrices of elements from the field F by M(mn, F). Thus∈ M(mn, F)
then it contains n columns which, as we have seen, are the images of the basis
vectors of the linear mapping which is being represented by the matrix. So The first
elementary column operation is to exchange column i with column j, for i ,= j. We
can write
_
_
_
a
11
a
1i
a
1j
a
1m
.
.
.
.
.
.
.
.
.
.
.
.
a
m1
a
mi
a
mj
a
mm
_
_
_
S
ij
−→
_
_
_
a
11
a
1j
a
1i
a
1m
.
.
.
.
.
.
.
.
.
.
.
.
a
m1
a
mj
a
mi
a
mm
_
_
_
So this column operation is denoted by S
ij
. It can be thought of as being a mapping
S
ij
: M(mn, F) → M(mn, F).
Another way to imagine this is to say that S is the set of column vectors in the
matrix A considered as an ordered list. Thus S ⊂ F
m
. Then S
ij
is the same set of
column vectors, but with the positions of the i-th and the j-th vectors interchanged.
But obviously, as a subset of F
n
, the order of the vectors makes no difference.
Therefore we can say that the span of S is the same as the span of S
ij
. That is
[S] = [S
ij
].
The second elementary column operation, denoted S
i
(a), is that we form the
scalar product of the element a ,= 0 in F with the i-th vector in S. So the i-th
vector
_
_
_
a
1i
.
.
.
a
mi
_
_
_
is changed to
a
_
_
_
a
1i
.
.
.
a
mi
_
_
_
=
_
_
_
aa
1i
.
.
.
aa
mi
_
_
_
.
All the other column vectors in the matrix remain unchanged.
The third elementary column operation, denoted S
ij
(c) is that we take the j-th
column (where j ,= i) and multiply it with c ,= 0, then add it to the i-th column.
Therefore the i-th column is changed to
_
_
_
a
1i
+ ca
1j
.
.
.
a
mi
+ ca
mj
_
_
_
.
All the other columns — including the j-th column — remain unchanged.
18
Theorem 20. [S] = [S
ij
] = [S
i
(a)] = [S
ij
(c)], where i ,= j and a ,= 0 ,= c.
Proof. Let us say that S = ¦v
1
, . . . , v
n
¦ ⊂ F
m
. That is, v
i
is the i-th column vector
of the matrix A, for each i. We have already seen that [S] = [S
ij
] is trivially true.
But also, say v = x
1
v
1
+ + x
n
v
n
is some arbitrary vector in [S]. Then, since
a ,= 0, we can write
v = x
1
v
1
+ + a
−1
x
i
(av
i
) + + x
n
v
n
.
Therefore [S] ⊂ [S
i
(a)]. The other inclusion, [S
i
(a)] ⊂ [S] is also quite trivial so
that we have [S] = [S
i
(a)].
Similarly we can write
v = x
1
v
1
+ + x
n
v
n
= x
1
v
1
+ + x
i
(v
i
+ cv
j
) + + (x
j
−x
i
c)v
j
+ + x
n
v
n
.
Therefore [S] ⊂ [S
ij
(c)], and again, the other inclusion is similar.
Let us call [S] the column space (Spaltenraum), which is a subspace of F
m
.
Then we see that the column space remains invariant under the three types of
elementary column operations. In particular, the dimension of the column space
remains invariant.
Elementary Row Operations
Again, looking at the mn matrix A in a purely abstract way, we can say that it is
made up of m row vectors, which are just the rows of the matrix. Let us call them
w
1
, . . . , w
m
∈ F
n
. That is,=
_
_
_
w
1
= (a
11
a
1n
)
.
.
.
w
m
= (a
m1
a
mn
)
_
_
_
.
Again, we define the three elementary row operations analogously to the way we
defined the elementary column operations. Clearly we have the same results. Namely
if R = ¦w
1
, . . . , w
m
¦ are the original rows, in their proper order, then we have
[R] = [R
ij
] = [R
i
(a)] = [R
ij
(c)].
But it is perhaps easier to think about the row operations when changing a
matrix into a form which is easier to think about. We would like to change the
matrix into a step form (Zeilenstufenform).
Definition. The m n matrix A is in step form if there exists some r with 0 ≤
r ≤ m and indices 1 ≤ j
1
< j
2
< < j
r
≤ m with a
ij
i
= 1 for all i = 1, . . . , r and
a
st
= 0 for all s, t with t < j
s
or s > j
r
. That is:
A =
_
_
_
_
_
_
_
_
_
_
_
1 a
1j
1
+1
a
1n
0 1 a
2j
2
+1
a
2n
0 0 1 a
3j
3
+1
a
2n
.
.
.
.
.
.
0 0 1 a
rjr+1
a
rn
0 0 0 0 0
.
.
.
.
.
.
.
.
.
_
_
_
_
_
_
_
_
_
_
_
.
19
Theorem 21. By means of a finite sequence of elementary row operations, every
matrix can be transformed into a matrix in step form.
Proof. Induction on m, the number of rows in the matrix. We use the technique
of "Gaussian elimination", which is simply the usual way anyone would go about
solving a system of linear equations. This will be dealt with in the next section.
The induction step in this proof, which uses a number of simple ideas which are
easy to write on the blackboard, but overly tedious to compose here in T
E
X, will be
described in the lecture.
Now it is obvious that the row space (Zeilenraum), that is [R] ⊂ F
n
, has the
dimension r, and in fact the non-zero row vectors of a matrix in step form provide
us with a basis for the row space. But then, looking at the column vectors of this
matrix in step form, we see that the columns j
1
, j
2
, and so on up to j
r
are all linearly
independent, and they generate the column space. (This is discussed more fully in
the lecture!)
Definition. Given an mn matrix, the dimension of the column space is called the
column rank; similarly the dimension of the row space is the row rank.
So, using theorem 21 and exercise 6.3, we conclude that:
Theorem 22. For any matrix A, the column rank is equal to the row rank. This
common dimension is simply called the rank — written Rank(A) — of the matrix.
Definition. Let A be a quadratic nn matrix. Then A is called regular if Rank(A) =
n, otherwise A is called singular.
Theorem 23. The n n matrix A is regular ⇔ the linear mapping f : F
n
→
F
n
, represented by the matrix A with respect to the canonical basis of F
n
is an
isomorphism.
Proof. '⇒' If A is regular, then the rank of A — namely the dimension of the column
space [S] — is n. Since the dimension of F
n
is n, we must therefore have [S] = F
n
.
The linear mapping f : F
n
→ F
n
is then both an injection (since S must be linearly
independent) and also a surjection.
'⇐' Since the set of column vectors S is the set of images of the canonical basis
vectors of F
n
under f, they must be linearly independent. There are n column
vectors; thus the rank of A is n.
8 Systems of Linear Equations
We now take a small diversion from our idea of linear algebra as being a method
of describing geometry, and instead we will consider simple linear equations. In
particular, we consider a system of m equations in n unknowns.
a
11
x
1
+ + a
1n
x
n
= b
1
.
.
.
a
m1
x
1
+ + a
mn
x
n
= b
m
20
We can also think about this as being a vector equation. That is,
and x =
_
_
_
x
1
.
.
.
x
n
_
_
_
∈ F
n
and b =
_
_
_
b
1
.
.
.
b
m
_
_
_
∈ F
m
, then our system of linear equations is
just the single vector equation
A x = b.
But what is the most obvious way to solve this system of equations? It is a
simple matter to write down an algorithm, as follows. The numbers a
ij
and b
k
are
given (as elements of F), and the problem is to find the numbers x
l
.
1. Let i := 1 and j := 1.
2. if a
ij
= 0 then if a
kj
= 0 for all i < k ≤ m, set j := j + 1. Otherwise find the
smallest index k > i such that a
kj
,= 0 and exchange the i-th equation with
the k-th equation.
3. Multiply both sides of the (possibly new) i-th equation by a
−1
ij
. Then for
each i < k ≤ m, subtract a
kj
times the i-th equation from the k-th equation.
Therefore, at this stage, after this operation has been carried out, we will have
a
kj
= 0, for all k > i.
4. Set i := i + 1. If i ≤ n then return to step 2.
So at this stage, we have transformed the system of linear equations into a system
in step form.
The next thing is to solve the system of equations in step form. The problem is
that perhaps there is no solution, or perhaps there are many solutions. The easiest
way to decide which case we have is to reorder the variables — that is the various
x
i
— so that the steps start in the upper left-hand corner, and they are all one unit
wide. That is, things then look like this:
x
1
+ a
12
x
2
+ a
13
x
3
+ + + a
1n
x
n
= b
1
x
2
+ a
23
x
3
+ a
24
x
4
+ + a
2n
x
n
= b
2
x
3
+ a
34
x
4
+ + a
3n
x
n
= b
3
.
.
.
x
k
+ + a
kn
x
k
= b
k
0 = b
k+1
.
.
.
0 = b
m
(Note that this reordering of the variables is like our first elementary column oper-
ation for matrices.)
So now we observe that:
21
• If b
l
,= 0 for some k +1 ≤ l ≤ m, then the system of equations has no solution.
• Otherwise, if k = n then the system has precisely one single solution. It
is obtained by working backwards through the equations. Namely, the last
equation is simply x
n
= b
n
, so that is clear. But then, substitute b
n
for x
n
in the n − 1-st equation, and we then have x
n−1
= b
n−1
− a
n−1 n
b
n
. By this
method, we progress back to the first equation and obtain values for all the
x
j
, for 1 ≤ j ≤ n.
• Otherwise, k < n. In this case we can assign arbitrary values to the variables
x
k+1
, . . . , x
n
, and then that fixes the value of x
k
. But then, as before, we
progressively obtain the values of x
k−1
, x
k−2
and so on, back to x
1
.
This algorithm for finding solutions of systems of linear equations is called "Gaussian
Elimination".
All of this can be looked at in terms of our matrix notation. Let us call the
following mn+1 matrix the augmented matrix for our system of linear equations:
A =
_
_
_
_
_
a
11
a
12
a
1n
b
1
a
21
a
22
a
2n
b
2
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
a
m1
a
m2
a
mn
b
m
_
_
_
_
_
.
Then by means of elementary row and column operations, the matrix is transformed
into the new matrix which is in simple step form
A
′
=
_
_
_
_
_
_
_
_
_
_
_
1 a
′
12
a
′
1 k+1
a
′
1n
b
′
1
0 1 a
′
23
a
′
2 k+1
a
′
2n
b
′
2
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0 0 1 a
′
k k+1
a
′
kn
b
′
k
0 0 0 0 0 b
′
k+1
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0 0 0 0 0 b
′
m
_
_
_
_
_
_
_
_
_
_
_
.
Finding the eigenvectors of linear mappings
Definition. Let V be a vector space over a field F, and let f : V → V be a linear
mapping of V into itself. An eigenvector of f is a non-zero vector v ∈ V (so we
have v ,= 0) such that there exists some λ ∈ F with f(v) = λv. The scalar λ is then
called the eigenvalue associated with this eigenvector.
So if f is represented by the nn matrix A (with respect to some given basis of
V), then the problem of finding eigenvectors and eigenvalues is simply the problem
of solving the equation
Av = λv.
But here both λ and v are variables. So how should we go about things? Well, as
we will see, it is necessary to look at the characteristic polynomial of the matrix, in
22
order to find an eigenvalue λ. Then, once an eigenvalue is found, we can consider it to
be a constant in our system of linear equations. And they become the homogeneous
7
system
(a
11
−λ)x
1
+ a
12
x
2
+ + a
1n
x
n
= 0
a
21
x
1
+ (a
22
−λ)x
2
+ + a
2n
x
n
= 0
.
.
.
a
n1
x
1
+ a
n2
x
2
+ + (a
nn
−λ)x
n
= 0
which can be easily solved to give us the (or one of the) eigenvector(s) whose eigen-
value is λ.
Now the n n identity matrix is
E =
_
_
_
_
_
1 0 0
0 1 0
.
.
.
.
.
.
.
.
.
.
.
.
0 0 1
_
_
_
_
_
Thus we see that an eigenvalue is any scalar λ ∈ F such that the vector equation
(A−λE)v = 0
has a solution vector v ∈ V, such that v ,= 0.
8
9 Invertible Matrices
Let f : V → W be a linear mapping, and let ¦v
1
, . . . , v
n
¦ ⊂ V and ¦w
1
, . . . , w
m
¦ ⊂
W be bases for V and W, respectively. Then, as we have seen, the mapping f can
be uniquely described by specifying the values of f(v
j
), for each j = 1, . . . , n. We
have
f(v
j
) =
m
i=1
a
ij
w
i
,
And the resulting matrix A =
_
_
_
a
11
a
1n
.
.
.
.
.
.
.
.
.
a
m1
a
mn
_
_
_ is the matrix describing f with
respect to these given bases.
A particular case
This is the case that V = W. So we have the linear mapping f : V → V. But now,
we only need a single basis for V. That is, ¦v
1
, . . . , v
n
¦ ⊂ V is the only basis we
7
That is, all the b
i
are zero. Thus a homogeneous system with matrix A has the form Av = 0.
8
Given any solution vector v, then clearly we can multiply it with any scalar κ ∈ F, and we
have
(A −λE)(κv) = κ(A −λE)v = κ0 = 0.
Therefore, as long as κ ,= 0, we can say that κv is also an eigenvector whose eigenvalue is λ.
23
need. Thus the matrix for f with respect to this single basis is determined by the
specifications
f(v
j
) =
m
i=1
a
ij
v
i
.
A trivial example
For example, one particular case is that we have the identity mapping
f = id : V → V.
Thus f(v) = v, for all v ∈ V. In this case it is obvious that the matrix of the
mapping is the n n identity matrix I
n
.
Regular matrices
Let us now assume that A is some regular n n matrix. As we have seen in theo-
rem 23, there is an isomorphism f : V → V, such that A is the matrix representing
f with respect to the given basis of V. According to theorem 17, the inverse map-
ping f
−1
is also linear, and we have f
−1
◦ f = id. So let f
−1
be represented by the
matrix B (again with respect to the same basis ¦v
1
, . . . , v
n
¦). Then we must have
the matrix equation
B A = I
n
.
Or, put another way, in the multiplication system of matrix algebra we must have
B = A
−1
. That is, the matrix A is invertible.
Theorem 24. Every regular matrix is invertible.
Definition. The set of all regular nn matrices over the field F is denoted GL(n, F).
Theorem 25. GL(n, F) is a group under matrix multiplication. The identity ele-
ment is the identity matrix.
Proof. We have already seen in an exercise that matrix multiplication is associative.
The fact that the identity element in GL(n, F) is the identity matrix is clear. By
definition, all members of GL(n, F) have an inverse. It only remains to see that
GL(n, F) is closed under matrix multiplication. So let A, C ∈ GL(n, F). Then
there exist A
−1
, C
−1
∈ GL(n, F), and we have that C
−1
A
−1
is itself an n n
matrix. But then
_
C
−1
A
−1
_
AC = C
−1
_
A
−1
A
_
C = C
−1
I
n
C = C
−1
C = I
n
.
Therefore, according to the definition of GL(n, F), we must also have AC ∈ GL(n, F).
24
Simplifying matrices using multiplication with regular matri-
ces
Theorem 26. Let A be an m n matrix. Then there exist regular matrices C ∈
GL(m, F) and D ∈ GL(n, F) such that the matrix A
′
= CAD
−1
consists simply of
zeros, except possibly for a block in the upper lefthand corner, which is an identity
matrix. That is
A
′
=
_
_
_
_
_
_
_
_
_
1 0
.
.
.
.
.
.
.
.
.
0 1
0 0
.
.
.
.
.
.
.
.
.
0 0
0 0
.
.
.
.
.
.
.
.
.
0 0
0 0
.
.
.
.
.
.
.
.
.
0 0
_
_
_
_
_
_
_
_
_
(Note that A
′
is also an mn matrix. That is, it is not necessarily square.)
Proof. A is the representation of a linear mapping f : V → W, with respect to
bases ¦v
1
, . . . , v
n
¦ and ¦w
1
, . . . , w
m
¦ of V and W, respectively. The idea of the
proof is to now find new bases ¦x
1
, . . . , x
n
¦ ⊂ V and ¦y
1
, . . . , y
m
¦ ⊂ W, such that
the matrix of f with respect to these new bases is as simple as possible.
So to begin with, let us look at ker(f) ⊂ V. It is a subspace of V, so its dimension
is at most n. In general, it might be less than n, so let us write dim(ker(f)) = n−p,
for some integer 0 ≤ p ≤ n. Therefore we choose a basis for ker(f), and we call it
¦x
p+1
, . . . , x
n
¦ ⊂ ker(f) ⊂ V.
Using the extension theorem (theorem 12), we extend this to a basis
¦x
1
, . . . , x
p
, x
p+1
, . . . , x
n
¦
for V.
Now at this stage, we look at the images of the vectors ¦x
1
, . . . , x
p
¦ under f in
W. We find that the set ¦f(x
1
), . . . , f(x
p
)¦ ⊂ W is linearly independent. To see
this, let us assume that we have the vector equation
0 =
p
i=1
a
i
f(x
i
) = f
_
p
i=1
a
i
x
i
_
for some choice of the scalars a
i
. But that means that
j=p+1
b
j
x
j
for appropriate choices of scalars b
j
. But ¦x
1
, . . . , x
p
, x
p+1
, . . . , x
n
¦ is a basis for V.
Thus it is itself linearly independent and therefore we must have a
i
= 0 and b
j
= 0
for all possible i and j. In particular, since the a
i
's are all zero, we must have the
set ¦f(x
1
), . . . , f(x
p
)¦ ⊂ W being linearly independent.
25
To simplify the notation, let us call f(x
i
) = y
i
, for each i = 1, . . . , p. Then we
can again use the extension theorem to find a basis
¦y
1
, . . . , y
p
, y
p+1
, . . . , y
m
¦
of W.
So now we define the isomorphism g : V → V by the rule
g(x
i
) = v
i
, for all i = 1, . . . , n.
Similarly the isomorphism h : W → W is defined by the rule
h(y
j
) = w
j
, for all j = 1, . . . , m.
Let D be the matrix representing the mapping g with respect to the basis ¦v
1
, . . . , v
n
¦
of V, and also let C be the matrix representing the mapping h with respect to the
basis ¦w
1
, . . . , w
m
¦ of W.
Let us now look at the mapping
h f g
−1
: V → W.
For the basis vector v
i
∈ V, we have
hfg
−1
(v
i
) = hf(x
i
) =
_
h(y
i
) = w
i
, for i ≤ p
h(0) = 0, otherwise.
This mapping must therefore be represented by a matrix in our simple form, consist-
ing of only zeros, except possibly for a block in the upper lefthand corner which is
an identity matrix. Furthermore, the rule that the composition of linear mappings
is represented by the product of the respective matrices leads to the conclusion that
the matrix A
′
= CAD
−1
must be of the desired form.
10 Similar Matrices; Changing Bases
Definition. Let A and A
′
be nn matrices. If a matrix C ∈ GL(n, F) exists, such
that A
′
= C
−1
AC then we say that the matrices A and A
′
are similar.
Theorem 27. Let f : V → V be a linear mapping and let ¦u
1
, . . . , u
n
¦, ¦v
1
, . . . , v
n
¦
be two bases for V. Assume that A is the matrix for f with respect to the ba-
sis ¦v
1
, . . . , v
n
¦ and furthermore A
′
is the matrix for f with respect to the basis
¦u
1
, . . . , u
n
¦. Let u
i
=
n
j=1
c
ji
v
j
we see that it represents a mapping g : V → V such that g(v
i
) = u
i
for all i, ex-
pressed in terms of the original basis ¦v
1
, . . . , v
n
¦. So we see that a similarity
27
transformation, taking a square matrix A to a similar matrix A
′
= C
−1
AC is always
associated with a change of basis for the vector space V .
Much of the theory of linear algebra is concerned with finding a simple basis
(with respect to a given linear mapping of the vector space into itself), such that
the matrix of the mapping with respect to this simpler basis is itself simple — for
example diagonal, or at least trigonal.
11 Eigenvalues, Eigenspaces, Matrices which can
be Diagonalized
Definition. Let f : V → V be a linear mapping of an n-dimensional vector space
into itself. A subspace U ⊂ V is called invariant with respect to f if f(U) ⊂ U.
That is, f(u) ∈ U for all u ∈ U.
Theorem 28. Assume that the r dimensional subspace U ⊂ V is invariant with
respect to f : V → V. Let A be the matrix representing f with respect to a given
basis ¦v
1
, . . . , v
n
¦ of V. Then A is similar to a matrix A
′
which has the following
form
A
′
=
_
_
_
_
_
_
_
_
_
_
a
′
11
. . . a
′
1r
.
.
.
.
.
.
a
′
r1
. . . a
′
rr
a
′
1(r+1)
. . . a
′
1n
.
.
.
.
.
.
a
′
r(r+1)
. . . a
′
rn
0
a
′
(r+1)(r+1)
. . . a
′
(r+1)n
.
.
.
.
.
.
a
′
n(r+1)
. . . a
′
nn
_
_
_
_
_
_
_
_
_
_
Proof. Let ¦u
1
, . . . , u
r
¦ be a basis for the subspace U. Then extend this to a basis
¦u
1
, . . . , u
r
, u
r+1
, . . . , u
n
¦ of V. The matrix of f with respect to this new basis has
the desired form.
Definition. Let U
1
, . . . , U
p
⊂ V be subspaces. We say that V is the direct sum of
these subspaces if V = U
1
+ + U
p
, and furthermore if v = u
1
+ + u
p
such
that u
i
∈ U
i
, for each i, then this expression for v is unique. In other words, if
v = u
1
+ +u
p
= u
′
1
+ +u
′
p
with u
′
i
∈ U
i
for each i, then u
i
= u
′
i
, for each i.
In this case, one writes V = U
1
⊕ ⊕U
p
This immediately gives the following result:
Theorem 29. Let f : V → V be such that there exist subspaces U
i
⊂ V, for
i = 1, . . . , p, such that V = U
1
⊕ ⊕ U
p
and also f is invariant with respect to
each U
i
. Then there exists a basis of V such that the matrix of f with respect to
this basis has the following block form.28
where each block A
i
is a square matrix, representing the restriction of f to the
subspace U
i
.
Proof. Choose the basis to be a union of bases for each of the U
i
.
A special case is when the invariant subspace is an eigenspace.
Definition. Assume that λ ∈ F is an eigenvalue of the mapping f : V → V. The
set ¦v ∈ V : f(v) = λv¦ is called the eigenspace of λ with respect to the mapping
f. That is, the eigenspace is the set of all eigenvectors (and with the zero vector 0
included) with eigenvalue λ.
Theorem 30. Each eigenspace is a subspace of V.
Proof. Let u, w ∈ V be in the eigenspace of λ. Let a, b ∈ F be arbitrary scalars.
Then we have
f(au + bw) = af(u) + bf(w) = aλu + bλw = λ(au + bw).
Obviously if λ
1
and λ
2
are two different (λ
1
,= λ
2
) eigenvalues, then the only
common element of the eigenspaces is the zero vector 0. Thus if every vector in V is
an eigenvector, then we have the situation of theorem 29. One very particular case
is that we have n different eigenvalues, where n is the dimension of V.
Theorem 31. Let λ
1
, . . . , λ
n
be eigenvalues of the linear mapping f : V → V,
where λ
i
,= λ
j
for i ,= j. Let v
1
, . . . , v
n
be eigenvectors to these eigenvalues. That
is, v
i
,= 0 and f(v
i
) = λ
i
v
i
, for each i = 1, . . . , n. Then the set ¦v
1
, . . . , v
n
¦ is
linearly independent.
Proof. Assume to the contrary that there exist a
1
, . . . , a
n
, not all zero, with
a
1
v
1
+ + a
n
v
n
= 0.
Assume further that as few of the a
i
as possible are non-zero. Let a
p
be the first
non-zero scalar. That is, a
i
= 0 for i < p, and a
p
,= 0. Obviously some other a
k
is
non-zero, for some k ,= p, for otherwise we would have the equation 0 = a
p
v
p
, which
would imply that v
p
= 0, contrary to the assumption that v
p
is an eigenvector.
Therefore we have
0 = f(0) = f
_
n
i=1
a
i
v
i
_
=
n
i=1
a
i
f(v
i
) =
n
i=1
a
i
λ
i
v
i
.
Also
0 = λ
p
0 = λ
p
_
n
i=1
a
i
v
i
_
.
Therefore
0 = 0 −0 = λ
p
_
n
i=1
a
i
v
i
_
−
n
i=1
a
i
λ
i
v
i
=
n
i=1
a
i
(λ
p
−λ
i
)v
i
.
29
But, remembering that λ
i
,= λ
j
for i ,= j, we see that the scalar term for v
p
is zero,
yet all other non-zero scalar terms remain non-zero. Thus we have found a new
sum with fewer non-zero scalars than in the original sum with the a
i
s. This is a
contradiction.
Therefore, in this particular case, the given set of eigenvectors ¦v
1
, . . . , v
n
¦ form
a basis for V. With respect to this basis, the matrix of the mapping is diagonal,
with the diagonal elements being the eigenvalues12 The Elementary Matrices
These are n n matrices which we denote by S
ij
, S
i
(a), and S
ij
(c). They are such
that when any n n matrix A is multiplied on the right by such an S, then the
given elementary column operation is performed on the matrix A. Furthermore, if
the matrix A is multiplied on the left by such an elementary matrix, then the given
row operation on the matrix is performed. It is a simple matter to verify that the
following matrices are the ones we are looking for.
S
ij
=
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
1
.
.
. 0
1
0
i−th row
−→ 1
1
↓
.
.
. ↑
1
1 ←−
j−th row
0
1
0
.
.
.
1
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
Here, everything is zero except for the two elements at the positions ij and ji, which
have the value 1. Also the diagonal elements are all 1 except for the elements at ii
and jj, which are zero.
Then we have
S
i
(a) =
_
_
_
_
_
_
_
_
_
_
_
1
.
.
. 0
1
a
1
0
.
.
.
1
_
_
_
_
_
_
_
_
_
_
_
30
That is, S
i
(a) is a diagonal matrix, all of whose diagonal elements are 1 except for
the single element at the position ii, which has the value a.
Finally we have
S
ij
(c) =
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
1 0
.
.
.
1
i−th row
−→ c
1
.
.
. ↑
j−th column
1
1
0
.
.
.
1
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
So this is again just the n n identity matrix, but this time we have replaced the
zero in the ij-th position with the scalar c. It is an elementary exercise to see that:
Theorem 32. Each of the n n elementary matrices are regular.
And thus we can prove that these elementary matrices generate the group GL(n, F).
Furthermore, for every elementary matrix, the inverse matrix is again elementary.
Theorem 33. Every matrix in GL(n, F) can be represented as a product of elemen-
tary matrices.
Proof∈ GL(n, F) be some arbitrary regular matrix. We
have already seen that A can be transformed into a matrix in step form by means of
elementary row operations. That is, there is some sequence of elementary matrices:
S
1
, . . . , S
p
, such that the product
A
∗
= S
p
S
1
A
is an nn matrix in step form. However, since A was a regular matrix, the number
of steps must be equal to n. That is, A
∗
must be a triangular matrix whose diagonal
elements are all equal to 1.
A
∗
=
_
_
_
_
_
_
_
_
_
_
_
1 a
∗
12
a
∗
13
⋆ ⋆ a
∗
1n
0 1 a
∗
23
⋆ ⋆ a
∗
2n
0 0 1 ⋆ ⋆ a
∗
3n
.
.
.
.
.
.
.
.
. ⋆ ⋆
.
.
.
0 0 1 a
∗
(n−2)(n−1)
a
∗
(n−2)n
0 0 1 a
∗
(n−1)n
0 0 1
_
_
_
_
_
_
_
_
_
_
_
But now it is obvious that the elements above the diagonal can all be reduced to
zero by elementary row operations of type S
ij
(c). These row operations can again
31
be realized by multiplication of A
∗
on the right by some further set of elementary
matrices: S
p+1
, . . . , S
q
. This gives us the matrix equation
S
q
S
p+1
S
p
S
1
A = I
n
or
A = S
−1
1
S
−1
p
S
−1
p+1
S
−1
q
.
Since the inverse of each elementary matrix is itself elementary, we have thus ex-
pressed A as a product of elementary matrices.
This proof also shows how we can go about programming a computer to cal-
culate the inverse of an invertible matrix. Namely, through the process of Gauss
elimination, we convert the given matrix into the identity matrix I
n
. During this
process, we keep multiplying together the elementary matrices which represent the
respective row operations. In the end, we obtain the inverse matrix
A
−1
= S
q
S
p+1
S
p
S
1
.
We also note that this is the method which can be used to obtain the value of the
determinant function for the matrix. But first we must find out what the definition
of determinants of matrices is!
13 The Determinant
Let M(n n, F) be the set of all n n matrices of elements of the field F.
Definition. A mapping det : M(n n, F) → F is called a determinant function if
it satisfies the following three conditions.
1. det(I
n
) = 1, where I
n
is the identity matrix.
2. If A ∈ M(nn, F) is changed to the matrix A
′
by multiplying all the elements
in a single row with the scalar a ∈ F, then det(A
′
) = a det(A). (This is our
row operation S
i
(a).)
3. If A
′
is obtained from A by adding one row to a different row, then det(A
′
) =
det(A). (This is our row operation S
ij
(1).)
Simple consequences of this definition
Let A ∈ M(n n, F) be an arbitrary n n matrix, and let us say that A is
transformed into the new matrix A
′
by an elementary row operation. Then we have:
• If A
′
is obtained by multiplying row i by the scalar a ∈ F, then det(A
′
) =
a det(A). This is completely obvious! It is just part of the definition of
"determinants".
• Therefore, if A
′
is obtained from A by multiplying a row with −1 then we have
det(A
′
) = −det(A).
32
• Also, it follows that a matrix containing a row consisting of zeros must have
zero as its determinant.
• If A has two identical rows, then its determinant must also be zero. For can
we multiply one of these rows with -1, then add it to the other row, obtaining
a matrix with a zero row.
• If A
′
is obtained by exchanging rows i and j, then det(A
′
) = −det(A). This
is a bit more difficult to see. Let us say that A = (u
1
, . . . , u
i
, . . . , u
j
, . . . u
n
),
where u
k
is the k-th row of the matrix, for each k. Then we can write
det(A) = det(u
1
, . . . , u
i
, . . . , u
j
, . . . u
n
)
= det(u
1
, . . . , u
i
+u
j
,−(u
i
+u
j
), . . . , u
n
)
= det(u
1
, . . . , u
i
+u
j
, . . . , −u
i
, . . . , u
n
)
= det(u
1
, . . . , (u
i
+u
j
) −u
i
, . . . , −u
i
, . . . , u
n
)
= det(u
1
, . . . , u
j
, . . . , −u
i
, . . . , u
n
)
= −det(u
1
, . . . , u
j
, . . . , u
i
, . . . , u
n
)
(This is the elementary row operation S
ij
.)
• If A
′
is obtained from A by an elementary row operation of the form S
ij
(c),
then det(A
′
) = det(A). For we have:
det(A) = det(u
1
, . . . , u
i
, . . . , u
j
, . . . , u
n
)
= c
−1
det(u
1
, . . . , u
i
, . . . , cu
j
, . . . , u
n
)
= c
−1
det(u
1
, . . . , u
i
+ cu
j
, . . . , cu
j
, . . . , u
n
)
= det(u
1
, . . . , u
i
+ cu
j
, . . . , u
j
, . . . , u
n
)
Therefore we see that each elementary row operation has a well-defined effect on
the determinant of the matrix. This gives us the following algorithm for calculating
the determinant of an arbitrary matrix in M(n n, F).
How to find the determinant of a matrix
Given: An arbitrary matrix A ∈ M(n n, F).
Find: det(A).
Method:
1. Using elementary row operations, transform A into a matrix in step form,
keeping track of the changes in the determinant at each stage.
33
2. If the bottom line of the matrix we obtain only consists of zeros, then the
determinant is zero, and thus the determinant of the original matrix was zero.
3. Otherwise, the matrix has been transformed into an upper triangular matrix,
all of whose diagonal elements are 1. But now we can transform this matrix
into the identity matrix I
n
by elementary row operations of the type S
ij
(c).
Since we know that det(I
n
) must be 1, we then find a unique value for the
determinant of the original matrix A. In particular, in this case det(A) ,= 0.
Note that in both this algorithm, as well as in the algorithm for finding the
inverse of a regular matrix, the method of Gaussian elimination was used. Thus we
can combine both ideas into a single algorithm, suitable for practical calculations in
a computer, which yields both the matrix inverse (if it exists), and the determinant.
This algorithm also proves the following theorem.
Theorem 34. There is only one determinant function and it is uniquely given by
our algorithm. Furthermore, a matrix A ∈ M(n n, F) is regular if and only if
det(A) ,= 0.
In particular, using these methods it is easy to see that the following theorem is
true.
Theorem 35. Let A, B ∈ M(nn, F). Then we have det(A B) = det(A) det(B).
Proof. If either A or B is singular, then A B is singular. This can be seen by
thinking about the linear mappings V → V which A and B represent. At least one
of these mappings is singular. Thus the dimension of the image is less than n, so
the dimension of the image of the composition of the two mappings must also be
less than n. Therefore A B must be singular. That means, on the one hand, that
det(A B) = 0. And on the other hand, that either det(A) = 0 or else det(B) = 0.
Either way, the theorem is true in this case.
If both A and B are regular, then they are both in GL(n, F). Therefore, as we
have seen, they can be written as products of elementary matrices. It suffices then
to prove that det(S
1
)det(S
2
) = det(S
1
S
2
), where S
1
and S
2
are elementary matrices.
But our arguments above show that this is, indeed, true.
Remembering that A is regular if and only if A ∈ GL(n, F), we have:
Corollary. If A ∈ GL(n, F) then det(A
−1
) = (det(A))
−1
.
In particular, if det(A) = 1 then we also have det(A
−1
) = 1. The set of all such
matrices must then form a group.
Another simple corollary is the following.
Corollary. Assume that the matrix A is in block form, so that the linear mapping
which it represents splits into a direct sum of invariant subspaces (see theorem 29).
Then det(A) is the product of the determinants of the blocks.
34
Proof. Ifthen for each i = 1, . . . , p let
A
∗
i
=
_
_
_
_
_
_
_
_
1 0 . . . 0
0
.
.
. 0
.
.
. 0 A
i
0
.
.
.
0
.
.
. 0
0 . . . 0 1
_
_
_
_
_
_
_
_
.
That is, for the matrix A
∗
i
, all the blocks except the i-th block are replaced with
identity-matrix blocks. Then A = A
∗
1
A
∗
p
, and it is easy to see that det(A
∗
i
) =
det(A
i
) for each i.
Definition. The special linear group of order n is defined to be the set
SL(n, F) = ¦A ∈ GL(n, F) : det(A) = 1¦.
Theorem 36. Let A
′
= C
−1
AC. Then det(A
′
) = det(A).
Proof. This follows, since det(C
−1
) = (det(C))
−1
.
14 Leibniz Formula
Definition. A permutation of the numbers ¦1, . . . , n¦ is a bijection
σ : ¦1, . . . , n¦ → ¦1, . . . , n¦.
The set of all permutations of the numbers ¦1, . . . , n¦ is denoted S
n
. In fact, S
n
is
a group: the symmetric group of order n. Given a permutation σ ∈ S
n
, we will say
that a pair of numbers (i, j), with i, j ∈ ¦1, . . . , n¦ is a "reversed pair" if i < j, yet
σ(i) > σ(j). Let s(σ) be the total number of reversed pairs in σ. Then the sign of
sigma is defined to be the number
sign(σ) = (−1)
s(σ)
.
Theorem 37 (Leibniz). Let the elements in the matrix A be a
ij
, for i, j between 1
and n. Then we have
det(A) =
σ∈Sn
sign(σ)
n
i=1
a
σ(i)i
.
As a consequence of this formula, the following theorems can be proved:
35
Theorem 38. Let A be a diagonal matrix λ
n
_
_
_
_
_
Then det(A) = λ
1
λ
2
λ
n
.
Theorem 39. Let A be a triangular matrix.
_
_
_
_
_
_
_
a
11
a
12
⋆ ⋆
0 a
22
⋆ ⋆
0 0
.
.
.
.
.
.
.
.
. 0 a
(n−1)(n−1)
a
(n−1)n
0 0 0 a
nn
_
_
_
_
_
_
_
Then det(A) = a
11
a
22
a
nn
.
Leibniz formula also gives:
Definition. Let A ∈ M(n n, F). The transpose A
t
of A is the matrix consisting
of elements a
t
ij
such that for all i and j we have a
t
ij
= a
ji
, where a
ji
are the elements
of the original matrix A.
Theorem 40. det(A
t
) = det(A).
14.1 Special rules for 2 2 and 3 3 matrices
Let A =
_
a
11
a
12
a
21
a
22
_
. Then Leibniz formula reduces to the simple formula
det(A) = a
11
a
22
−a
12
a
21
.
For 33 matrices, the formula is a little more complicated. Let A =
_
_
a
11
a
12
a
13
a
21
a
22
a
23
a
31
a
32
a
33
_
_
.
Then we have
det(A) = a
11
a
22
a
33
+ a
12
a
23
a
33
+ a
13
a
21
a
32
−a
11
a
23
a
32
−a
12
a
21
a
33
−a
11
a
23
a
32
.
14.2 A proof of Leibniz Formula
Let the rows of the n n identity matrix be ǫ
1
, . . . , ǫ
n
. Thus
ǫ
1
= (1 0 0 0), ǫ
2
= (0 1 0 0), . . . , ǫ
n
= (0 0 0 1).
Therefore, given that the i-th row in a matrix is
ξ
i
= (a
i1
a
i2
a
in
),
36
then we have
ξ
i
=
n
i=1
a
iσ(i)
.
15 Why is the Determinant Important?
I am sure there are many points which could be advanced in answer to this question.
But here I will concentrate on only two special points.
• The transformation formula for integrals in higher-dimensional spaces.
This is a theorem which is usually dealt with in the Analysis III lecture. Let
G ⊂ R
n
be some open region, and let f : G → R be a continuous function.
Then the integral
_
G
f(x)d
(n)
x
has some particular value (assuming, of course, that the integral converges).
Now assume that we have a continuously differentiable injective mapping φ :
G →R
n
and a continuous function F : φ(G) →R. Then we have the formula
_
φ(G)
F(u)d
(n)
u =
_
G
F(φ(x))[detD(φ(x))[d
(n)
x.
Here, D(φ(x)) is the Jacobi matrix of φ at the point x.
This formula reflects the geometric idea that the determinant measures the
change of the volume of n-dimensional space under the mapping φ.
If φ is a linear mapping, then take Q ⊂ R
n
to be the unit cube: Q =
¦(x
1
, . . . , x
n
) : 0 ≤ x
i
≤ 1, ∀i¦. Then the volume of Q, which we can de-
note by vol(Q) is simply 1. On the other hand, we have vol(φ(Q)) = det(A),
where A is the matrix representing φ with respect to the canonical coordinates
for R
n
. (A negative determinant — giving a negative volume — represents an
orientation-reversing mapping.)
• The characteristic polynomial.
Let f : V → V be a linear mapping, and let v be an eigenvector of f with
f(v) = λv. That means that (f − λid)(v) = 0; therefore the mapping (f −
λid) : V → V is singular. Now consider the matrix A, representing f with
respect to some particular basis of V. Since λI
n
is the matrix representing the
mapping λid, we must have that the difference A − λI
n
is a singular matrix.
In particular, we have det(A−λI
n
) = 0.
Another way of looking at this is to take a "variable" x, and then calculate
(for example, using the Leibniz formula) the polynomial in x
P(x) = det(A −xI
n
).
This polynomial is called the characteristic polynomial for the matrix A.
Therefore we have the theorem:
38
Theorem 41. The zeros of the characteristic polynomial of A are the eigen-
values of the linear mapping f : V → V which A represents.
Obviously the degree of the polynomial is n for an n n matrix A. So let us
write the characteristic polynomial in the standard form
P(x) = c
n
x
n
+ c
n−1
x
n−1
+ + c
1
x + c
0
.
The coefficients c
0
, . . . , c
n
are all elements of our field F.
Now the matrix A represents the mapping f with respect to a particular choice
of basis for the vector space V. With respect to some other basis, f is repre-
sented by some other matrix A
′
, which is similar to A. That is, there exists
some C ∈ GL(n, F) with A
′
= C
−1
AC. But we have
det(A
′
−xI
n
) = det(C
−1
AC −xC
−1
I
n
C)
= det(C
−1
(A−xI
n
)C)
= det(C
−1
)det(A−xI
n
)det(C)
= det(A−xI
n
)
= P(x).
Therefore we have:
Theorem 42. The characteristic polynomial is invariant under a change of
basis; that is, under a similarity transformation of the matrix.
In particular, each of the coefficients c
i
of the characteristic polynomial P(x) =
c
n
x
n
+c
n−1
x
n−1
+ +c
1
x+c
0
remains unchanged after a similarity transfor-
mation of the matrix A.
What is the coefficient c
n
? Looking at the Leibniz formula, we see that the
term x
n
can only occur in the product
(a
11
−x)(a
22
−x) (a
nn
−x) = (−1)x
n
−(a
11
+ a
22
+ + a
nn
)x
n−1
+ .
Therefore c
n
= 1 if n is even, and c
n
= −1 if n is odd. This is not particularly
interesting.
So let us go one term lower and look at the coefficient c
n−1
. Where does x
n−1
occur in the Leibniz formula? Well, as we have just seen, there certainly is the
term
(−1)
n−1
(a
11
+ a
22
+ + a
nn
)x
n−1
,
which comes from the product of the diagonal elements in the matrix A−xI
n
.
Do any other terms also involve the power x
n−1
? Let us look at Leibniz formula
more carefully in this situation. We have
det(A −xI
n
) = (a
11
−x)(a
22
−x) (a
nn
−x)
+
σ∈Sn
σ=id
sign(σ)
n
i=1
_
a
σ(i)i
−xδ
σ(i)i
_
39
Here, δ
ij
= 1 if i = j. Otherwise, δ
ij
= 0. Now if σ is a non-trivial permutation
— not just the identity mapping — then obviously we must have two different
numbers i
1
and i
2
, with σ(i
1
) ,= i
1
and also σ(i
2
) ,= i
2
. Therefore we see that
these further terms in the sum can only contribute at most n −2 powers of x.
So we conclude that the (n −1)-st coefficient is
c
n−1
= (−1)
n−1
(a
11
+ a
22
+ + a
nn
).
Definitionbe an n n matrix. The trace of A
(in German, the spur of A) is the sum of the diagonal elements:
tr(A) = a
11
+ a
22
+ + a
nn
.
Theorem 43. tr(A) remains unchanged under a similarity transformation.
An example
Let f : R
2
→ R
2
be a rotation through the angle θ. Then, with respect to the
canonical basis of R
2
, the matrix of f is
A =
_
cos θ −sin θ
sin θ cos θ
_
.
Therefore the characteristic polynomial of A is
det
__
cos θ −sin θ
sin θ cos θ
_
−x
_
1 0
0 1
__
= det
_
cos θ −x −sin θ
sin θ cos θ −x
_
= x
2
−2xcos θ + 1.
That is to say, if λ ∈ R is an eigenvalue of f, then λ must be a zero of the charac-
teristic polynomial. That is,
λ
2
−2λ cos θ + 1 = 0.
But, looking at the well-known formula for the roots of quadratic polynomials, we
see that such a λ can only exist if [ cos θ[ = 1. That is, θ = 0 or π. This reflects the
obvious geometric fact that a rotation through any angle other than 0 or π rotates
any vector away from its original axis. In any case, the two possible values of θ give
the two possible eigenvalues for f, namely +1 and −1.
16 Complex Numbers
On the other hand, looking at the characteristic polynomial, namely x
2
−2xcos θ+1
in the previous example, we see that in the case θ = ±π this reduces to x
2
+1. And
in the realm of the complex numbers C, this equation does have zeros, namely ±i.
Therefore we have the seemingly bizarre situation that a "complex" rotation through
40
a quarter of a circle has vectors which are mapped back onto themselves (multiplied
by plus or minus the "imaginary" number i). But there is no need for panic here!
We need not follow the example of numerous famous physicists of the past, declaring
the physical world to be "paradoxical", "beyond human understanding", etc. No.
What we have here is a purely algebraic result using the abstract mathematical
construction of the complex numbers which, in this form, has nothing to do with
rotations of real physical space!
So let us forget physical intuition and simply enjoy thinking about the artificial
mathematical game of extending the system of real numbers to the complex numbers.
I assume that you all know that the set of complex numbers C can be thought of as
being the set of numbers of the form x +yi, where x and y are elements of the real
numbers R and i is an abstract symbol, introduced as a "solution" to the equation
x
2
+ 1 = 0. Thus i
2
= −1. Furthermore, the set of numbers of the form x + 0 i
can be identified simply with x, and so we have an embedding R ⊂ C. The rules of
addition and multiplication in C are
(x
1
+ y
1
i) + (x
2
+ y
2
i) = (x
1
+ x
2
) + (y
1
+ y
2
)i
and
(x
1
+ y
1
i) (x
2
+ y
2
i) = (x
1
x
2
−y
1
y
2
) + (x
1
y
2
+ x
2
y
1
)i.
Let z = x+yi be some complex number. Then the absolute value of z is defined
to be the (non-negative) real number [z[ =
_
x
2
+ y
2
. The complex conjugate of z
is z = x −yi. Therefore [z[ =
√
zz.
It is a simple exercise to show that C is a field. The main result — called (in
German) the Hauptsatz der Algebra — is that C is an algebraically closed field.
That is, let C[z] be the set of all polynomials with complex numbers as coefficients.
Thus, for P(z) ∈ C[z] we can write P(z) = c
n
z
n
+ + c
1
z + c
0
, where c
j
∈ C, for
all j = 0, . . . , n. Then we have:
Theorem 44 (Hauptsatz der Algebra). Let P(z) ∈ C[z] be an arbitrary polynomial
with complex coefficients. Then P has a zero in C. That is, there exists some λ ∈ C
with P(λ) = 0.
The theory of complex numbers (Funktionentheorie in German) is an extremely
interesting and pleasant subject. Complex analysis is quite different from the real
analysis which you are learning in the Analysis I and II lectures. If you are interested,
you might like to have a look at my lecture notes on the subject (in English), or
look at any of the many books in the library with the title "Funktionentheorie".
Unfortunately, owing to a lack of time in this summer semester, I will not be
able to describe the proof of theorem 44 here. Those who are interested can find a
proof in my other lecture notes on linear algebra. In any case, the consequence is
Theorem 45. Every complex polynomial can be completely factored into linear fac-
tors. That is, for each P(z) ∈ C[z] of degree n, there exist n complex numbers
(perhaps not all different) λ
1
, . . . , λ
n
, and a further complex number c, such that
P(z) = c(λ
1
−z) (λ
n
−z).
41
Proof. Given P(z), theorem 44 tells us that there exists some λ
1
∈ C, such that
P(λ
1
) = 0. Let us therefore divide the polynomial P(z) by the polynomial (λ
1
−z).
We obtain
P(z) = (λ
1
−z) Q(z) + R(z),
where both Q(z) and R(z) are polynomials in C[z]. However, the degree of R(z) is
less than the degree of the divisor, namely (λ
1
−z), which is 1. That is, R(z) must
be a polynomial of degree zero, i.e. R(z) = r ∈ C, a constant. But what is r? If we
put λ
1
into our equation, we obtain
0 = P(λ
1
) = (λ
1
−λ
1
)Q(z) + r = 0 +r.
Therefore r = 0, and so
P(z) = (λ
1
−z)Q(z),
where Q(z) must be a polynomial of degree n−1. Therefore we apply our argument in
turn to Q(z), again reducing the degree, and in the end, we obtain our factorization
into linear factors.
So the consequence is: let V be a vector space over the field of complex numbers
C. Then every linear mapping f : V → V has at least one eigenvalue, and thus at
least one eigenvector.
17 Scalar Products, Norms, etc.
So now we have arrived at the subject matter which is usually taught in the second
semester of the beginning lectures in mathematics — that is in Linear Algebra II
— namely, the properties of (finite dimensional) real and complex vector spaces.
Finally now, we are talking about geometry. That is, about vector spaces which
have a distance function. (The word "geometry" obviously has to do with the
measurement of physical distances on the earth.)
So let V be some finite dimensional vector space over R, or C. Let v ∈ V be
some vector in V. Then, since V
∼
= R
n
, or C
n
, we can write v =
n
j=1
a
j
e
j
, where
¦e
1
, . . . , e
n
¦ is the canonical basis for R
n
or C
n
, and a
j
∈ R or C, respectively, for
all j. Then the length of v is defined to be the non-negative real number
|v| =
_
[a
1
[
2
+ +[a
n
[
2
.
Of course, as these things always are, we will not simply confine ourselves to
measurements of normal physical things on the earth. We have already seen that the
idea of a complex vector space defies our normal powers of geometric visualization.
Also, we will not always restrict things to finite dimensional vector spaces. For
example, spaces of functions — which are almost always infinite dimensional — are
also very important in theoretical physics. Therefore, rather than saying that |v|
is the "length" of the vector v, we use a new word, and we say that |v| is the
norm of v. In order to define this concept in a way which is suitable for further
developments, we will start with the idea of a scalar product of vectors.
42
Definition. Let F = R or C and let V, W be two vector spaces over F. A bilinear
form is a mapping s : V W → F satisfying the following conditions with respect
to arbitrary elements v, v
1
and v
2
∈ V, w, w
1
and w
2
∈ W, and a ∈ F.
1. s(v
1
+v
2
, w) = s(v
1
, w) + s(v
2
, w),
2. s(av, w) = as(v, w),
3. s(v, w
1
+w
2
) = s(v, w
1
) + s(v, w
2
) and
4. s(v, aw) = as(v, w).
If V = W, then we say that a bilinear form s : V V → F is symmetric, if
we always have s(v
1
, v
2
) = s(v
2
, v
1
). Also the form is called positive definite if
s(v, v) > 0 for all v ,= 0.
On the other hand, if F = C and f : V → W is such that we always have
1. f(v
1
+v
2
) = f(v
1
) + f(v
1
) and
2. f(av) = af(v)
Then f is a semi-linear (not a linear) mapping. (Note: if F = R then semi-linear
is the same as linear.)
A mapping s : VW → F such that
1. The mapping given by s(, w) : V → F, where v → s(v, w) is semi-linear for
all w ∈ W, whereas
2. The mapping given by s(v, ) : W → F, where w → s(v, w) is linear for all
v ∈ V
is called a sesqui-linear form.
In the case V = W, we say that the sesqui-linear form is Hermitian (or Eu-
clidean, if we only have F = R), if we always have s(v
1
, v
2
) = s(v
2
, v
1
). (Therefore,
if F = R, an Hermitian form is symmetric.)
Finally, a scalar product is a positive definite Hermitian form s : V V → F.
Normally, one writes ¸v
1
, v
2
), rather than s(v
1
, v
2
).
Well, these are a lot of new words. To be more concrete, we have the inner
products, which are examples of scalar products.
Inner products
Let u =
_
_
_
_
_
u
1
u
2
.
.
.
u
n
_
_
_
_
_
, v =
_
_
_
_
_
v
1
v
2
.
.
.
v
n
_
_
_
_
_
∈ C
n
. Thus, we are considering these vectors as column
vectors, defined with respect to the canonical basis of C
n
. Then define (using matrix
43
multiplication)
¸u, v) = u
t
v = (u
1
u
2
u
n
)
_
_
_
_
_
v
1
v
2
.
.
.
v
n
_
_
_
_
_
=
n
j=1
u
j
v
j
.
It is easy to check that this gives a scalar product on C
n
. This particular scalar
product is called the inner product.
Remark. One often writes u v for the inner product. Thus, considering it to be a
scalar product, we just have u v = ¸u, v).
This inner product notation is often used in classical physics; in particular in
Maxwell's equations. Maxwell's equations also involve the "vector product" u v.
However the vector product of classical physics only makes sense in 3-dimensional
space. Most physicists today prefer to imagine that physical space has 10, or even
more — perhaps even a frothy, undefinable number of — dimensions. Therefore
it appears to be the case that the vector product might have gone out of fashion
in contemporary physics. Indeed, mathematicians can imagine many other possible
vector-space structures as well. Thus I shall dismiss the vector product from further
discussion here.
Definition. A real vector space (that is, over the field of the real numbers R),
together with a scalar product is called a Euclidean vector space. A complex vector
space with scalar product is called a unitary vector space.
Now, the basic reason for making all these definitions is that we want to define
the length — that is the norm — of the vectors in V. Given a scalar product, then
the norm of v ∈ V — with respect to this scalar product — is the non-negative real
number
|v| =
_
¸v, v).
More generally, one defines a norm-function on a vector space in the following
way.
Definition. Let V be a vector space over C (and thus we automatically also include
the case R ⊂ C as well). A function | | : V → R is called a norm on V if it
satisfies the following conditions.
1. |av| = [a[|v| for all v ∈ V and for all a ∈ C,
2. |v
1
+v
2
| ≤ |v
1
| +|v
2
| for all v
1
, v
2
∈ V (the triangle inequality), and
3. |v| = 0 ⇔ v = 0.
Theorem 46 (Cauchy-Schwarz inequality). Let V be a Euclidean or a unitary vector
space, and let |v| =
_
¸v, v) for all v ∈ V. Then we have
[¸u, v)[ ≤ |u| |v|
for all u and v ∈ V. Furthermore, the equality [¸u, v)[ = |u| |v| holds if, and
only if, the set ¦u, v¦ is linearly dependent.
44
Proof. It suffices to show that [¸u, v)[
2
≤ ¸u, u)¸v, v). Now, if v = 0, then — using
the properties of the scalar product — we have both ¸u, v) = 0 and ¸v, v) = 0.
Therefore the theorem is true in this case, and we may assume that v ,= 0. Thus
¸v, v) > 0. Let
a =
¸u, v)
¸v, v)
∈ C.
Then we have
0 ≤ ¸u −av, u −av)
= ¸u, u −av) +¸−av, u −av)
= ¸u, u) +¸u, −av) +¸−av, u) +¸−av, −av)
= ¸u, u) − a¸u, v)
. ¸¸ .
u,vu,v
v,v
− a¸u, v)
. ¸¸ .
u,vu,v
v,v
+aa¸v, v)
. ¸¸ .
u,vu,v
v,v
.
Therefore,
0 ≤ ¸u, u)¸v, v) −¸u, v)¸u, v).
But
¸u, v)¸u, v) = [¸u, v)[
2
,
which gives the Cauchy-Schwarz inequality. When do we have equality?
If v = 0 then, as we have already seen, the equality [¸u, v)[ = |u||v| is trivially
true. On the other hand, when v ,= 0, then equality holds when ¸u−av, u−av) = 0.
But since the scalar product is positive definite, this holds when u −av = 0. So in
this case as well, ¦u, v¦ is linearly dependent.
Theorem 47. Let V be a vector space with scalar product, and define the non-
negative function | | : V →R by |v| =
_
¸v, v). Then | | is a norm function on
V.
Proof. The first and third properties in our definition of norms are obviously sat-
isfied. As far as the triangle inequality is concerned, begin by observing that for
arbitrary complex numbers z = x + yi ∈ C we have
z + z = (x + yi) + (x −yi) = 2x ≤ 2[x[ ≤ 2[z[.
Therefore, let u and v ∈ V be chosen arbitrarily. Then we have
|u +v|
2
= ¸u +v, u +v)
= ¸u, u) +¸u, v) +¸v, u) +¸v, v)
= ¸u, u) +¸u, v) +¸u, v) +¸v, v)
≤ ¸u, u) + 2[¸u, v)[ +¸v, v)
≤ ¸u, u) + 2|u| |v| +¸v, v) (Cauchy-Schwarz inequality)
= |u|
2
+ 2|u| |v| +|v|
2
= (|u| +|v|)
2
.
Therefore |u +v| ≤ |u| +|v|.
45
18 Orthonormal Bases
Our vector space V is now assumed to be either Euclidean, or else unitary — that
is, it is defined over either the real numbers R, or else the complex numbers C. In
either case we have a scalar product ¸, ) : VV → F (here, F = R or C).
As always, we assume that V is finite dimensional, and thus it has a basis
¦v
1
, . . . , v
n
¦. Thinking about the canonical basis for R
n
or C
n
, and the inner
product as our scalar product, we see that it would be nice if we had
• ¸v
j
, v
j
) = 1, for all j (that is, the basis vectors are normalized), and further-
more
• ¸v
j
, v
k
) = 0, for all j ,= k (that is, the basis vectors are an orthogonal set in
V).
9
That is to say, ¦v
1
, . . . , v
n
¦ is an orthonormal basis of V. Unfortunately, most
bases are not orthonormal. But this doesn't really matter. For, starting from any
given basis, we can successively alter the vectors in it, gradually changing it into an
orthonormal basis. This process is often called the Gram-Schmidt orthonormaliza-
tion process. But first, to show you why orthonormal bases are good, we have the
following theorem.
Theorem 48. Let V have the orthonormal basis ¦v
1
, . . . , v
n
¦, and let x ∈ V be
arbitrary. Then
x =
n
j=1
¸v
j
, x)v
j
.
That is, the coefficients of x, with respect to the orthonormal basis, are simply the
scalar products with the respective basis vectors.
Proof. This follows simply because if x =
n
j=1
a
j
v
j
, then we have for each k,
¸v
k
, x) = ¸v
k
,
n
j=1
a
j
v
j
) =
n
j=1
a
j
¸v
k
, v
j
) = a
k
.
So now to the Gram-Schmidt process. To begin with, if a non-zero vector v ∈ V
is not normalized — that is, its norm is not one — then it is easy to multiply it by a
9
Note that any orthogonal set of non-zero vectors ¦u
1
, . . . , u
m
¦ in V is linearly independent.
This follows because if
0 =
m
j=1
a
j
u
j
then
0 = ¸u
k
, 0) = ¸u
k
,
m
j=1
a
j
u
j
) =
m
j=1
a
j
¸u
k
, u
j
) = a
k
¸u
k
, u
k
)
since ¸u
k
, u
j
) = 0 if j ,= k, and otherwise it is not zero. Thus, we must have a
k
= 0. This is true
for all the a
k
.
46
scalar, changing it into a vector with norm one. For we have ¸v, v) > 0. Therefore
|v| =
_
¸v, v) > 0 and we have
_
_
_
_
v
|v|
_
_
_
_
=
¸
_
v
|v|
,
v
|v|
_
=
¸
¸v, v)
¸v, v)
=
|v|
|v|
= 1.
In other words, we simply multiply the vector by the inverse of its norm.
Theorem 49. Every finite dimensional vector space V which has a scalar product
has an orthonormal basis.
Proof. The proof proceeds by constructing an orthonormal basis ¦u
1
, . . . , u
n
¦ from
a given, arbitrary basis ¦v
1
, . . . , v
n
¦. To describe the construction, we use induction
on the dimension, n. If n = 1 then there is almost nothing to prove. Any non-zero
vector is a basis for V, and as we have seen, it can be normalized by dividing by the
norm. (That is, scalar multiplication with the inverse of the norm.)
So now assume that n ≥ 2, and furthermore assume that the Gram-Schmidt pro-
cess can be constructed for any n−1 dimensional space. Let U ⊂ V be the subspace
spanned by the first n − 1 basis vectors ¦v
1
, . . . , v
n−1
¦. Since U is only n − 1 di-
mensional, our assumption is that there exists an orthonormal basis ¦u
1
, . . . , u
n−1
¦
for U. Clearly
10
, adding in v
n
gives a new basis ¦u
1
, . . . , u
n−1
, v
n
¦ for V. Unfor-
tunately, this last vector, v
n
, might disturb the nice orthonormal character of the
other vectors. Therefore, we replace v
n
with the new vector
11
u
∗
n
= v
n
−
n−1
_
_
_
a
11
a
1n
.
.
.
.
.
.
.
.
.
a
n1
a
nn
_
_
_
= I
n
.
Thus we conclude that A
−1
= A
t
. (Note: this was only the proof that f orthogo-
nal ⇒ A
−1
= A
t
. The proof in the other direction, going backwards through our
argument, is easy, and is left as an exercise for you.)
20.2 Unitary matrices
Theorem 51. The n n matrix A is unitary ⇔ A
−1
= A
t
. (The matrix A is
obtained by taking the complex conjugates of all its elements.)
Proof. Entirely analogous with the case of orthogonal matrices. One must note
however, that the inner product in the complex case is
¸u, w) = u
t
w = (u
1
u
n
)
_
_
_
w
1
.
.
.
w
n
_
_
_
=
n
j=1
u
j
w
j
.
20.3 Hermitian and symmetric matrices
Theorem 52. The n n matrix A is Hermitian ⇔ A = A
t
.
Proof. This is again a matter of translating the condition ¸v
j
, f(v
k
)) = ¸f(v
j
), v
k
)
into matrix notation, where f is the linear mapping which is represented by the
matrix A, with respect to the orthonormal basis ¦v
1
, . . . , v
n
¦. We have
¸v
j
, f(v
k
)) = v
t
j
Av
k
= v
t
j
_
_
_
a
1k
.
.
.
a
n
k
_
_
_
= a
jk
.
On the other hand
¸f(v
j
), v
k
) = Av
t
j
v
k
= (a
1j
a
nj
) v
k
= a
kj
.
In particular, we see that in the real case, self-adjoint matrices are symmetric.
50
21 Which Matrices can be Diagonalized?
The complete answer to this question is a bit too complicated for me to explain to
you in the short time we have in this semester. It all has to do with a thing called
the "minimal polynomial".
Now we have seen that not all orthogonal matrices can be diagonalized. (Think
about the rotations of R
2
.) On the other hand, we can prove that all unitary, and
also all Hermitian matrices can be diagonalized.
Of course, a matrix M is only a representation of a linear mapping f : V → V
with respect to a given basis ¦v
1
, . . . , v
n
¦ of the vector space V. So the idea that
the matrix can be diagonalized is that it is similar to a diagonal matrix. That is,
there exists another matrix S, such that S
−1
MS is diagonal.
S
−1
MSBut this means that there must be a basis for V, consisting entirely of eigenvectors.
In this section we will consider complex vector spaces — that is, V is a vector
space over the complex numbers C. The vector space V will be assumed to have a
scalar product associated with it, and the bases we consider will be orthonormal.
We begin with a definition.
Definition. Let W ⊂ V be a subspace of V. Let
W
⊥
= ¦v ∈ V : ¸v, w) = 0, ∀w ∈ W¦.
Then W
⊥
is called the perpendicular space to W.
It is a rather trivial matter to verify that W
⊥
is itself a subspace of V, and
furthermore W∩ W
⊥
= ¦0¦. In fact, we have:
Theorem 53. V = W⊕W
⊥
.
Proof. Let ¦w
1
, . . . , w
m
¦ be some orthonormal basis for the vector space W. This
can be extended to a basis ¦w
1
, . . . , w
m
, w
m+1
, . . . , w
n
¦ of V. Assuming the Gram-
Schmidt process has been used, we may assume that this is an orthonormal basis.
The claim is then that ¦w
m+1
, . . . , w
n
¦ is a basis for W
⊥
.
Now clearly, since ¸w
j
, w
k
) = 0, for j ,= k, we have that ¦w
m+1
, . . . , w
n
¦ ⊂ W
⊥
.
If u ∈ W
⊥
is some arbitrary vector in W
⊥
, then we have
u =
n
j=1
¸w
j
, u)w
j
=
n
j=m+1
¸w
j
, u)w
j
,
since ¸w
j
, u) = 0 if j ≤ m. (Remember, u ∈ W
⊥
.) Therefore, ¦w
m+1
, . . . , w
n
¦ is a
linearly independent, orthonormal set which generates W
⊥
, so it is a basis. And so
we have V = W⊕W
⊥
.
51
Theorem 54. Let f : V → V be a unitary mapping (V is a vector space over the
complex numbers C). Then there exists an orthonormal basis ¦v
1
, . . . , v
n
¦ for V
consisting of eigenvectors under f. That is to say, the matrix of f with respect to
this basis is a diagonal matrix.
Proof. If the dimension of V is zero or one, then obviously there is nothing to prove.
So let us assume that the dimension n is at least two, and we prove things by
induction on the number n. That is, we assume that the theorem is true for spaces
of dimension less than n.
Now, according to the fundamental theorem of algebra, the characteristic poly-
nomial of f has a zero, λ say, which is then an eigenvalue for f. So there must be
some non-zero vector v
n
∈ V, with f(v
n
) = λv
n
. By dividing by the norm of v
n
if
necessary, we may assume that |v
n
| = 1.
Let W ⊂ V be the 1-dimensional subspace generated by the vector v
n
. Then
W
⊥
is an n−1 dimensional subspace. We have that W
⊥
is invariant under f. That
is, if u ∈ W
⊥
is some arbitrary vector, then f(u) ∈ W
⊥
as well. This follows since
λ¸f(u), v
n
) = ¸f(u), λv
n
) = ¸f(u), f(v
n
)) = ¸u, v
n
) = 0.
But we have already seen that for an eigenvalue λ of a unitary mapping, we must
have [λ[ = 1. Therefore we must have ¸f(u), v
n
) = 0.
So we can consider f, restricted to W
⊥
, and using the inductive hypothesis,
we obtain an orthonormal basis of eigenvectors ¦v
1
, . . . , v
n−1
¦ for W
⊥
. There-
fore, adding in the last vector v
n
, we have an orthonormal basis of eigenvectors
¦v
1
, . . . , v
n
¦ for V.
Theorem 55. All Hermitian matrices can be diagonalized.
Proof. This is similar to the last one. Again, we use induction on n, the dimension
of the vector space V. We have a self-adjoint mapping f : V → V. If n is zero or
one, then we are finished. Therefore we assume that n ≥ 2.
Again, we observe that the characteristic polynomial of f must have a zero, hence
there exists some eigenvalue λ, and an eigenvector v
n
of f, which has norm equal
to one, where f(v
n
) = λv
n
. Again take W to be the one dimensional subspace of
V generated by v
n
. Let W
⊥
be the perpendicular subspace. It is only necessary to
show that, again, W
⊥
is invariant under f. But this is easy. Let u ∈ W
⊥
be given.
Then we have
¸f(u), v
n
) = ¸u, f(v
n
)) = ¸u, λv
n
) = λ¸u, v
n
) = λ 0 = 0.
The rest of the proof follows as before.
In the particular case where we have only real numbers (which of course are a
subset of the complex numbers), then we have a symmetric matrix.
Corollary. All real symmetric matrices can be diagonalized.
Note furthermore, that even in the case of a unitary matrix, the symmetry con-
dition, namely a
jk
= a
kj
, implies that on the diagonal, we have a
jj
= a
jj
for all j.
That is, the diagonal elements are all real numbers. But these are the eigenvalues.
Therefore we have:
52
Corollary. The eigenvalues of a self-adjoint matrix — that is, a symmetric or a
Hermitian matrix — are all real numbers.
Orthogonal matrices revisited
Let A be an n n orthogonal matrix. That is, it consists of real numbers, and we
have A
t
= A
−1
. In general, it cannot be diagonalized. But on the other hand, it can
be brought into the following form by means of similarity transformations.
A
′′
=
_
_
_
_
_
_
_
_
_
±1
.
.
. 0
±1
R
1
0
.
.
.
R
p
_
_
_
_
_
_
_
_
_
,
where each R
j
is a 2 2 block of the form
_
cos θ ±sin θ
sin θ ∓cos θ
_
.
To see this, start by imagining that A represents the orthogonal mapping f :
R
n
→ R
n
with respect to the canonical basis of R
n
. Now consider the symmetric
matrix
B = A+ A
t
= A+ A
−1
.
This matrix represents another linear mapping, call it g : R
n
→ R
n
, again with
respect to the canonical basis of R
n
.
But, as we have just seen, B can be diagonalized. In particular, there exists some
vector v ∈ R
n
with g(v) = λg(v), for some λ ∈ R. We now proceed by induction
on the number n. There are two cases to consider:
• v is also an eigenvector for f, or
• it isn't.
The first case is easy. Let W ⊂ V be simply W = [v]. i.e. this is just the set of all
scalar multiples of v. Let W
⊥
be the perpendicular space to W. (That is, w ∈ W
⊥
means that ¸w, v) = 0.) But it is easy to see that W
⊥
is also invarient under f.
This follows by observing first of all that f(v) = αv, with α = ±1. (Remember that
the eigenvalues of orthogonal mappings have absolute value 1.) Now take w ∈ W
⊥
.
Then ¸f(w), v) = α
−1
¸f(w), αv) = α
−1
¸f(w), f(v)) = α
−1
¸w, v) = α
−1
0 = 0.
Thus, by changing the basis of R
n
to being an orthonormal basis, starting with v
(which we can assume has been normalized), we obtain that the original matrix is
similar to the matrix
_
α 0
0 A
∗
_
,
where A
∗
is an (n−1)(n−1) orthogonal matrix, which, according to the inductive
hypothesis, can be transformed into the required form.
53
If v is not an eigenvector of f, then, still, we know it is an eigenvector of g, and
furthermore g = f + f
−1
. In particular, g(v) = λv = f(v) + f
−1
(v). That is,
f(f(v)) = λf(v) −v.
So this time, let W = [v, f(v)]. This is a 2-dimensional subspace of V. Again,
consider W
⊥
. We have V = W ⊕ W
⊥
. So we must show that W
⊥
is invarient
under f. Now we have another two cases to consider:
• λ = 0, and
• λ ,= 0.
So if λ = 0 then we have f(f(v)) = −v. Therefore, again taking w ∈ W
⊥
, we have
¸f(w), v) = ¸f(w), −f(f(v))) = −¸w, f(v)) = 0. (Remember that w ∈ W
⊥
, so
that ¸w, f(v)) = 0.) Of course we also have ¸f(w), f(v)) = ¸w, v) = 0.
On the other hand, if λ ,= 0 then we have v = λf(v) − f(f(v)) so that
¸f(w), v) = ¸f(w), λf(v) − f(f(v))) = λ¸f(w), f(v)) − ¸f(w), f(f(v))), and we
have seen that both of these scalar products are zero. Finally, we again have
¸f(w), f(v)) = ¸w, v) = 0.
Therefore we have shown that V = W ⊕ W
⊥
, where both of these subspaces
are invariant under the orthogonal mapping f. By our inductive hypothesis, there
is an orthonormal basis for f restricted to the n−2 dimensional subspace W
⊥
such
that the matrix has the required form. As far as W is concerned, we are back in
the simple situation of an orthogonal mapping R
2
→R
2
, and the matrix for this has
the form of one of our 2 2 blocks.
22 Dual Spaces
Again let V be a vector space over a field F (and, although its not really necessary
here, we continue to take F = R or C).
Definition. The dual space to V is the set of all linear mappings f : V → F. We
denote the dual space by V
∗
.
Examples
• Let V = R
n
. Then let f
i
be the projection onto the i-th coordinat. That is, if
e
j
is the j-th canonical basis vector, then
f
i
(e
j
) =
_
1, if i = j,
0, otherwise.
So each f
i
is a member of V
∗
, for i = 1, . . . , n, and as we will see, these dual
vectors form a basis for the dual space.
54
• More generally, let V be any finite dimensional vector space, with some basis
¦v
1
, . . . , v
n
¦. Let f
i
: V → F be defined as follows. For an arbitrary vector
v ∈ V there is a unique linear combination
v = a
1
v
1
+ + a
n
v
n
.
Then let f
i
(v
i
) = a
i
. Again, f
i
∈ V
∗
, and we will see that the n vectors,
f
1
, . . . , f
n
form a basis of the dual space.
• Let C
0
([0, 1]) be the space of continuous functions f : [0, 1] → R. As we have
seen, this is a real vector space, and it is not finite dimensional. For each
f ∈ C
0
([0, 1]) let
Λ(f) =
_
1
0
f(x)dx.
This gives us a linear mapping Λ : C
0
([0, 1]) →R. Thus it belongs to the dual
space of C
0
([0, 1]).
• Another vector in the dual space to C
0
([0, 1]) is given as follows. Let x ∈ [0, 1]
be some fixed point. Then let Γ
x
: C
0
([0, 1]) →R is defined to be Γ(f) = f(x),
for all f ∈ C
0
([0, 1]).
• For this last example, let us assume that V is a vector space with scalar
product. (Thus F = R or C.) For each v ∈ V, let φ
v
(u) = ¸v, u). Then
φ
v
∈ V
∗
.
Theorem 56. Let V be a finite dimensional vector space (over C) and let V
∗
be
the dual space. For each v ∈ V, let φ
v
: V → C be given by φ
v
(u) = ¸v, u). Then
given an orthonormal basis ¦v
1
, . . . , v
n
¦ of V, we have that ¦φ
v
1
, . . . , φ
vn
¦ is a basis
of V
∗
. This is called the dual basis to ¦v
1
, . . . , v
n
¦.
Proof. Let φ ∈ V
∗
be an arbitrary linear mapping φ : V → C. But, as always, we
remember that φ is uniquely determined by vectors (which in this case are simply
complex numbers) φ(v
1
), . . . , φ(v
n
). Say φ(v
j
) = c
j
∈ C, for each j. Now take some
arbitrary vector v ∈ V. There is the unique expression
v = a
1
v
1
+ + a
n
v
n
.
But then we have
φ(v) = φ(a
1
v
1
+ + a
n
v
n
)
= a
1
φ(v
1
) + + a
n
φ(v
n
)
= a
1
c
1
+ + a
n
c
n
= c
1
φ
v
1
(v) + + c
n
φ
vn
(v)
= (c
1
φ
v
1
+ + c
n
φ
vn
)(v).
Therefore, φ = c
1
φ
v
1
+ + c
n
φ
vn
, and so ¦φ
v
1
, . . . , φ
vn
¦ generates V
∗
.
To show that ¦φ
v
1
, . . . , φ
vn
¦ is linearly independent, let φ = c
1
φ
v
1
+ +c
n
φ
vn
be
some linear combination, where c
j
,= 0, for at least one j. But then φ(v
j
) = c
j
,= 0,
and thus φ ,= 0 in V
∗
.
55
Corollary. dim(V
∗
) = dim(V).
Corollary. More specifically, we have an isomorphism V → V
∗
, such that v → φ
v
for each v ∈ V.
But somehow, this isomorphism doesn't seem to be very "natural". It is defined
in terms of some specific basis of V. What if V is not finite dimensional so that we
have no basis to work with? For this reason, we do not think of V and V
∗
as being
"really" just the same vector space.
13
On the other hand, let us look at the dual space of the dual space (V
∗
)
∗
. (Perhaps
this is a slightly mind-boggling concept at first sight!) We imagine that "really" we
just have (V
∗
)
∗
= V. For let Φ ∈ (V
∗
)
∗
. That means, for each φ ∈ V
∗
we have
Φ(φ) being some complex number. On the other hand, we also have φ(v) being
some complex number, for each V ∈ V. Can we uniquely identify each V ∈ V with
some Φ ∈ (V
∗
)
∗
, in the sense that both always give the same complex numbers, for
all possible φ ∈ V
∗
?
Let us say that there exists a v ∈ V such that Φ(φ) = φ(v), for all φ ∈ V
∗
. In
fact, if we define Φ
v
to be Φ(φ) = φ(v), for each φ ∈ V
∗
, then we certainly have a
linear mapping, V
∗
→ C. On the other hand, given some arbitrary Φ ∈ (V
∗
)
∗
, do
we have a unique v ∈ V such that Φ(φ) = φ(v), for all φ ∈ V
∗
? At least in the case
where V is finite dimensional, we can affirm that it is true by looking at the dual
basis.
Dual mappings
Let V and W be two vector spaces (where we again assume that the field is C).
Assume that we have a linear mapping f : V → W. Then we can define a linear
mapping f
∗
: W
∗
→ V
∗
in a natural way as follows. For each φ ∈ W
∗
, let f
∗
(φ) =
φ ◦ f. So it is obvious that f
∗
(φ) : V →C is a linear mapping. Now assume that V
and W have scalar products, giving us the mappings s : V → V
∗
and t : W → W
∗
.
So we can draw a little "diagram" to describe the situation.
V
f
−→ W
s ↓ ↓ t
V
∗
f
∗
←− W
∗
The mappings s and t are isomorphisms, so we can go around the diagram, using
the mapping f
adj
= s
−1
◦ f
∗
◦ t : W → V. This is the adjoint mapping to f. So
we see that in the case V = W, we have that a self-adjoint mapping f : V → V is
such that f
adj
= f.
Does this correspond with our earlier definition, namely that ¸u, f(v)) = ¸f(u), v)
for all u and v ∈ V? To answer this question, look at the diagram, which now has
the form
V
f
−→ V
s ↓ ↓ s
V
∗
f
∗
←− V
∗
13
In case we have a scalar product, then there is a "natural" mapping V → V
∗
, where v → φ
v
,
such that φ
v
(u) = ¸v, u), for all u ∈ V.
56
where s(v) ∈ V
∗
is such that s(v)(u) = ¸v, u), for all u ∈ V. Now f
adj
= s
−1
◦f
∗
◦s;
that is, the condition f
adj
= f becomes s
−1
◦ f
∗
◦ s = f. Since s is an isomorphism,
we can equally say that the condition is that f
∗
◦s = s◦f. So let v be some arbitrary
vector in V. We have s ◦ f(v) = f
∗
◦ s(v). However, remembering that this is an
element of V
∗
, we see that this means
(s ◦ f(v))(u) = (f
∗
◦ s)(v)(u),
for all u ∈ V. But (s◦f(v))(u) = ¸f(v), u) and (f
∗
◦s)(v)(u) = ¸v, f(u)). Therefore
we have
¸f(v), u) = ¸v, f(u))
for all v and u ∈ V, as expected.
23 The End
This is the end of the semester, and thus the end of what I have to say about
"linear algebra in physics" here. But that is not to say that there is nothing more
that you have to know about the subject. For example, when studying the theory
of relativity you will encounter tensors, which are combinations of linear mappings
and dual mappings. One speaks of "covariant" and "contravariant" tensors. That
is, linear mappings and dual mappings.
But then, proceeding to the general theory of relativity, these tensors are used
to describe differential geometry. That is, we no longer have a linear (that is, a
vector) space. Instead, we imagine that space is curved, and in order to describe
this curvature, we define a thing called the tangent vector space which you can
think of as being a kind of linear approximation to the spacial structure near a
given point. And so it goes on, leading to more and more complicated mathematical
constructions, taking us away from the simple "linear" mathematics which we have
seen in this semester.
After a few years of learning the mathematics of contemporary theoretical physics,
perhaps you will begin to ask yourselves whether it really makes so much sense after
all. Can it be that the physical world is best described by using all of the latest
techniques which pure mathematicians happen to have been playing around with
in the last few years — in algebraic topology, functional analysis, the theory of
complex functions, and so on and so forth? Or, on the other hand, could it be
that physics has been loosing touch with reality, making constructions similar to the
theory of epicycles of the medieval period, whose conclusions can never be verified
using practical experiments in the real world?
In his book "The Meaning of Relativity", Albert Einstein wrote
"One can give good reasons why reality cannot at all be represented by
a continuous field. From the quantum phenomena it apears to follow
with certainty that a finite system of finite energy can be completely
described by a finite set of numbers (quantum numbers). This does not
seem to be in accordance with a continuum theory, and must lead to an
attempt to find a purely algebraic theory for the description of reality.
But nobody knows how to obtain the basis of such a theory."
57
Definition. A group is a non-empty set G, together with an operation1 , which is a mapping ' · ' : G × G → G, such that the following conditions are satisfied. 1. For all a, b, c ∈ G, we have (a · b) · c = a · (b · c), 2. There exists a particular element (the "neutral" element), often called e in group theory, such that e · g = g · e = g, for all g ∈ G. 3. For each g ∈ G, there exists an inverse element g −1 ∈ G such that g · g −1 = g −1 · g = e. If, in addition, we have a · b = b · a for all a, b ∈ G, then G is called an "Abelian" group. Definition. A field is a non-empty set F , having two arithmetical operations, denoted by '+' and '·', that is, addition and multiplication2. Under addition, F is an Abelian group with a neutral element denoted by '0'. Furthermore, there is another element, denoted by '1', with 1 = 0, such that F \ {0} (that is, the set F , with the single element 0 removed) is an Abelian group, with neutral element 1, under multiplication. In addition, the distributive property holds: a · (b + c) = a · b + a · c for all a, b, c ∈ F . The simplest example of a field is the set consisting of just two elements {0, 1} with the obvious multiplication. This is the field Z/2Z. Also, as we have seen in the analysis lectures, for any prime number p ∈ N, the set Z/pZ of residues modulo p is a field. The following theorem, which should be familiar from the analysis lectures, gives some elementary general properties of fields. Theorem 1. Let F be a field. Then for all a, b ∈ F , we have: 1. a · 0 = 0 · a = 0, 2. a · (−b) = −(a · b) = (−a) · b, 3. −(−a) = a, 4. (a−1 )−1 = a, if a = 0, 5. (−1) · a = −a, 6. (−a) · (−b) = a · b, 7. a · b = 0 ⇒ a = 0 or b = 0.
The operation is usually called "multiplication" in abstract group theory, but the sets we will deal with are also groups under "addition". 2 Of course, when writing a multiplication, it is usual to simply leave the '·' out, so that the expression a · b is simplified to ab.
1
and
(a + b) · c = a · c + b · c,
2
Proof. An exercise (dealt with in the analysis lectures). So the theory of abstract vector spaces starts with the idea of a field as the underlying arithmetical system. But in physics, and in most of mathematics (at least the analysis part of it), we do not get carried away with such generalities. Instead we will usually be confining our attention to one of two very particular fields, namely either the field of real numbers R, or else the field of complex numbers C. Despite this, let us adopt the usual generality in the definition of a vector space. Definition. A vector space V over a field F is an Abelian group — with vector addition denoted by v + w, for vectors v, w ∈ V. The neutral element is the "zero vector" 0. Furthermore, there is a scalar multiplication F × V → V satisfying (for arbitrary a, b ∈ F and v, w ∈ V): 1. a · (v + w) = a · v + a · w, 2. (a + b) · v = a · v + b · v, 3. (a · b) · v = a · (b · v), and 4. 1 · v = v for all v ∈ V.
Examples
• Given any field F , then we can say that F is a vector space over itself. The vectors are just the elements of F . Vector addition is the addition in the field. Scalar multiplication is multiplication in the field. • Let Rn be the set of n-tuples, for some n ∈ N. That is, the set of ordered lists of n real numbers. One can also say that this is Rn = R × R × · · · × R,
n times
Let V be a vector space over a field F and let W ⊂ V be some subset. If W is itself a vector space over F , considered using the addition and scalar multiplication in V, then we say that W is a subspace of V. Analogously, a subset H of a group G, which is itself a group using the multiplication operation from G, is called a subgroup of G. Subfields are similarly defined. Theorem 2. Let W ⊂ V be a subset of a vector space over the field F . Then W is a subspace of V ⇔ a · v + b · w ∈ W, for all v, w ∈ W and a, b ∈ F . Proof. The direction '⇒' is trivial. For '⇐', begin by observing that 1 · v + 1 · w = v + w ∈ W, and a · v + 0 · w = a · v ∈ W, for all v, w ∈ W and a ∈ F . Thus W is closed under vector addition and scalar multiplication. Is W a group with respect to vector addition? We have 0 · v = 0 ∈ W, for v ∈ W; therefore the neutral element 0 is contained in W. For an arbitrary v ∈ W we have v + (−1) · v = = = = 1 · v + (−1) · v (1 + (−1)) · v 0·v 0.
Therefore (−1) · v is the inverse element to v under addition, and so we can simply write (−1) · v = −v. The other axioms for a vector space can be easily checked. The method of this proof also shows that we have similar conditions for subsets of groups or fields to be subgroups, or subfields, respectively. Theorem 3. Let H ⊂ G be a (non-empty) subset of the group G. Then H is a subgroup of G ⇔ ab−1 ∈ H, for all a, b ∈ H. 4
Proof. The direction '⇒' is trivial. As for '⇐', let a ∈ H. Then aa−1 = e ∈ H. Thus the neutral element of the group multiplication is contained in H. Also ea−1 = a−1 ∈ H. Furthermore, for all a, b ∈ H, we have a(b−1 )−1 = ab ∈ H. Thus H is closed under multiplication. The fact that the multiplication is associative (a(bc) = (ab)c, for all a, b and c ∈ H) follows since G itself is a group; thus the multiplication throughout G is associative. Theorem 4. Let U, W ⊂ V be subspaces of the vector space V over the field F . Then U ∩ W is also a subspace. Proof. Let v, w ∈ U ∩ W be arbitrary vectors in the intersection, and let a, b ∈ F be arbitrary elements of the field F . Then, since U is a subspace of V, we have a · v + b · w ∈ U. This follows from theorem 2. Similarly a · v + b · w ∈ W. Thus it is in the intersection, and so theorem 2 shows that U ∩ W is a subspace.
4
Linear Independence and Dimension
such that not all ai ∈ F are simply zero. If no such non-trivial equation exists, then the set {v1 , . . . , vn } ⊂ V is said to be linearly independent. This definition is undoubtedly the most important idea that there is in the theory of linear algebra!
Definition. Let v1 , . . . , vn ∈ V be finitely many vectors in the vector space V over the field F . We say that the vectors are linearly dependent if there exists an equation of the form a1 · v1 + · · · + an · vn = 0,
Examples
• In R2 let v1 = (1, 0), v2 = (0, 1) and v3 = (1, 1). Then the set {v1 , v2 , v3 } is linearly dependent, since we have v1 + v2 − v1 = 0. On the other hand, the set {v1 , v2 } is linearly independent. • In C0 ([0, 1], R), let f1 : [0, 1] → R be given by f1 (x) = 1 for all x ∈ [0, 1]. Similarly, let f2 be given by f2 (x) = x, and f3 is f3 (x) = 1 − x. Then the set {f1 , f2 , f3 } is linearly dependent. Now take some vector space V over a field F , and let S ⊂ V be some subset of V. (The set S can be finite or infinite here, although we will usually be dealing with finite sets.) Let v1 , . . . , vn ⊂ S be some finite collection of vectors in S, and let a1 , . . . , an ∈ F be some arbitrary collection of elements of the field. Then the sum a1 · v1 + · · · + an · vn is a linear combination of the vectors v1 , . . . , vn in S. The set of all possible linear combinations of vectors in S is denoted by span(S), and it is called the linear span 5
0) .
In general such function spaces — which play a big role in quantum field theory. then [S] is a subspace of V. let e1 = (1. We will say that this representation of w is unique if. . . 0) e2 = (0. . . we will mostly be concerned with finitely generated vector spaces. . . an and b1 . such that a1 v1 + · · · + an vn = b1 v1 + · · · + bn vn . . the vector space C0 ([0. From now on in these discussions. . . where aj = bj . . . we will assume that such sets are finite unless stated otherwise. Let w = a1 v1 + · · · an vn be some vector in [S] ⊂ V. . . . .
3
6
. . n. for at least one j between 1 and n. If S is finite. Since this representation of the zero vector is unique. . then we have that the set S is linearly independent ⇔ the representation of all vectors in the span of S as linear combinations of vectors in S is unique. Theorem 6. • On the other hand.3 So let S = {v1 . e2 .
Examples
• For any n ∈ N. 1) Then S = {e1 . then we say that S is a generating set for V . Proof. . . 1.of S. R) is clearly not finitely generated. . Given S ⊂ V. '⇐' We certainly have 0 · v1 + · · · 0 · vn = 0. . and yet there exists a vector in the span of S which is not uniquely represented as a linear combination of the vectors in S? Assume that there exist elements a1 . . bn of the field F . . given some other linear combination. . 0. and it generates V. 1]. Given this. . and which are studied using the mathematical theory of functional analysis — are not finitely generated. . But then (a1 − b1 )v1 + · · · + (aj − bj ) vi + · · · + (an − bn )vn = 0
=0
shows that S cannot be a linearly independent set. . S is the generating set of [S]. . A simple consequence of theorem 2. Therefore if [S] = V. then we must have bi = ai for all i = 1. Proof. . it follows S is linearly independent. . . . then we say that the vector space V is finitely generated. . vn } ⊂ V be a finite set. . '⇒' Can it be that S is linearly independent. . . en = (0. w = b1 v1 + · · · bn vn . However in this lecture. Theorem 5. en } is a generating set for Rn . One also writes [S]. . . an are arbitrarily given elements of the field F . 0. where a1 .
we see that since w = 0. . Proof. and take some arbitrary non-zero vector w ∈ V. Then S is called a basis for V. . leaving us with a still smaller generating set for V. Writing w = a1 v1 + · · · + an vn . . vj+1. . . . vn } be a basis for the vector space V. Since S is linearly dependent. such that vj =
i=j
ai vi . Let S = {v1 . Then [S] = [S ′ ]. . then we could remove some element. as in the lemma. . . Theorem 8. Assume that S = {v1 . vn } ⊂ V be linearly dependent. . . . Let S = {v1 . Lemma. Proof.
n
bi vi = 0. Since V is finitely generated. Assume that S ⊂ V is a finite. there exists a finite generating set.
Corollary.
i=1
such that bj = 0. . Assume that the vector space V is finitely generated. linearly independent subset with [S] = V. vj−1 . Then there exists a basis for V. such that S ′ = {v1 . vj−1 . w. Let S ′ = {v1 . we write vj = a−1 w + j
i=j
−
ai aj
vi . . vn } be S. . . Then there exists some j ∈ {1. . . . Then bj vj = − and so vj =
i=j
bi vi
i=j
−
bi bj
vj . . . and elements ai ∈ F . for i = j. there exists some non-trivial linear combination of the elements of S.
7
. and let vj be as in the lemma above. . . Let S be such a finite generating set which has as few elements as possible. . for at least one of the j. vn } ⊂ V is linearly dependent. . . . . n}. summing to the zero vector. Taking that j. Theorem 7. n}. . . This is a contradiction. . . vn } is also a basis of V.Definition. Then there exists some j ∈ {1. . vj+1. at least one aj = 0.
Proof. Take such a one. with the element vj removed. . . . If S were linearly dependent. Therefore S must be a basis for V.
in fact. If [S] = V then we simply take B = S. Y ⊂ V. Therefore T would be linearly dependent. . Proof of Theorem 13. since any basis generates V. Definition. . . there exist extensions T = {x1 . Corollary. According to theorem 10. according to theorem 9. thus giving us the basis U. Proof. Then U is also finitely generated. written X ⊕ Y. Therefore U must be finitely generated. . dim(X ⊕ Y) = dim(X) + dim(Y). Theorem 11. written dim(V). vn } be a basis of X ∩ Y. Assume the vector space V has a basis consisting of n elements. The number of vectors in a basis of the vector space V is called the dimension of V. Then we have dim(X + Y) = dim(X) + dim(Y) − dim(X ∩ Y). namely am = am+1 = · · · = an = 0. Definition. Assume that the vector space V is finitely generated and that we have a linearly independent subset S ⊂ V. . Let U be a subspace of the (finitely generated) vector space V. and each possible basis for U has no more elements than any basis for V. We will now show that. . Y ⊂ V. xm } and U = {y1 . Then there exists a basis B of V with S ⊂ B. The subspace X + Y = [X ∪ Y ] is called the sum of X and Y. . Theorem 13 (A Dimension Formula). start with some given basis A ⊂ V and apply theorem 9 successively. Then. Proof. which is not true. Theorem 10 (Extension Theorem). Assume there is a basis B of V containing n vectors. Let S = {v1 . Theorem 12.
9
. Let V be a finite dimensional vector space with subspaces X. Otherwise. then we would have wm being expressed as a linear combination of the other vectors in T . .If we had all the coefficients of the vectors from S being zero. then it is the direct sum. S ∪ T ∪ U is a basis for X + Y. Thus one of the aj = 0. there cannot exist more than n linearly independent vectors in U. If X ∩ Y = {0}. . . . Then every basis of V also has precisely n elements. such that any basis for U has at most n elements. yr }. Using theorem 8. Let V be a vector space with subspaces X. we may exchange wm for the vector vj in U ′ . This follows directly from theorem 11. Proof. . for j ≥ m. such that S ∪ T is a basis for X and S ∪ U is a basis for Y. which is a subspace of itself.
. y ∈ Y. Then there exists another subspace Y ⊂ V. The mapping f is called a linear mapping if f (au + bv) = af (u) + bf (v) for all a. for all a ∈ F and for all u and v ∈ V. r. Let V be a finite dimensional vector space. Definition. Proof. Let f : V → W be a mapping from the vector space V to the vector space W. . Thus y ∈ X. we conclude that all the bj are zero. we must have ck = 0 for k = 1. Otherwise. Proof.
5
Linear Mappings
Definition. By choosing a and b to be either 0 or 1. If [S] = V then we are finished. v ∈ V. use the extension theorem (theorem 10) to find a basis B of V. Also it is obvious that f (0) = 0 always. But clearly we also have. if the zero vector in V is the only vector which is mapped into the zero vector in W under f . The converse is of course trivial. If ker(f ) = {0}. Theorem 15. is the set of vectors in V which are mapped by f into the zero vector in W. Then4 Y = [B \ S] satisfies the condition of the theorem.
say. This gives the dimension formula. Let f : V → W be a linear mapping. b ∈ F and all u. . The kernel of the mapping. we see that the dim(X) = n + m. looking at the vector x and applying the same argument. But then all the ai must also be zero since the set S is linearly independent. then we must have u = v. Take a basis S of X. But f (u) = f (v)
4
⇒
0 = f (u) − f (v) = f (u − v).
The notation B \ S denotes the set of elements of B which are not in S
10
. Theorem 14. Therefore y ∈ X ∩ Y. dim(Y) = n + r and dim(X ∩ Y) = n. That is. denoted by ker(f ). with S ⊂ B. such that V = X ⊕ Y. both over the field F .To begin with. then f is an injection (monomorphism). Let V and W be vector spaces. we immediately see that a linear mapping always has both f (av) = af (v) and f (u + v) = f (u) + f (v). Is the set S ∪ T ∪ U linearly independent? Let
n m r
0 =
i=1
ai vi +
j=1
bj xj +
k=1
ck yk
= v + x + y.
Then we have y = −v −x. Putting this all together. and let X ⊂ V be a subspace. it is clear that X + Y = [S ∪ T ∪ U]. . that is. Thus y can be expressed as a linear combination of vectors in S alone. and since S ∪ U is is a basis for Y . we must show that if u and v are two vectors in V with the property that f (u) = f (v). Similarly.
. . 0. . . Now define the mapping f : V → W by the rule f (vi ) = wi . . . Since A and B are both bases. let A = {u1 . . . and let f : V → W be a linear mapping. . vn } be a basis for V. Let B = {v1 . . . . Clearly the mapping is uniquely determined. . "⇐" Take B = {v1 . Let V and W be two finite dimensional vector spaces over a field F . Therefore we see that if the values of f (v1 ). . . 1
i-th Position
. . .Theorem 18. Proof. . with ai ∈ F . and it contains precisely n elements. We have f (vi ) = ui for all the basis vectors vi ∈ B. then the value of f (v) is uniquely determined. Then. Then we have V ∼ W ⇔ dim(V) = dim(W). since v is uniquely determined as a linear combination of the basis vectors B. · · · . Then clearly F n is also a vector space of dimension n over F . vn } ⊂ V to again be a basis of V and let A = {w1 . . it follows that f must be a bijection. This immediately gives us a complete classification of all finite-dimensional vector spaces. where v = a1 v1 + · · · an vn . Therefore A is a basis of W. . . Theorem 19. On the other hand. . Furthermore. On the other hand. vn } ⊂ V be a basis for V. we can think that it is "really" just F n . . . and let B = {v1 . By theorem 18 we see that a linear mapping f is thus uniquely determined. . . The canonical basis is the set of vectors {e1 . since the mapping f is linear. . when thinking about V. . wn } ⊂ W be some basis of W (with n elements). . = Proof. It is a trivial matter to verify that the mapping which is so defined is also linear. 0}
for each i. . for each arbitrarily given vector v ∈ V. for all i. we have f (v) = f (a1 v1 + · · · an vn ) = f (a1 v1 ) + · · · + f (an vn ) = a1 f (v1 ) + · · · + an f (vn ). . . For let V be a vector space of dimension n over the field F . where ei = (0. Therefore. . . . en }. we have A = {f (v1 ). Let v ∈ V be an arbitrary vector in V. Since B is a basis for V. Then f is uniquely determined by the n vectors {f (v1 ). for each i. un } be a set of n arbitrarily given vectors in W. f (vn )} ⊂ W being linearly independent. . . . f (vn )} in W. f (vn ) are given.. "⇒" Let f : V → W be an isomorphism. we have [B] = V. thus dim(V) = dim(W). 0. . Then. Let V and W be finite dimensional vector spaces over a field F . since B is a basis of V. for each v ∈ V. as shown in our Remark above. Then let a mapping f : V → W be defined by the rule f (v) = a1 u1 + · · · an un . Thus [A] = W also. we can uniquely write v = a1 v1 + · · · an vn . the central idea in the theory of linear algebra is that 12
.
This is a reflection of the 2-dimensional plane into itself. But then. the question is. x1 sin φ + x2 cos φ). if we remember the formulas for cosines and sines of sums. with the axis of reflection being the diagonal axis x1 = x2 . we note that cos(φ + π/2) = − sin(φ) and sin(φ + π/2) = cos(φ). where r = x2 + x2 . with various possible bases. such that x1 = r cos θ and x2 = r sin θ. Then v = (r cos θ. for some real number φ ∈ R. The space F n seems to have a preferred. cos φ) = (x1 cos φ − x2 sin φ.we can look at things using different possible bases (or "frames of reference" in physics). (0. 1). namely the canonical basis. turns out to be (r(cos(θ) cos(φ) − sin(θ) sin(φ)). r(sin(θ) cos(φ) − cos(θ) sin(φ)). That is. x1 sin φ + x2 cos φ). sin φ) and f1 (e2 ) = (− sin φ. Thus it is better to think about an abstract V.
13
. fixed frame of reference. we will consider the 2-dimensional real vector space R2 . In particular we have cos(θ + φ) sin(θ + φ) = sin(θ) cos(φ) − cos(θ) sin(φ). x2 ) be some arbitrary point of the plane R2 .
• f1 : R2 → R2 with f1 (e1 ) = (−1. assuming that they are not both zero. we find two unique real numbers r ≥ 0 and θ ∈ [0. 0). cos φ).
Examples
For these examples. given any two real numbers x1 and x2 then. remembering that x1 = r cos θ and x2 = r sin θ. • f3 : R2 → R2 with f3 (e1 ) = (cos φ. = cos(θ) cos(φ) − sin(θ) sin(φ). 0) and f1 (e2 ) = (0. sin φ) + x2 (− sin φ. what happens to the vector v when it is rotated through the angle φ while preserving its length? Perhaps the best way to look at this is to think about v in polar coordinates. together with its canonical basis B = {e1 . 1)}. r sin θ). 2π). • f2 : R2 → R2 with f2 (e1 ) = e2 and f1 (e2 ) = e1 .
Taking θ = π/2. with the axis of reflection being the second coordinate axis. we learn about the formulas of trigonometry. Looking at this from the point of view of geometry. Then we have f3 (v) = x1 f3 (e1 ) + x2 f (e2 ) = x1 (cos φ. So a rotation of v through 1 2 the angle φ must bring it to the new vector (r cos(φ + θ).
5
In analysis. r sin(φ + θ)) which.5 For let v = (x1 . This is a reflection of the 2-dimensional plane into itself. through an angle of φ. x2 ) ∈ R2 with x1 = 0. This is a rotation of the plane about its middle point. we see that the rotation brings the vector v into the new vector (x1 cos φ − x2 sin φ. that is the set of points (x1 . e2 } = {(1.
we have a very nice system for writing down the coordinates of the vectors after they have been mapped by a linear mapping. x1 sin φ + x2 cos φ). consider the rotation of the plane through the angle φ. x2 ) being rotated into the new vector (x1 cos φ − x2 sin φ. 14
. This is a great improvement! So whereas before. To illustrate this system. x1 sin φ + x2 cos φ But then. leaving hardly any room left over to describe symbolically what we want to do with the vector.which was precisely the specification for f3 (v). which was described in the last section. Thus we have
A · v = f (v). But if we change into the column vector notation. In three and more dimensions. xn
It is true that we use up lots of vertical space on the page in this way. x1 sin φ + x2 cos φ cos φ − sin φ sin φ cos φ represents the mapping x1 x2
So we can say that the 2 × 2 matrix A = f3 : R2 → R2 . and the vertical lists are column vectors. we wrote v = (x1 . v = . Now. remembering how we multiplied matrices. . things become even worse! Thus it is obvious that we need a more sensible system for describing these linear mappings. we see that this is just cos φ − sin φ sin φ cos φ x1 x2 = x1 cos φ − x2 sin φ . xn ) run over the page. the most obvious problem with our previous notation for vectors was that the lists of the coordinates (x1 . but rather as vertical lists. we have (x1 . · · · . . In terms of row vectors. That is. The solution to this problem is to write vectors not as horizontal lists. The usual system is to use matrices. In addition. now we will write x1 . we can afford to waste this vertical space. · · · . We say that the horizontal lists are row vectors.
6
Linear Mappings and Matrices
This last example of a linear mapping of R2 into itself — which should have been simple to describe — has brought with it long lines of lists of coordinates which are difficult to think about. but since the rest of the writing is horizontal. matrix multiplication gives the result of the linear mapping. xn ). and the 2 × 1 matrix x1 x2
represents the vector v. we have v= being rotated to x1 cos φ − x2 sin φ .
we conclude that: Theorem 22. Now it is obvious that the row space (Zeilenraum). looking at the column vectors of this matrix in step form. There are n column vectors. and instead we will consider simple linear equations. The n × n matrix A is regular ⇔ the linear mapping f : F n → F n . using theorem 21 and exercise 6. and so on up to jr are all linearly independent. The linear mapping f : F n → F n is then both an injection (since S must be linearly independent) and also a surjection. Let A be a quadratic n×n matrix. This will be dealt with in the next section. Then A is called regular if Rank(A) = n. This common dimension is simply called the rank — written Rank(A) — of the matrix.3. But then.
8
Systems of Linear Equations
We now take a small diversion from our idea of linear algebra as being a method of describing geometry. but overly tedious to compose here in TEX. the column rank is equal to the row rank. every matrix can be transformed into a matrix in step form. Proof. and in fact the non-zero row vectors of a matrix in step form provide us with a basis for the row space. Since the dimension of F n is n. we see that the columns j1 . we consider a system of m equations in n unknowns. . Given an m × n matrix. represented by the matrix A with respect to the canonical basis of F n is an isomorphism. '⇒' If A is regular. . and they generate the column space. thus the rank of A is n. Proof. '⇐' Since the set of column vectors S is the set of images of the canonical basis vectors of F n under f . otherwise A is called singular. The induction step in this proof. We use the technique of "Gaussian elimination". By means of a finite sequence of elementary row operations. j2 . then the rank of A — namely the dimension of the column space [S] — is n. similarly the dimension of the row space is the row rank. we must therefore have [S] = F n . a11 x1 + · · · + a1n xn = b1 . the number of rows in the matrix. which is simply the usual way anyone would go about solving a system of linear equations. Theorem 23. they must be linearly independent. will be described in the lecture. the dimension of the column space is called the column rank. In particular. For any matrix A. Definition. So.Theorem 21. that is [R] ⊂ F n . am1 x1 + · · · + amn xn = bm 20
. which uses a number of simple ideas which are easy to write on the blackboard. has the dimension r. Induction on m. (This is discussed more fully in the lecture!) Definition.
subtract akj times the i-th equation from the k-th equation. 1. Otherwise find the smallest index k > i such that akj = 0 and exchange the i-th equation with the k-th equation. and the problem is to find the numbers xl . if aij = 0 then if akj = 0 for all i < k ≤ m. 2. xk b1 b2 b3
+ · · · + akn xk = bk 0 = bk+1 .) So now we observe that: 21
. then our system of linear equations is and x = . The next thing is to solve the system of equations in step form. am1 · · · amn x1 b1 . ∈ F n and b = . . . . Multiply both sides of the (possibly new) i-th equation by a−1 . . 4. The problem is that perhaps there is no solution. 0 = bm
(Note that this reordering of the variables is like our first elementary column operation for matrices. ∈ F m . or perhaps there are many solutions. . we will have akj = 0. . . The easiest way to decide which case we have is to reorder the variables — that is the various xi — so that the steps start in the upper left-hand corner. The numbers aij and bk are given (as elements of F ). we have transformed the system of linear equations into a system in step form.We can also think about this as being a vector equation. at this stage. things then look like this: x1 + a12 x2 + a13 x3 + · · · + · · · + a1n xn = x2 + a23 x3 + a24 x4 + · · · + a2n xn = x3 + a34 x4 + · · · + a3n xn = . That is. . and they are all one unit wide.
But what is the most obvious way to solve this system of equations? It is a simple matter to write down an algorithm. Set i := i + 1. 3. for all k > i.. If i ≤ n then return to step 2. set j := j + 1. . Therefore. as follows. . after this operation has been carried out. xn bm just the single vector equation A · x = b. . That is. . . A= . Then for ij each i < k ≤ m. .
So at this stage. Let i := 1 and j := 1. if a11 · · · a1n .
. we progress back to the first equation and obtain values for all the xj . substitute bn for xn in the n − 1-st equation. . . It is obtained by working backwards through the equations. . it is necessary to look at the characteristic polynomial of the matrix.. . then the problem of finding eigenvectors and eigenvalues is simply the problem of solving the equation Av = λv. . Let V be a vector space over a field F . . . . Namely. . . . This algorithm for finding solutions of systems of linear equations is called "Gaussian Elimination". .. . for 1 ≤ j ≤ n.• If bl = 0 for some k + 1 ≤ l ≤ m. • Otherwise. as before. 0 · · · 0 b′k+1 0 · · · 0 0 . • Otherwise. we progressively obtain the values of xk−1 . . . am1 am2 · · · amn bm Then by means of elementary row and column operations. But then. so that is clear. So how should we go about things? Well. . .. In this case we can assign arbitrary values to the variables xk+1 . xk−2 and so on. . All of this can be looked at in terms of our matrix notation. as we will see. . back to x1 . . . . An eigenvector of f is a non-zero vector v ∈ V (so we have v = 0) such that there exists some λ ∈ F with f (v) = λv. if k = n then the system has precisely one single solution. and then that fixes the value of xk . . . The scalar λ is then called the eigenvalue associated with this eigenvector. . . . . xn . Let us call the following m × n + 1 matrix the augmented matrix for our system of linear equations: a11 a12 · · · a1n b1 a21 a22 · · · a2n b2 A= . . in 22
. But then. . A′ = 0 · · · 0 1 a′k k+1 · · · a′kn b′k . So if f is represented by the n × n matrix A (with respect to some given basis of V). . · .. But here both λ and v are variables. and let f : V → V be a linear mapping of V into itself. and we then have xn−1 = bn−1 − an−1 n bn . . . . . . the matrix is transformed into the new matrix which is in simple step form 1 a′12 · · · · a′1 k+1 · · · a′1n b′1 0 1 a′23 · a′2 k+1 · · · a′2n b′2 . By this method. . . . the last equation is simply xn = bn . . k < n. . . then the system of equations has no solution. . . . . . . ′ 0 ··· 0 0 0 ··· 0 bm
Finding the eigenvectors of linear mappings
Definition. . .
such that v = 0.
23
. But now. vn } ⊂ V is the only basis we
That is.8
which can be easily solved to give us the (or one value is λ. So we have the linear mapping f : V → V. once an eigenvalue is found. .. we can say that κv is also an eigenvector whose eigenvalue is λ. {v1 . . . . n. .
0 0 ··· 1
0 0 . .. . all the bi are zero. . . and let {v1 . . Then. . . Then. . we only need a single basis for V.
8 7
Therefore. respectively. And the resulting matrix A = . . . wm } ⊂ W be bases for V and W. is the matrix describing f with . Now the n × n identity matrix is 1 0 ··· 0 1 · · · E = . . and we have (A − λE)(κv) = κ(A − λE)v = κ0 = 0. . . . then clearly we can multiply it with any scalar κ ∈ F . for each j = 1. . . Thus a homogeneous system with matrix A has the form Av = 0. That is. . am1 · · · amn respect to these given bases. Given any solution vector v.
A particular case
This is the case that V = W. we can consider it to be a constant in our system of linear equations. And they become the homogeneous 7 system (a11 − λ)x1 + a12 x2 + ··· + a21 x1 + (a22 − λ)x2 + · · · + an1 x1 + an2 x2 a1n xn a2n xn = 0 = 0 .
+ · · · + (ann − λ)xn = 0 of the) eigenvector(s) whose eigen-
Thus we see that an eigenvalue is any scalar λ ∈ F such that the vector equation (A − λE)v = 0 has a solution vector v ∈ V. We have m f (vj ) =
i=1
aij wi .
9
Invertible Matrices
Let f : V → W be a linear mapping. . . . .order to find an eigenvalue λ. . . as long as κ = 0. .
a11 · · · a1n . . . . as we have seen. the mapping f can be uniquely described by specifying the values of f (vj ). vn } ⊂ V and {w1 .
vn }). F ). Definition. The fact that the identity element in GL(n. F ) is the identity matrix is clear. . F ) is closed under matrix multiplication. and we have f −1 ◦ f = id.
24
. C ∈ GL(n. . F ). and we have that C −1 · A−1 is itself an n × n matrix. F ) is a group under matrix multiplication. It only remains to see that GL(n. F ). Or. . put another way. According to theorem 17. But then C −1 A−1 AC = C −1 A−1 A C = C −1 In C = C −1 C = In . the inverse mapping f −1 is also linear. Thus f (v) = v.
Regular matrices
Let us now assume that A is some regular n × n matrix. The set of all regular n×n matrices over the field F is denoted GL(n. F ). As we have seen in theorem 23. In this case it is obvious that the matrix of the mapping is the n × n identity matrix In . one particular case is that we have the identity mapping f = id : V → V. the matrix A is invertible. The identity element is the identity matrix. GL(n. Every regular matrix is invertible. We have already seen in an exercise that matrix multiplication is associative.need. we must also have AC ∈ GL(n. .
A trivial example
For example. such that A is the matrix representing f with respect to the given basis of V. in the multiplication system of matrix algebra we must have B = A−1 . C −1 ∈ GL(n. So let f −1 be represented by the matrix B (again with respect to the same basis {v1 . Then we must have the matrix equation B · A = In . Then there exist A−1 . Theorem 25. So let A. That is. Theorem 24. Therefore. there is an isomorphism f : V → V. Proof. for all v ∈ V. according to the definition of GL(n. F ) have an inverse. By definition. F ). Thus the matrix for f with respect to this single basis is determined by the specifications
m
f (vj ) =
i=1
aij vi . all members of GL(n.
Proof. . . then the only common element of the eigenspaces is the zero vector 0. Thus if every vector in V is an eigenvector.
Also 0 = λp 0 = λp
n
ai vi . . Let λ1 . Definition. The set {v ∈ V : f (v) = λv} is called the eigenspace of λ with respect to the mapping f . ai = 0 for i < p. vn } is linearly independent. . . That is. λn be eigenvalues of the linear mapping f : V → V. Theorem 30. where n is the dimension of V. That is. for each i = 1. Then the set {v1 . . Therefore we have
n n n
0 = f (0) = f
i=1
ai vi
=
i=1
ai f (vi ) =
i=1
ai λi vi . where λi = λj for i = j. . vi = 0 and f (vi ) = λi vi . an . . Let a. representing the restriction of f to the subspace Ui . w ∈ V be in the eigenspace of λ. b ∈ F be arbitrary scalars. with a1 v1 + · · · + an vn = 0. which would imply that vp = 0. Assume further that as few of the ai as possible are non-zero. vn be eigenvectors to these eigenvalues. Each eigenspace is a subspace of V. That is. . not all zero. . . Let v1 . for otherwise we would have the equation 0 = ap vp . Theorem 31. . then we have the situation of theorem 29. the eigenspace is the set of all eigenvectors (and with the zero vector 0 included) with eigenvalue λ. for some k = p. .where each block Ai is a square matrix.
29
. Proof. and ap = 0. . n. . Let u. . Let ap be the first non-zero scalar. A special case is when the invariant subspace is an eigenspace. Assume to the contrary that there exist a1 . Then we have f (au + bw) = af (u) + bf (w) = aλu + bλw = λ(au + bw). contrary to the assumption that vp is an eigenvector.
Obviously if λ1 and λ2 are two different (λ1 = λ2 ) eigenvalues. . Proof. Assume that λ ∈ F is an eigenvalue of the mapping f : V → V.
i=1
Therefore
n n n
0 = 0 − 0 = λp
ai vi
i=1
−
ai λi vi =
i=1 i=1
ai (λp − λi )vi . Choose the basis to be a union of bases for each of the Ui . . . . One very particular case is that we have n different eigenvalues. Obviously some other ak is non-zero.
. . . remembering that λi = λj for i = j. 0 1 30
. . It is a simple matter to verify that the following matrices are the ones we are looking for. 0 1 Here. Also the diagonal elements are all 1 except for the elements at ii and jj. . λ1 0 · · · 0 0 λ2 · · · 0 A=. vn } form a basis for V... Thus we have found a new sum with fewer non-zero scalars than in the original sum with the ai s. This is a contradiction. ↑ ↓ 1 1 ←− 0 j−th row 1 . . 0 1 i−th row 0 −→ 1 1 . . . . Then we have 1 . if the matrix A is multiplied on the left by such an elementary matrix. 1 . Furthermore. 0 1 Si (a) = a 1 . . the matrix of the mapping is diagonal. . . which are zero. Si (a). then the given row operation on the matrix is performed. the given set of eigenvectors {v1 . in this particular case. 0 0 · · · λn
12
The Elementary Matrices
These are n × n matrices which we denote by Sij . . everything is zero except for the two elements at the positions ij and ji. . which have the value 1. and Sij (c). Therefore. yet all other non-zero scalar terms remain non-zero. . With respect to this basis..But. .. They are such that when any n × n matrix A is multiplied on the right by such an S. .... with the diagonal elements being the eigenvalues. then the given elementary column operation is performed on the matrix A. Sij = . . we see that the scalar term for vp is zero.
We also note that this is the method which can be used to obtain the value of the determinant function for the matrix. Namely. if A′ is obtained from A by multiplying a row with −1 then we have det(A′ ) = −det(A). Then we have: • If A′ is obtained by multiplying row i by the scalar a ∈ F . If A ∈ M(n × n. This proof also shows how we can go about programming a computer to calculate the inverse of an invertible matrix. we convert the given matrix into the identity matrix In . and let us say that A is transformed into the new matrix A′ by an elementary row operation. But first we must find out what the definition of determinants of matrices is!
−1 −1 −1 −1 A = S1 · · · · · · Sp Sp+1 · · · Sq . we keep multiplying together the elementary matrices which represent the respective row operations.)
Simple consequences of this definition
Let A ∈ M(n × n. This gives us the matrix equation Sq · · · Sp+1 Sp · · · S1 A = In or Since the inverse of each elementary matrix is itself elementary. Definition. (This is our row operation Sij (1).be realized by multiplication of A∗ on the right by some further set of elementary matrices: Sp+1 . then det(A′ ) = a · det(A). F ) is changed to the matrix A′ by multiplying all the elements in a single row with the scalar a ∈ F .) 3. . F ) → F is called a determinant function if it satisfies the following three conditions. det(In ) = 1. 32
. F ) be an arbitrary n × n matrix.
13
The Determinant
Let M(n × n. A mapping det : M(n × n. During this process. then det(A′ ) = det(A). Sq . 1. (This is our row operation Si (a). . we have thus expressed A as a product of elementary matrices. This is completely obvious! It is just part of the definition of "determinants". then det(A′ ) = a · det(A). If A′ is obtained from A by adding one row to a different row. through the process of Gauss elimination. . . we obtain the inverse matrix A−1 = Sq · · · Sp+1 Sp · · · S1 . where In is the identity matrix. 2. • Therefore. In the end. F ) be the set of all n × n matrices of elements of the field F .
Furthermore. Then we have det(A · B) = det(A) · det(B). That means. F ) then det(A−1 ) = (det(A))−1 . all of whose diagonal elements are 1. using these methods it is easy to see that the following theorem is true.2. Either way. In particular. that det(A · B) = 0. F ). F ). we then find a unique value for the determinant of the original matrix A. Thus we can combine both ideas into a single algorithm. If A ∈ GL(n. There is only one determinant function and it is uniquely given by our algorithm. Proof. If both A and B are regular. and thus the determinant of the original matrix was zero. indeed. so that the linear mapping which it represents splits into a direct sum of invariant subspaces (see theorem 29). which yields both the matrix inverse (if it exists). so the dimension of the image of the composition of the two mappings must also be less than n. If the bottom line of the matrix we obtain only consists of zeros. if det(A) = 1 then we also have det(A−1 ) = 1. Thus the dimension of the image is less than n. And on the other hand. then they are both in GL(n. F ). Remembering that A is regular if and only if A ∈ GL(n. In particular. Another simple corollary is the following. B ∈ M(n × n. Otherwise. that either det(A) = 0 or else det(B) = 0.
34
. Therefore. Therefore A · B must be singular. The set of all such matrices must then form a group. Since we know that det(In ) must be 1. then A · B is singular. Corollary. Assume that the matrix A is in block form. and the determinant. where S1 and S2 are elementary matrices. In particular. It suffices then to prove that det(S1 )det(S2 ) = det(S1 S2 ). But now we can transform this matrix into the identity matrix In by elementary row operations of the type Sij (c). Then det(A) is the product of the determinants of the blocks. then the determinant is zero. the theorem is true in this case. Note that in both this algorithm. Theorem 34. true. At least one of these mappings is singular. we have: Corollary. the method of Gaussian elimination was used. they can be written as products of elementary matrices. Let A. on the one hand. Theorem 35. as we have seen. the matrix has been transformed into an upper triangular matrix. But our arguments above show that this is. This algorithm also proves the following theorem. suitable for practical calculations in a computer. in this case det(A) = 0. F ) is regular if and only if det(A) = 0. as well as in the algorithm for finding the inverse of a regular matrix. a matrix A ∈ M(n × n. This can be seen by thinking about the linear mappings V → V which A and B represent. 3. If either A or B is singular.
Let G ⊂ Rn be some open region. and let f : G → R be a continuous function. of course. • The transformation formula for integrals in higher-dimensional spaces. we have vol(φ(Q)) = det(A).
Let f : V → V be a linear mapping. Now consider the matrix A. Then the integral f (x)d(n) x
G
has some particular value (assuming.Therefore we obtain Leibniz formula
n
det(A) =
σ∈Sn
sign(σ)
i=1
aiσ(i) . representing f with respect to some particular basis of V. Therefore we have the theorem: 38
. D(φ(x)) is the Jacobi matrix of φ at the point x. where A is the matrix representing φ with respect to the canonical coordinates for Rn . That means that (f − λid)(v) = 0. xn ) : 0 ≤ xi ≤ 1. we have det(A − λIn ) = 0. using the Leibniz formula) the polynomial in x P (x) = det(A − xIn ). If φ is a linear mapping.) • The characteristic polynomial. But here I will concentrate on only two special points. . Then the volume of Q.
Another way of looking at this is to take a "variable" x. . and then calculate (for example. therefore the mapping (f − λid) : V → V is singular. (A negative determinant — giving a negative volume — represents an orientation-reversing mapping.
This is a theorem which is usually dealt with in the Analysis III lecture. Now assume that we have a continuously differentiable injective mapping φ : G → Rn and a continuous function F : φ(G) → R. In particular. which we can denote by vol(Q) is simply 1. This formula reflects the geometric idea that the determinant measures the change of the volume of n-dimensional space under the mapping φ. that the integral converges). . On the other hand.
15
Why is the Determinant Important?
I am sure there are many points which could be advanced in answer to this question. Then we have the formula F (u)d(n) u =
φ(G) G
F (φ(x))|detD(φ(x))|d(n)x. ∀i}. Since λIn is the matrix representing the mapping λid.
Here. we must have that the difference A − λIn is a singular matrix.
This polynomial is called the characteristic polynomial for the matrix A. . then take Q ⊂ Rn to be the unit cube: Q = {(x1 . and let v be an eigenvector of f with f (v) = λv.
. Where does xn−1 occur in the Leibniz formula? Well. . So let us write the characteristic polynomial in the standard form P (x) = cn xn + cn−1 xn−1 + · · · + c1 x + c0 . F ) with A′ = C −1 AC. f is represented by some other matrix A′ . and cn = −1 if n is odd. So let us go one term lower and look at the coefficient cn−1 . under a similarity transformation of the matrix. We have det(A − xIn ) = (a11 − x)(a22 − x) · · · (ann − x)
n
+
σ=id
σ∈Sn
sign(σ)
i=1
aσ(i)i − xδσ(i)i
39
. This is not particularly interesting. which is similar to A. But we have det(A′ − xIn ) = = = = = Therefore we have: Theorem 42. each of the coefficients ci of the characteristic polynomial P (x) = cn xn + cn−1 xn−1 + · · · + c1 x + c0 remains unchanged after a similarity transformation of the matrix A. that is. as we have just seen.Theorem 41. The zeros of the characteristic polynomial of A are the eigenvalues of the linear mapping f : V → V which A represents. cn are all elements of our field F . there exists some C ∈ GL(n. det(C −1 AC − xC −1 In C) det(C −1 (A − xIn )C) det(C −1 )det(A − xIn )det(C) det(A − xIn ) P (x). Therefore cn = 1 if n is even. we see that the term xn can only occur in the product (a11 − x)(a22 − x) · · · (ann − x) = (−1)xn − (a11 + a22 + · · · + ann )xn−1 + · · · . . What is the coefficient cn ? Looking at the Leibniz formula. In particular. Now the matrix A represents the mapping f with respect to a particular choice of basis for the vector space V. there certainly is the term (−1)n−1 (a11 + a22 + · · · + ann )xn−1 . With respect to some other basis.
which comes from the product of the diagonal elements in the matrix A − xIn . . Obviously the degree of the polynomial is n for an n × n matrix A. That is. The characteristic polynomial is invariant under a change of basis. Do any other terms also involve the power xn−1 ? Let us look at Leibniz formula more carefully in this situation. The coefficients c0 .
But. with respect to the canonical basis of R2 . Then. . That is. namely +1 and −1. . This reflects the obvious geometric fact that a rotation through any angle other than 0 or π rotates any vector away from its original axis. looking at the characteristic polynomial. the spur of A) is the sum of the diagonal elements: tr(A) = a11 + a22 + · · · + ann . tr(A) remains unchanged under a similarity transformation. a11 · · · a1n .
That is to say.
An example
Let f : R2 → R2 be a rotation through the angle θ. with σ(i1 ) = i1 and also σ(i2 ) = i2 . sin θ cos θ
Therefore the characteristic polynomial of A is det 1 0 cos θ − sin θ −x 0 1 sin θ cos θ = det cos θ − x − sin θ sin θ cos θ − x
= x2 − 2x cos θ + 1. namely ±i. if λ ∈ R is an eigenvalue of f . So we conclude that the (n − 1)-st coefficient is cn−1 = (−1)n−1 (a11 + a22 + · · · + ann ). λ2 − 2λ cos θ + 1 = 0. Now if σ is a non-trivial permutation — not just the identity mapping — then obviously we must have two different numbers i1 and i2 .Here. we see that such a λ can only exist if | cos θ| = 1. this equation does have zeros. That is. . the matrix of f is A= cos θ − sin θ . In any case. then λ must be a zero of the characteristic polynomial. . θ = 0 or π. δij = 1 if i = j. namely x2 −2x cos θ + 1 in the previous example. Therefore we have the seemingly bizarre situation that a "complex" rotation through 40
. we see that in the case θ = ±π this reduces to x2 + 1.
16
Complex Numbers
On the other hand. The trace of A . Theorem 43. . be an n × n matrix. . Otherwise. Therefore we see that these further terms in the sum can only contribute at most n − 2 powers of x. δij = 0. And in the realm of the complex numbers C. Let A = . Definition. an1 · · · ann (in German. the two possible values of θ give the two possible eigenvalues for f . looking at the well-known formula for the roots of quadratic polynomials.
Then the absolute value of z is defined to be the (non-negative) real number |z| = x2 + y 2 . That is. declaring the physical world to be "paradoxical". and so we have an embedding R ⊂ C. . It is a simple exercise to show that C is a field. In any case. Then P has a zero in C. The main result — called (in German) the Hauptsatz der Algebra — is that C is an algebraically closed field. Those who are interested can find a proof in my other lecture notes on linear algebra. Therefore |z| = zz. Thus. the consequence is Theorem 45. in this form. . etc.a quarter of a circle has vectors which are mapped back onto themselves (multiplied by plus or minus the "imaginary" number i). What we have here is a purely algebraic result using the abstract mathematical construction of the complex numbers which. you might like to have a look at my lecture notes on the subject (in English). for each P (z) ∈ C[z] of degree n. No. If you are interested. for P (z) ∈ C[z] we can write P (z) = cn z n + · · · + c1 z + c0 . where x and y are elements of the real numbers R and i is an abstract symbol. I assume that you all know that the set of complex numbers C can be thought of as being the set of numbers of the form x + yi. the set of numbers of the form x + 0 · i can be identified simply with x. Furthermore. introduced as a "solution" to the equation x2 + 1 = 0. and a further complex number c. I will not be able to describe the proof of theorem 44 here. Every complex polynomial can be completely factored into linear factors. Let P (z) ∈ C[z] be an arbitrary polynomial with complex coefficients. . 41
. Let z = x + yi be some complex number. The rules of addition and multiplication in C are (x1 + y1 i) + (x2 + y2 i) = (x1 + x2 ) + (y1 + y2 )i and (x1 + y1 i) · (x2 + y2 i) = (x1 x2 − y1 y2 ) + (x1 y2 + x2 y1 )i. there exists some λ ∈ C with P (λ) = 0. let C[z] be the set of all polynomials with complex numbers as coefficients. where cj ∈ C. . has nothing to do with rotations of real physical space! So let us forget physical intuition and simply enjoy thinking about the artificial mathematical game of extending the system of real numbers to the complex numbers. n. The theory of complex numbers (Funktionentheorie in German) is an extremely interesting and pleasant subject. or look at any of the many books in the library with the title "Funktionentheorie". That is. Complex analysis is quite different from the real analysis which you are learning in the Analysis I and II lectures. for all j = 0. such that P (z) = c(λ1 − z) · · · (λn − z). λn . . . "beyond human understanding". Unfortunately. . there exist n complex numbers (perhaps not all different) λ1 . That is. owing to a lack of time in this summer semester. Thus i2 = −1. The complex conjugate of z √ is z = x − yi. . But there is no need for panic here! We need not follow the example of numerous famous physicists of the past. Then we have: Theorem 44 (Hauptsatz der Algebra).
rather than saying that v is the "length" of the vector v. We have already seen that the idea of a complex vector space defies our normal powers of geometric visualization. Also. That is. Therefore we apply our argument in turn to Q(z). So the consequence is: let V be a vector space over the field of complex numbers C. we will not simply confine ourselves to measurements of normal physical things on the earth. the properties of (finite dimensional) real and complex vector spaces. (The word "geometry" obviously has to do with the measurement of physical distances on the earth. Then every linear mapping f : V → V has at least one eigenvalue. a constant. and we say that v is the norm of v. Norms. Given P (z).
So now we have arrived at the subject matter which is usually taught in the second semester of the beginning lectures in mathematics — that is in Linear Algebra II — namely. the degree of R(z) is less than the degree of the divisor.) So let V be some finite dimensional vector space over R. spaces of functions — which are almost always infinite dimensional — are also very important in theoretical physics. But what is r? If we put λ1 into our equation. Then the length of v is defined to be the non-negative real number v = |a1 |2 + · · · + |an |2 . where = j=1 {e1 . . Finally now. R(z) must be a polynomial of degree zero.
42
. which is 1. However. In order to define this concept in a way which is suitable for further developments. we will start with the idea of a scalar product of vectors. For example. and in the end. namely (λ1 − z). again reducing the degree. . Therefore. Let v ∈ V be some vector in V. . We obtain P (z) = (λ1 − z) · Q(z) + R(z). we obtain our factorization into linear factors. such that P (λ1) = 0. where both Q(z) and R(z) are polynomials in C[z]. since V ∼ Rn . Then. or C. i. we will not always restrict things to finite dimensional vector spaces. and so P (z) = (λ1 − z)Q(z). etc. . Therefore r = 0. for all j.
Of course. we are talking about geometry. where Q(z) must be a polynomial of degree n−1. and aj ∈ R or C. R(z) = r ∈ C. about vector spaces which have a distance function. respectively.e. we use a new word. as these things always are. theorem 44 tells us that there exists some λ1 ∈ C.Proof. and thus at least one eigenvector. Let us therefore divide the polynomial P (z) by the polynomial (λ1 − z). That is. en } is the canonical basis for Rn or Cn . or Cn . we obtain 0 = P (λ1 ) = (λ1 − λ1 )Q(z) + r = 0 + r. we can write v = n aj ej .
17
Scalar Products.
1. v for all v ∈ V. Given a scalar product. Then we have | u. 44
. Furthermore. and 3. Indeed. and let v = v. .multiplication) v1 v2 · · · un ) . v . v = ut v = (u1 u2
uj vj . Remark. considering it to be a scalar product. then the norm of v ∈ V — with respect to this scalar product — is the non-negative real number v. A real vector space (that is. undefinable number of — dimensions. Definition.
j=1
It is easy to check that this gives a scalar product on Cn . the equality | u. Most physicists today prefer to imagine that physical space has 10. A complex vector space with scalar product is called a unitary vector space. v = More generally. Thus I shall dismiss the vector product from further discussion here. v | ≤ u · v for all u and v ∈ V. v | = u · v holds if. v} is linearly dependent. Now. vn
n
u. av = |a| v for all v ∈ V and for all a ∈ C. This inner product notation is often used in classical physics. the basic reason for making all these definitions is that we want to define the length — that is the norm — of the vectors in V. v = 0 ⇔ v = 0. we just have u · v = u. Let V be a Euclidean or a unitary vector space. Therefore it appears to be the case that the vector product might have gone out of fashion in contemporary physics. the set {u. mathematicians can imagine many other possible vector-space structures as well. Theorem 46 (Cauchy-Schwarz inequality). and only if. or even more — perhaps even a frothy. together with a scalar product is called a Euclidean vector space. 2. This particular scalar product is called the inner product. However the vector product of classical physics only makes sense in 3-dimensional space. Definition. one defines a norm-function on a vector space in the following way. over the field of the real numbers R). Thus. v1 + v2 ≤ v1 + v2 for all v1 . v2 ∈ V (the triangle inequality). Maxwell's equations also involve the "vector product" u × v. in particular in Maxwell's equations. A function · : V → R is called a norm on V if it satisfies the following conditions. One often writes u · v for the inner product. = . Let V be a vector space over C (and thus we automatically also include the case R ⊂ C as well). v .
. x vj . In either case we have a scalar product ·. . This follows because if m
0=
j=1
aj uj
then 0 = uk . x = vk .18
Orthonormal Bases
Our vector space V is now assumed to be either Euclidean. aj vk . then we have for each k. .
j=1
aj vj =
j=1
So now to the Gram-Schmidt process. . . vn }. But first. . and thus it has a basis {v1 . Proof. and furthermore • vj . for all j = k (that is. .
m
m
aj uj =
j=1 j=1
aj uk . As always. gradually changing it into an orthonormal basis. we must have ak = 0.
That is. if a non-zero vector v ∈ V is not normalized — that is. and let x ∈ V be arbitrary. it is defined over either the real numbers R. the coefficients of x. . and the inner product as our scalar product. most bases are not orthonormal. vj = 1. we see that it would be nice if we had • vj . F = R or C). we can successively alter the vectors in it.
46
. · : V × V → F (here. Let V have the orthonormal basis {v1 . This process is often called the Gram-Schmidt orthonormalization process. Unfortunately. are simply the scalar products with the respective basis vectors. . This is true for all the ak . vj = ak . . But this doesn't really matter. {v1 . its norm is not one — then it is easy to multiply it by a
9 Note that any orthogonal set of non-zero vectors {u1 . um } in V is linearly independent. For. Theorem 48. vn } is an orthonormal basis of V. uj = 0 if j = k. . uk
since uk . Then
n
x=
j=1
vj . . This follows simply because if x =
n n j=1 n
aj vj . Thus. we assume that V is finite dimensional. or else the complex numbers C. . for all j (that is. .9 That is to say. the basis vectors are an orthogonal set in V). we have the following theorem.
vk . uj = ak uk . . To begin with. 0 = uk . vk = 0. to show you why orthonormal bases are good. the basis vectors are normalized ). Thinking about the canonical basis for Rn or Cn . starting from any given basis. with respect to the orthonormal basis. or else unitary — that is. and otherwise it is not zero. vn }. .
changing it into a vector with norm one. we can write each vj as a linear combination of the uk 's. Let U ⊂ V be the subspace spanned by the first n − 1 basis vectors {v1 . . and as we have seen. v > 0 and we have v = v v = v v .
Thus the basis {u1 . vn uk . vn . it must be a basis. vn−1 } and {u1 . Also. For we have v. we simply multiply the vector by the inverse of its norm. v. this can be easily changed by taking the normalized vector un = u∗ n . u∗ } is orthogonal. .
10
47
. un−1 . . . we replace vn with the new vector11
n−1
u∗ = vn − n
uj . . and since the dimension is n. . adding in vn gives a new basis {u1 . but n n as we have seen. v v
In other words. . . v v = v. v > 0. . un−1 . . u∗ n
Since both {v1 . u∗ } is a basis of V. for k < n. this last vector. and furthermore assume that the Gram-Schmidt process can be constructed for any n−1 dimensional space. u∗ n
=
uk . Theorem 49. . . . The proof proceeds by constructing an orthonormal basis {u1 . vn = 0. vn − uk .scalar. we have n
n−1
uk . . . scalar multiplication with the inverse of the norm. . . un−1 . vn } for V. . it can be normalized by dividing by the norm. . Clearly10 . . Unfortunately. might disturb the nice orthonormal character of the other vectors. . vn −
uj . Therefore v. 11 A linearly independent set remains linearly independent if one of the vectors has some linear combination of the other vectors added on to it. . our assumption is that there exists an orthonormal basis {u1 . we use induction on the dimension. . . Since U is only n − 1 dimensional. To describe the construction. . vn − uk . vn uj .
j=1
Thus the new set {u1 . vn } spans V. . . v v = = 1. arbitrary basis {v1 . Therefore {u1 . . un−1 } are bases for U. Proof. (That is. . . . . un−1 } for U. Perhaps u∗ is not normalized. vn }.) So now assume that n ≥ 2. . un } from a given. . . uj
j=1
uk . vn−1 }. vn uj
j=1 n−1
= =
uj . . If n = 1 then there is almost nothing to prove. un−1 . Therefore. . . . . Every finite dimensional vector space V which has a scalar product has an orthonormal basis. Any non-zero vector is a basis for V. n.
Physicists call this Minkowski space. in our normal 3-dimensional space of physical reality. A linear map∗ ∗ ping f : M 4 → M 4 is called a Lorentz transformation if. for f (v) = (t∗ . u . v . f (v) = f (v).
12
48
. In either case. On the other hand. • The special orthogonal group SO(n): This is the subgroup of O(n). v v we have
2 2 ∗ ∗ • −(t∗ )2 + (x∗ )2 + (yv )2 + (zv )2 = −t2 + x2 + yv + zv . the subgroup of U(n) with determinant +1. containing all orthogonal mappings whose matrices have determinant +1. • The special unitary group SU(n): Again. the matrices for such mappings are symmetric in the real case. and a typical point is v = (tv . Since v is an eigenvector. and Hermitian in the complex case. any rotating object — for example the Earth rotating in space — has an axis of rotation. for all u. for all v ∈ M 4 . xv . Examples of Hermitian matrices are the Pauli spin-matrices: 0 1 . 0 −1
We also have the Lorentz group. for every unitary mapping Cn → Cn . f (v) = λv. On the other hand. To see this. 1 0 0 −i . x∗ . zv ). "most" orthogonal matrices cannot be diagonalized. all eigenvalues — if they exist — must have absolute value 1. we must have |λ| = 1. v = f (u). and also v v v v
For example. which is an eigenvector. We think of this as being all possible rotations and inversions (Spiegelungen) of n-dimensional Euclidean space. and thus v = 0. We will prove that all unitary matrices can be diagonalized. v ∈ Cn . u. Note that for orthogonal. yv . the matrices can be diagonalized. As we will see. i 0 1 0 .12 • The self-adjoint mappings f (of Rn → Rn or Cn → Cn ) are such that u. which they often denote by M 4 . v = |λ|2 v. v = f (v). f (v) . or unitary mappings. Let us imagine that physical space is R4 . λv = λλ v. • The unitary group U(n): The analog of O(n). v = f (u). That is. let v be an eigenvector with eigenvalue λ. for all u. v in Rn or Cn . as we have already seen in the case of simple rotations of 2-dimensional space. has at least one eigenvector. where the vector space is ndimensional complex space Cn . which is important in the Theory of Relativity. there exists a basis consisting of eigenvectors. v ∈ Rn . Then we have v. for all u. f (v) . we can prove that every orthogonal mapping Rn → Rn . yv . That is.19
Some "Classical Groups" Often Seen in Physics
• The orthogonal group O(n): This is the set of all linear mappings f : Rn → Rn such that u. zv ). respectively. where n is an odd number.
. 0 0 · · · λn
u=
j=1
wj . Let {w1 . .) Therefore. . then we have
n n
The complete answer to this question is a bit too complicated for me to explain to you in the short time we have in this semester. . wm } be some orthonormal basis for the vector space W. Assuming the GramSchmidt process has been used. . . . Now we have seen that not all orthogonal matrices can be diagonalized. and also all Hermitian matrices can be diagonalized. Proof. wk = 0. . vn } of the vector space V. we may assume that this is an orthonormal basis. consisting entirely of eigenvectors. Now clearly. . . . . The claim is then that {wm+1 . wn } of V. . . (Think about the rotations of R2 . Let W ⊂ V be a subspace of V. That is. . . So the idea that the matrix can be diagonalized is that it is similar to a diagonal matrix. Then W⊥ is called the perpendicular space to W. 51
. . V is a vector space over the complex numbers C.
since wj . . wn } is a linearly independent. we can prove that all unitary.21
Which Matrices can be Diagonalized?
But this means that there must be a basis for V. It all has to do with a thing called the "minimal polynomial". and furthermore W ∩ W⊥ = {0}. so it is a basis. u w j . . . . wm . . such that S −1 MS is diagonal. . . . We begin with a definition. This can be extended to a basis {w1 . for j = k. . If u ∈ W⊥ is some arbitrary vector in W⊥ . (Remember. a matrix M is only a representation of a linear mapping f : V → V with respect to a given basis {v1 . w = 0. orthonormal set which generates W⊥ . . there exists another matrix S. It is a rather trivial matter to verify that W⊥ is itself a subspace of V. . Definition.. u = 0 if j ≤ m. . V = W ⊕ W⊥ . wm+1 . . . The vector space V will be assumed to have a scalar product associated with it. In fact. we have that {wm+1 .) On the other hand. λ1 0 · · · 0 0 λ2 · · · 0 S −1 MS = . Let W⊥ = {v ∈ V : v. In this section we will consider complex vector spaces — that is. since wj . . . . ∀w ∈ W}. . . wn } ⊂ W⊥ . And so we have V = W ⊕ W⊥ . u ∈ W⊥ . . wn } is a basis for W⊥ . u w j =
j=m+1
wj . we have: Theorem 53. {wm+1 . Of course. and the bases we consider will be orthonormal. . .
But this is easy. Proof. if u ∈ W⊥ is some arbitrary vector. the dimension of the vector space V. Then we have f (u). The rest of the proof follows as before. . This is similar to the last one. we have ajj = ajj for all j. then obviously there is nothing to prove. that even in the case of a unitary matrix. where f (vn ) = λvn . with f (vn ) = λvn . we may assume that vn = 1. Therefore we must have f (u). adding in the last vector vn . hence there exists some eigenvalue λ. according to the fundamental theorem of algebra. . . we must have |λ| = 1. If the dimension of V is zero or one. That is to say. the diagonal elements are all real numbers. But we have already seen that for an eigenvalue λ of a unitary mapping. . . That is. . then f (u) ∈ W⊥ as well. Note furthermore. which has norm equal to one. then we are finished. vn = u. the characteristic polynomial of f has a zero. Let u ∈ W⊥ be given. Now. vn = f (u).Theorem 54. f (vn ) = u. Theorem 55. . vn } for V. That is. Let W⊥ be the perpendicular subspace. we have an orthonormal basis of eigenvectors {v1 . In the particular case where we have only real numbers (which of course are a subset of the complex numbers). All real symmetric matrices can be diagonalized. Let W ⊂ V be the 1-dimensional subspace generated by the vector vn . Therefore. restricted to W⊥ . λvn = λ u. W⊥ is invariant under f . the symmetry condition. So let us assume that the dimension n is at least two. This follows since λ f (u). It is only necessary to show that. Again. . λvn = f (u). . Again. and an eigenvector vn of f . and using the inductive hypothesis. we assume that the theorem is true for spaces of dimension less than n. Then W⊥ is an n − 1 dimensional subspace. and we prove things by induction on the number n. We have a self-adjoint mapping f : V → V. All Hermitian matrices can be diagonalized. By dividing by the norm of vn if necessary. we obtain an orthonormal basis of eigenvectors {v1 . . again. we observe that the characteristic polynomial of f must have a zero. If n is zero or one. vn = 0. So we can consider f . which is then an eigenvalue for f . But these are the eigenvalues. vn−1 } for W⊥ . vn = λ · 0 = 0. vn = 0. implies that on the diagonal. That is. . Let f : V → V be a unitary mapping (V is a vector space over the complex numbers C). Proof. then we have a symmetric matrix. f (vn ) = u. We have that W⊥ is invariant under f . vn } for V consisting of eigenvectors under f . Then there exists an orthonormal basis {v1 . we use induction on n. namely ajk = akj . Again take W to be the one dimensional subspace of V generated by vn . . Therefore we have: 52
. λ say. the matrix of f with respect to this basis is a diagonal matrix. Corollary. So there must be some non-zero vector vn ∈ V. Therefore we assume that n ≥ 2.
e. Let W⊥ be the perpendicular space to W. w ∈ W⊥ means that w. v = 0. with α = ±1. i. 53
. or • it isn't. it cannot be diagonalized. it can be brought into the following form by means of similarity transformations. B can be diagonalized. In general.) But it is easy to see that W⊥ is also invarient under f . we obtain that the original matrix is similar to the matrix α 0 .. (Remember that the eigenvalues of orthogonal mappings have absolute value 1. (That is. for some λ ∈ R. . start by imagining that A represents the orthogonal mapping f : R → Rn with respect to the canonical basis of Rn . f (v) = α−1 w. R1 . But on the other hand.
n
This matrix represents another linear mapping. The first case is easy. That is. Then f (w).Corollary. there exists some vector v ∈ Rn with g(v) = λg(v). But. again with respect to the canonical basis of Rn . In particular. We now proceed by induction on the number n.. and we have At = A−1 . sin θ ∓ cos θ To see this. according to the inductive hypothesis. ±1 . this is just the set of all scalar multiples of v. call it g : Rn → Rn . v = α−1 · 0 = 0. can be transformed into the required form. 0 ±1 ′′ A = . Let W ⊂ V be simply W = [v]. it consists of real numbers. Now consider the symmetric matrix B = A + At = A + A−1 . . starting with v (which we can assume has been normalized). as we have just seen. a symmetric or a Hermitian matrix — are all real numbers. which.
Orthogonal matrices revisited
Let A be an n × n orthogonal matrix. αv = α−1 f (w). There are two cases to consider: • v is also an eigenvector for f . 0 Rp where each Rj is a 2 × 2 block of the form cos θ ± sin θ . by changing the basis of Rn to being an orthonormal basis. Thus. 0 A∗ where A∗ is an (n − 1) × (n − 1) orthogonal matrix.) Now take w ∈ W⊥ . The eigenvalues of a self-adjoint matrix — that is. v = α−1 f (w). This follows by observing first of all that f (v) = αv.
then fi (ej ) = 1. f (v)]. Therefore. v = 0.
Examples
• Let V = Rn . otherwise. for i = 1. v = 0. By our inductive hypothesis. if λ = 0 then we have v = λf (v) − f (f (v)) so that f (w). So this time. Therefore we have shown that V = W ⊕ W⊥ . if i = j. and • λ = 0. if ej is the j-th canonical basis vector. and as we will see. we know it is an eigenvector of g. we have f (w). Now we have another two cases to consider: • λ = 0. In particular.If v is not an eigenvector of f . still. we are back in the simple situation of an orthogonal mapping R2 → R2 .
22
Dual Spaces
Again let V be a vector space over a field F (and. v = f (w). λf (v) − f (f (v)) = λ f (w). although its not really necessary here. (Remember that w ∈ W⊥ . these dual vectors form a basis for the dual space. f (v) = w. f (v) = 0. This is a 2-dimensional subspace of V. let W = [v. So if λ = 0 then we have f (f (v)) = −v. g(v) = λv = f (v) + f −1 (v). f (f (v)) . On the other hand. n. v = f (w). f (f (v)) = λf (v) − v. then. and the matrix for this has the form of one of our 2 × 2 blocks. −f (f (v)) = − w. Definition. again taking w ∈ W⊥ . consider W⊥ . f (v) = w. That is. 0. Finally. The dual space to V is the set of all linear mappings f : V → F .) Of course we also have f (w). We have V = W ⊕ W⊥ . f (v) = 0. . . and furthermore g = f + f −1 . That is. and we have seen that both of these scalar products are zero.
54
. we continue to take F = R or C). As far as W is concerned. f (v) − f (w). We denote the dual space by V∗. . so that w. Again. we again have f (w). . So we must show that W⊥ is invarient under f . Then let fi be the projection onto the i-th coordinat. where both of these subspaces are invariant under the orthogonal mapping f . there is an orthonormal basis for f restricted to the n − 2 dimensional subspace W⊥ such that the matrix has the required form.
So each fi is a member of V∗ .
On the other hand. look at the diagram. giving us the mappings s : V → V∗ and t : W → W∗. So it is obvious that f ∗ (φ) : V → C is a linear mapping. for all possible φ ∈ V∗? Let us say that there exists a v ∈ V such that Φ(φ) = φ(v). So we see that in the case V = W. u . On the other hand. For let Φ ∈ (V∗)∗ . Then we can define a linear mapping f ∗ : W∗ → V∗ in a natural way as follows. V∗ → C. This is the adjoint mapping to f . dim(V∗) = dim(V). this isomorphism doesn't seem to be very "natural". do we have a unique v ∈ V such that Φ(φ) = φ(v). we also have φ(v) being some complex number. So we can draw a little "diagram" to describe the situation. if we define Φv to be Φ(φ) = φ(v). for all φ ∈ V∗ .
Dual mappings
Let V and W be two vector spaces (where we again assume that the field is C). In fact. so we can go around the diagram. such that φv (u) = v. for all φ ∈ V∗ ? At least in the case where V is finite dimensional. for each φ ∈ V∗ we have Φ(φ) being some complex number. we have that a self-adjoint mapping f : V → V is such that f adj = f . namely that u. for each φ ∈ V∗ . Assume that we have a linear mapping f : V → W. More specifically. Does this correspond with our earlier definition. That means.Corollary. Corollary. using the mapping f adj = s−1 ◦ f ∗ ◦ t : W → V. (Perhaps this is a slightly mind-boggling concept at first sight!) We imagine that "really" we just have (V∗)∗ = V. in the sense that both always give the same complex numbers. It is defined in terms of some specific basis of V.13 On the other hand. f (v) = f (u). V −→ s↓
f∗ f
The mappings s and t are isomorphisms. let us look at the dual space of the dual space (V∗)∗ . for all u ∈ V. What if V is not finite dimensional so that we have no basis to work with? For this reason. then there is a "natural" mapping V → V∗ . For each φ ∈ W∗ . Can we uniquely identify each V ∈ V with some Φ ∈ (V∗)∗ . we can affirm that it is true by looking at the dual basis. v for all u and v ∈ V? To answer this question. we do not think of V and V∗ as being "really" just the same vector space. such that v → φv for each v ∈ V. given some arbitrary Φ ∈ (V∗ )∗ . which now has the form f V −→ V s↓ ↓s V∗ ←− V∗
f∗
V∗ ←− W∗
W ↓t
In case we have a scalar product. then we certainly have a linear mapping. for each V ∈ V. let f ∗ (φ) = φ ◦ f .
13
56
.
But somehow. we have an isomorphism V → V∗ . Now assume that V and W have scalar products. where v → φv .
That is. perhaps you will begin to ask yourselves whether it really makes so much sense after all. Albert Einstein wrote "One can give good reasons why reality cannot at all be represented by a continuous field. But that is not to say that there is nothing more that you have to know about the subject. That is. Instead. u and (f ∗ ◦s)(v)(u) = v. functional analysis. we can equally say that the condition is that f ∗ ◦s = s◦f . taking us away from the simple "linear" mathematics which we have seen in this semester. linear mappings and dual mappings. could it be that physics has been loosing touch with reality. One speaks of "covariant" and "contravariant" tensors. we imagine that space is curved. After a few years of learning the mathematics of contemporary theoretical physics. u = v. This does not seem to be in accordance with a continuum theory. the theory of complex functions.
23
The End
This is the end of the semester.where s(v) ∈ V∗ is such that s(v)(u) = v. which are combinations of linear mappings and dual mappings. proceeding to the general theory of relativity. remembering that this is an element of V∗ . f (u) . u . we no longer have a linear (that is. we see that this means for all u ∈ V. Can it be that the physical world is best described by using all of the latest techniques which pure mathematicians happen to have been playing around with in the last few years — in algebraic topology. f (u) for all v and u ∈ V. and thus the end of what I have to say about "linear algebra in physics" here. But nobody knows how to obtain the basis of such a theory. these tensors are used to describe differential geometry. But (s◦f (v))(u) = f (v). leading to more and more complicated mathematical constructions. and so on and so forth? Or. a vector) space. From the quantum phenomena it apears to follow with certainty that a finite system of finite energy can be completely described by a finite set of numbers (quantum numbers). the condition f adj = f becomes s−1 ◦ f ∗ ◦ s = f . And so it goes on. and in order to describe this curvature. However. when studying the theory of relativity you will encounter tensors. as expected. making constructions similar to the theory of epicycles of the medieval period. on the other hand. But then. (s ◦ f (v))(u) = (f ∗ ◦ s)(v)(u). whose conclusions can never be verified using practical experiments in the real world? In his book "The Meaning of Relativity"." 57
. for all u ∈ V. Since s is an isomorphism. Now f adj = s−1 ◦f ∗ ◦s. that is. For example. we define a thing called the tangent vector space which you can think of as being a kind of linear approximation to the spacial structure near a given point. So let v be some arbitrary vector in V. and must lead to an attempt to find a purely algebraic theory for the description of reality. We have s ◦ f (v) = f ∗ ◦ s(v). Therefore we have f (v). | 677.169 | 1 |
ISBN-10: 0321816196
ISBN-13: 9780321816191
Edition: 4 Friendly Introduction to Number Theory, Fourth Edition is designed to introduce readers to the overall themes and methodology of mathematics through the detailed study of one particular facet number theory. Starting with nothing more than basic high school algebra, readers are gradually led to the point of actively performing mathematical research while getting a glimpse of current mathematical frontiers. The writing is appropriate for the undergraduate audience and includes many numerical examples, which are analyzed for patterns and used to make conjectures. Emphasis is on the methods used for proving theorems rather than on specific | 677.169 | 1 |
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1 MB|3 (including 1 cover, 1 Maze, & 1 answer key)
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This maze assess the understanding of the relationships between the subsets of real numbers.
There are 15 questions where a different polynomial is given. From start to end, the student will be able to answer 11 questions out of the 15 provided to get to the end of the mazeThis maze focuses only and only on "Classifying Polynomials by degree and number of terms". More mazes related to polynomials are on the way. Stay tuned | 677.169 | 1 |
Common Core Editions are now available for HMH Fuse 006x10 divided by 12x0 3x0, helps students in need of basic skill review and provides enrichment for those ready to
Chemistry for christian schools 2nd edition homework help
Online and improves learning resources to encyclopedia entries and guarantees to help you over 90 of a christmas season to a student. Denison and Company, inc, george Saad Mathematics
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Do you have math anxiety? Do you feel a bit unsure about your arithmetic or algebra skills? Do you feel like you could use the practice to bolster your confidence? Do you have gaps in your developmental math skills?
The Spinelli Center is offering an eight-week series of workshops for students to change how they view their abilities and to improve their confidence through conversation and collaborative work in a safe, welcoming environment.
We encourage you to attend all eight sessions, but no registration is required.
All sessions are held in the Spinelli Center on the second level of Neilson Library. | 677.169 | 1 |
Grades 9-12 MHS
High School Mathematics 9-12
In high school, the content standards are not organized by grade level; they are separated into "conceptual categories." The high school content standards presented by conceptual categories specify the mathematics that all students should study in order to be college and career ready. | 677.169 | 1 |
Category Archives: mathOne of my private MCR3U classes finished on Wednesday, and another will be done next Wednesday. A third is running 3 times a week, so will take until February to finish. This will allow me to re-think some of my … Continue reading →
We got through trig and, I'll be honest, it was a bit of a rough ride. Looking back, I perhaps should have split it into two units instead of one long one. The students found this material the most challenging … Continue reading →
We finished the Exponential Functions unit, and I think the students enjoyed it, especially the last part where we were modelling real-life situations. I used Fry's Bank Account 3-Act Math, and it was a hit! In stark contrast, we have … Continue reading →
This past week brought the half-way point of two of my one-on-one Grade 11 Math classes, and we completed the quadratics unit. The unit start off strong, but with many outside factors for both students made the end of the … Continue reading →
There's only one lesson in the Quadratic Functions unit – units sure go by fast when math classes are 2.5 hours a day! Even though I cover about twice as much as I normally would in a 70-minute class, it seems … Continue reading → | 677.169 | 1 |
Polynomials
In this learning exercise, students solve and complete 32 various types of problems. Included are problems on simplifying polynomials, factoring, graphing equations by plotting points, and simplifying expressions using the rules of exponents. This is a good learning exercise for a review of basic algebra. | 677.169 | 1 |
High School Algebra Homework Answer Key
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Straight Lines. Math Tools:: Math Links Mathematics is commonly called Math in the US and Maths in the UK.Free math lessons, formulas, calculators and homework help, in calculus,. | 677.169 | 1 |
Math 155 - Worksheet 3. Limits
In this limits instructional activity, students use the intermediate value theorem to explore intervals. They sketch a graph of the function and determine the correct property of limits to evaluate the limit. This two-page instructional activity contains five problems. | 677.169 | 1 |
Foundations of Higher Mathematics Exploration and Proof
ISBN-10: 0201125870
ISBN-13: 9780201125870 Foundations of Higher Mathematics: Exploration and Proof is the ideal text to bridge the crucial gap between the standard calculus sequence and upper division mathematics courses. The book takes a fresh approach to the subject: it asks students to explore mathematical principles on their own and challenges them to think like mathematicians. Two unique featuresan exploration approach to mathematics and an intuitive and integrated presentation of logic based on predicate calculusdistinguish the book from the competition. Both features enable students to own the mathematics they're working on. As a result, your students develop a stronger motivation to tackle upper-level courses and gain a deeper understanding of concepts | 677.169 | 1 |
Test Generalizations Of Polygons
... contains important information and a detailed explanation about test generalizations of polygons ... which is also related with , name practice 10-8 problem solving: make and test generalizations, comments, working with polygons - vdoe :: virginia.
Make sense of problems and persevere in solving them. ... solution. In first grade, students realize that doing mathematics involves solving problems and discussing . Mathematically proficient students see repeated calculations and look for generalizations and How could you test your solution to see if it answers the.
ebook.dexcargas.com is a PDF Ebook search engine and unrelated to Adobe System Inc. No pdf files hosted in Our server. All trademarks and copyrights on this website are property of their respective owners. | 677.169 | 1 |
Intro to Sequences and Series
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551 KB|13 pages
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This Algebra II lesson is an introduction to sequences and series. It involves having students intuitively look at patterns of numbers and write rules, use rules to write sequences, graph sequences, and find small sums of sequences. | 677.169 | 1 |
Logarithms
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Data science courses contain math—no avoiding that! This course is designed to teach learners the basic math you will need in order to be successful in almost any data science math course and was created for learners who have basic math skills but may not have taken algebra or pre-calculus. Data Science Math Skills introduces the core math that data science is built upon, with no extra complexity, introducing unfamiliar ideas and math symbols one-at-a-time | 677.169 | 1 |
3D Graph OpenGL 3D graph is program for drawing mathematical equations in form of 3D graph. You can write whatever equation you like and use variables x and y as you likeMath Practice Maths Practice, Maths Workouts with addition, subtraction, division, multiplication, Tables up to 20,
You can practice maths and increase speed without pen and paper | 677.169 | 1 |
Succinct and understandable, this book is a step-by-step guide to the mathematics and construction of Electrical Load Forecasting models. Written by one of the world's foremost experts on the subject, Short and Long Term Electrical Load Forecasting provides a brief discussion of algorithms, there advantages and disadvantages and when they are best utilized. Supported by an online computer program, this book online arrangement allows readers construct, validate, and run short and long term models. The book begins with a brief discussion algorithm, there advantages and disadvantages and when they are best utilized. This is followed by a clear and rigorous exposition of the statistical techniques and algorithms such as regression, neural networks, fuzzy logic, and expert systems. In this book, readers find reliable and easy-to-use techniques designed to improve their forecasting techniques and construct more accurate models. The book begins with a good description of the basic theory and models needed to truly understand how the models are prepared so that they are not just blindly plugging and chugging numbers.
Step-by-step guide to model construction
Construct, verify, and run short and long term models
Accurately evaluate load shape and pricing
Creat regional specific electrical load models
"synopsis" may belong to another edition of this title.
Product Description:
Succinct and understandable, this book is a step-by-step guide to the mathematics and construction of electrical load forecasting models. Written by one of the world's foremost experts on the subject, Electrical Load Forecasting provides a brief discussion of algorithms, their advantages and disadvantages and when they are best utilized. The book begins with a good description of the basic theory and models needed to truly understand how the models are prepared so that they are not just blindly plugging and chugging numbers. This is followed by a clear and rigorous exposition of the statistical techniques and algorithms such as regression, neural networks, fuzzy logic, and expert systems. The book is also supported by an online computer program that allows readers to construct, validate, and run short and long term models. | 677.169 | 1 |
INTRODUCTION This Handbook aims to provide the general public – parents, students, researchers, and other stakeholders – an overview of the Mathematics program at the secondary level. Those in education, however, may use it as a reference for implementing the 2002 secondary education curriculum, or as a source document to inform policy and guide practice.
For quick reference, the Handbook is outlined as follows:
* The description defines the focus and the emphasis of the learning area as well as the language of instruction used. * The unit credit indicates the number of units assigned to a learning area computed on a 40-minute per unit credit basis and which shall be used to evaluate a student's promotion to the next year level.
* The time allotment specifies the number of minutes allocated to a learning area on a daily (or weekly, as the case may be) basis.
* The expectancies refer to the general competencies that the learners are expected to demonstrate at the end of each year level.
* The scope and sequence outlines the content, or the coverage of the learning area in terms of concepts or themes, as the case may be.
* The suggested strategies are those that are typically employed to develop the content, build skills, and integrate learning. * The materials include those that have been approved for classroom use. The application of information and communication technology is encouraged, where available. * The grading system specifies how learning outcomes shall be evaluated and the aspects of student performance which shall be rated.
* The learning competencies are the knowledge, skills, attitudes and values that the students are expected to develop or acquire during the teaching-learning situations. * Lastly, sample lesson plans are provided to illustrate the mode of integration, where appropriate, the application of life skills and higher order thinking skills, the valuing process and the differentiated activities to address the learning needs of students.
The Handbook is designed as a practical guide and is not intended to structure the operationalization of the curriculum or impose restrictions on how the curriculum shall be implemented. Decisions on how best to teach and how learning outcomes can be achieved most successfully rest with the school principals and teachers. They know the direction they need to take and how best to get there.
DESCRIPTION First Year is Elementary Algebra. It deals with life situations and problems involving measurement, real number system, algebraic expressions, first degree equations and inequalities in one variable, linear equations in two variables, special products and factoring.
Second Year is Intermediate Algebra. It deals with systems of linear equations and inequalities, quadratic equations, rational algebraic expressions, variation, integral exponents, radical expressions, and searching for patterns in sequences (arithmetic, geometric, etc) as applied in real-life situations.
Third Year is Geometry. It deals with the practical application to life of the geometry of shape and size, geometric relations, triangle congruence, properties of quadrilaterals, similarity, circles, and plane coordinate geometry.
Fourth Year is still the existing integrated ( algebra, geometry, statistics and a unit of trigonometry) spiral mathematics but in school year 2003-2004 the graduating students have the option to take up either Business Mathematics and Statistics or Trigonometry and Advanced Algebra.
UNIT CREDIT
Mathematics in each year level shall be given 1.5 units each.
TIME ALLOTMENT
The daily time allotment for Mathematics in all year levels is 60 minutes or 300 minutes weekly...
...Zero in Mathematics
Zero as a number is incredibly tricky to deal with. Though zero provides us with some useful mathematical tools, such as calculus, it presents some problems that if approached incorrectly, lead to a breakdown of mathematics as we know it.
Adding, subtracting and multiplying by zero are straightforward.
If c is a real number,
c+0=c
c-0=c
c x 0=0
These facts are widely known and regarded to hold true in every situation.
However, division by zero is a far more complicated matter. With most divisions, for example,
10/5=2
We can infer that
2 x 5=10
But if we try to do this with zero,
10/0=a
0 x a=10
Can you think of a number that, when multiplied by 0, equals 10? There is no such number that we have ever encountered that will satisfy this equation.
Another example will emphasise the mysteriousness of dividing by zero.
One may assume that
(c x 0)∕0=c
The zeroes should cancel, as would be done with any other number. But since we know that
c x 0=0
it follows that
(c x 0)/0=0/0=c
This does not seem to make sense. This also means that
1=0/0=2
1=2
since 1 and 2 are both real numbers. Actually, this means that 0/0 is equal to every real number!
In effect, there is no real answer to a division by zero. It cannot be done.
In fact, if we could divide by zero, it would be possible to prove anything that we could dream of. For example, imagine a student trying to prove to his teacher that he...
...Introduction
Mathematics is an indispensable subject of study. It plays an important role in forming the basis of all other sciences which deal with the material substance of space and time.
What is Mathematics?
Mathematics may be described as the fundamental science. It may be broadly described as the science of space, time and number. The universe exists in space and time, and is constituted of units of matter. To calculate the extension or composition of matter in space and time and to compute the units that make up the total mass of the material universe is the object of Mathematics. For the space-time quantum is everywhere full of matter and we have to know matter mathematically in the first instance.
Importance of Mathematics
Knowledge of Mathematics is absolutely necessary for the study of the physical sciences.
Computation and calculation are the bases of all studies that deal with matter in any form.
Even the physician who has to study biological cells and bacilli need to have a knowledge of Mathematics, if he means to reduce the margin of error which alone can make his diagnosis dependable.
To the mechanic and the engineer it is a constant guide and help, and without exact knowledge of Mathematics, they cannot proceed one step in coming to grips with any complicated problem.
Be it the airplane or the atom bomb,...
...Why study Mathematics?
The main reason for studying mathematics to an advanced level is that it is interesting and enjoyable. People like its challenge, its clarity, and the fact that you know when you are right. The solution of a problem has an excitement and a satisfaction. You will find all these aspects in a university degree course.
You should also be aware of the wide importance of Mathematics, and the way in which it is advancing at a spectacular rate. Mathematics is about pattern and structure; it is about logical analysis, deduction, calculation within these patterns and structures. When patterns are found, often in widely different areas of science and technology, the mathematics of these patterns can be used to explain and control natural happenings and situations. Mathematics has a pervasive influence on our everyday lives, and contributes to the wealth of the country.
The importance of mathematics
The everyday use of arithmetic and the display of information by means of graphs, are an everyday commonplace. These are the elementary aspects of mathematics. Advanced mathematics is widely used, but often in an unseen and unadvertised way.
• The mathematics of error-correcting codes is applied to CD players and to computers.
• The stunning pictures of far away planets sent by Voyager II could not have had their...
...-------------------------------------------------
Set (mathematics)
From Wikipedia, the free encyclopedia
This article is about what mathematicians call "intuitive" or "naive" set theory. For a more detailed account, see Naive set theory. For a rigorous modern axiomatic treatment of sets, see Set theory.
An example of a Venn diagram
The intersection of two sets is made up with the objects contained in both sets
In mathematics, a set is a collection of distinct objects, considered as an object in its own right. For example, the numbers 2, 4, and 6 are distinct objects when considered separately, but when they are considered collectively they form a single set of size three, written {2,4,6}. Sets are one of the most fundamental concepts in mathematics. Developed at the end of the 19th century, set theory is now a ubiquitous part of mathematics, and can be used as a foundation from which nearly all of mathematics can be derived. In mathematics education, elementary topics such as Venn diagrams are taught at a young age, while more advanced concepts are taught as part of a university degree.
-------------------------------------------------
Definition[edit]
A set is a well defined collection of objects. The objects that make up a set (also known as the elements or members of a set) can be anything: numbers, people, letters of the alphabet, other sets, and so on. Georg Cantor, the...
...Theory of Knowledge
Éanna OBoyle
ToK Mathematics
"... what the ordinary person in the street regards as mathematics is usually nothing more than the operations of counting with perhaps a little geometry thrown in for good measure. This is why banking or accountancy or architecture is regarded as a suitable profession for someone who is 'good at figures'. Indeed, this popular view of what mathematics is, and what is required to be good at it, is extremely prevalent; yet it would be laughed at by most professional mathematicians, some of whom rather like to boast of their ineptitude when it comes to totalling a column of numbers....Yet ... it is not the mathematics of the accountant that is of most interest. Rather, it is ... abstract structures and everyday intuition and experience" (p.173, Barrow).
2.1 Mathematical Propositions
2.1.1 Mathematics consist of A Priori Propositions (theorems)
We know mathematical propositions (or theorems) to be true independently of any particular experiences. No one ever checks empirically that, for example, 364.112 + 112.364 = 476.476 by counting objects of those numbers separately, adding them together, and then counting the result. The techical term to describe this independence of experiences is to say that the propositions are a priori. Therefore we say that mathematical propositions are a priori propositions.
2.1.2 Universality
When mathematical propositions...
...up the pillar and ran away. The man never saw him again.
Life without mathematics
Do any of us realize the importance of maths in our daily life? This is a subject that is applied to every field and profession. Without the application of maths, no field or profession is complete. To help us realize this why don't we imagine a world without maths?
Imagine living your days without a watch and a calendar. Both the watch and the calendar use numbers, the most basic and important of mathematic characters. How would you know the time of the day? Wouldn't you miss your own birthday without a calendar?
Consider this, you go to a shop to buy something but since this is a world without maths, you don't know what money is, you don't know measurements. So what do you do?
Whether it is a Zoologist assessing the number of animal species on earth or a doctor checking your heartbeat they have to know how to count. Without mathematics an engineer cannot build a bridge. A quantity checker chemist cannot prepare medicines if he cannot accurately measure the quantity of each chemical.
We wouldn't have had markets and businesses without math as the world of trade runs on money. And as a country's development depends heavily on its economic growth, wouldn't that be a problem?
There wouldn't be any more advancements of technology as each sector of technology directly or indirectly employs the application of mathematics. We are all...
...Mathematics of the Greeks and the Mayans
Mathematics is the study of time, space, structure, and quantity which is used to calculate almost anything in the world from the amount of atoms in an element to calculating the air pressure in a room. Although levels of math such as calculus are not taught until college, the use and study of mathematics have been around since the beginning of time and the world wouldn't be able to function without it. The term "mathematics" comes from the Greek word mathema which means study, knowledge, or learning. Along with philosophy and astronomy, the ancient Greeks were well known for their development and contribution as brilliant mathematicians. Despite their widely spread fame as the world's first greatest mathematicians, the Greeks had competitors. My contention is that the Mayans were one of the top mathematicians of the ancient world and were even more advanced than the Greeks.
Ancient Greek mathematics has been developed since the early seventh century B.C. which could also be called the period during the Hellenistic Mathematics. Some of the greatest Ancient Greek mathematicians were Pythagoras, Aristotle, Anaxagoras, Euclid, Archimedes, Thales, and Aristarchus. These Greek mathematicians were big on the development of geometry which is a subdivision of math that focuses on shapes, size, and the relativity of space. Although their number system was... | 677.169 | 1 |
Jorge Federico Osorio, an imaginative interpreter with a powerful technique (The New York Times), deftly pairs Brahmss final solo piano works with those by Schubert for an inventive program of richly satisfying works that capture the essence of each composers towering individuality. Here, Osorio records Brahmss Three Intermezzos, Op. 117, and Six Piano Pieces, Op 118, which he last recorded nearly two decades ago, to great acclaim: Quite marvelous, said BBC Music Magazine. Its clear that Jorge Federico Osorio is an important Brahmsian, proclaimed the Chicago Tribune. On his new album, Osorios penchant for color accentuates the individual character of these concentrated miniatures. Osorio has also chosen to record Schuberts final two Piano Sonatas, D. 959 and D. 960, epic in scale and brimming with melodic invention. His insights into the musics architecture yield eloquent performances of these spacious, ambitious masterworksThe Algebra 2 Tutor is a 6 hour course spread over 2 DVD disks that will aid the student in the core topics of Algebra 2. This DVD bridges the gap between Algebra 1 and Trigonometry, and contains essential material to do well in advanced mathematics. Many of the topics in contained in this DVD series are used in other Math courses, such as writing equations of lines, graphing equations, and solving systems of equations. These skills are used time again in more advanced courses such as Physics and CalculusThe Laplace Transform is one of the most powerful mathematical tools that can be used to solve a wide variety of problems in Math, Science, and Engineering. We begin this course by discussing what the Laplace Transform is and why it is important. Next, we show the Laplace Integral, and derive several fundamental transformations that we will use in the remainder of the course. We also discuss the Inverse Laplace Transform and derive several inverses. We discuss how to solve Ordinary Differential Equations (ODEs) with initial conditions and work several examples to give practice with real problemsThis month's SOMM release will no doubt be a special treat for the many fans of Peter Donohoe's recordings: a two-CD slimline set of Scriabin's Ten Piano Sonatas, ending with Vers la Flamme Op. 72, one of his last pieces for piano, written in 1914
Pollini's performances of Beethoven's last five piano sonatas have assumed almost legendary status, and this reissue at midprice in improved sound ought to win them many new friends. Sometimes considered a cold interpreter, Pollini here pays scrupulous attention to Beethoven's instructions, an attention that never gets in the way of sincere expression. There's a lot to be said for approaching this music with a maximum of clarity and simplicity, and a minimum of Romantic panting and heaving. In fact, Beethoven's instructions are so detailed, and the music itself is often so elaborately developed, that it's all most pianists can do to play it as he wrote it. Pollini does that, and much more. –David Hurwitz
Recorded over 13 years between 1975 and 1988, Murray Perahia's cycle of the complete piano concertos of Mozart, including the concert rondos and double concertos, remains perhaps the most enduring monument to his art. What is it about Perahia's art, some skeptics might ask, that is worth enduring? For one thing, as this 12-disc set amply demonstrates, there is his incredible tone. Clear as a bell, bright as the sky, and deep as the ocean, Perahia's tone is not only one of the wonders of the age, it's admirably suited to the pellucid loveliness of Mozart's music. | 677.169 | 1 |
Chapter 10
In this chapter we will review some of the materials from pre-calculus, Geometry and
Calculus. We will look at three different coordinates (Rectangular, Parametric and Polar)
plus two linear operators (Differentiation and integration).
Rectan
Chapter 10 Homework Solutions
1- When a plane crosses a double cone connected at their vertex, it traces
the graph of conics (ellipse, parabola, and hyperbola). Find a relationship
between the eccentricity and the angle of the
plane with the horizon.
L
Le
MATH-1C
Sec10.1 CURVES DEFINED BY PARAMETRIC EQUATIONS
We often find it convenient to describe the location of a point
( x, y )
in the plane in terms of a
parameter. For instance, in tracking the movement of a satellite, we would naturally want to give it
calculus Advice
Showing 1 to 1 of 1
This course requires a lot of self studying. Calculus itself is a hard subject which requires a lot of hard work and dedication, but this course makes it a lot harder since the class is based off power points.
Course highlights:
You will learn a lot about series and sequences, and will go over vector. This course is a lot like math 43, but calculus based.
Hours per week:
9-11 hours
Advice for students:
You should know your trigonometry, geometry, integration, differentiation, etc like the back of your hand. | 677.169 | 1 |
AP Calculus AB Questions & Answers
AP Calculus AB Flashcards
AP Calculus AB Advice
AP Calculus AB Advice
Showing 1 to 3 of 3
Mrs. Leonard makes a seemly hard course of calculus an easily understandable class while also challenging you so you come out of the class incredibly able to continue in your math career.
Course highlights:
I learned the in's and out's of calculus in a very thorough class. A great thing about this class is how open the teacher is to helping you understand it and making it relatable enough to understand easier.
Calculus can be applied to so many things in everyday life. My teacher made this subject so easy to learn and created such an impeccable learning experience for me on such a crucial subject.
Course highlights:
I learned how to understand calculus so easily it almost came as second nature. A good teacher did that for me and now I can apply that to many everyday things as well as my intended major of engineering.
Hours per week:
9-11 hours
Advice for students:
Definitely take it. It will help you so much in succeeding academically and becoming better at math.
I am very passionate about math, so I thoroughly enjoyed this math-heavy class.
Course highlights:
It's a course that teaches you first level calculus with the following three main topics: integrals, derivatives, and limits. In this course, you are able to apply basic calculus skills to real life situations. For example, I was able to design the optimal soda can, a can that uses the least amount of metal but can hold the most amount of soda.
Hours per week:
6-8 hours
Advice for students:
I would suggest to have an open mind because it can be super tough at times, so don't get discouraged. If you stay positive and work hard, you will succeed! The skills developed in this class will be very helpful in college. | 677.169 | 1 |
STAAR Algebra 1 EOC Reporting Category 3 Practice Booklet
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this file type before downloading and/or purchasing.
4 MB|18 pages
Product Description
This Independent Practice Booklet is specifically aligned with the STAAR Algebra 1 End of Course Exam but can also be used to review for most state Algebra 1 EOC exams as content standards across the U.S. are similar.
Independent practice booklets can be incorporated in a variety of ways: Interactive Notebooks, Homework, Warm-Ups, Stations/Centers and many more creative ways! Its compact design makes it easier for students to take class work to and from school making the booklets a great alternative to assigning homework from a heavy textbook.
Each of the 12 problems is coded with the specific TEKS alignment for easy lesson planning.
Readiness Standards: A.2(A) determine the domain and range of a linear function in mathematical problems; determine reasonable domain and range values for real-world situations, both continuous and discrete; and represent domain and range using inequalities A.2(C) write linear equations in two variables given a table of values, a graph, and a verbal description A.2(I) write systems of two linear equations given a table of values, a graph, and a verbal description A.5(A) solve linear equations in one variable, including those for which the application of the distributive property is necessary and for which variables are included on both sides A.5(C) solve systems of two linear equations with two variables for mathematical and real-world problems
Supporting Standards: A.2(B) write linear equations in two variables in various forms, including y = mx + b, Ax + By = C, and y – y1 = m(x – x1), given one point and the slope and given two points A.2(D) write and solve equations involving direct variation A.2(E) write the equation of a line that contains a given point and is parallel to a given line A.2(F) write the equation of a line that contains a given point and is perpendicular to a given line A.2(G) write an equation of a line that is parallel or perpendicular to the x- or y-axis and determine whether the slope of the line is zero or undefined A.2(H) write linear inequalities in two variables given a table of values, a graph, and a verbal description A.5(B) solve linear inequalities in one variable, including those for which the application of the distributive property is necessary and for which variables are included on both sides | 677.169 | 1 |
intermediate algebra.
This package includes MyMathLab®Personalize learning with MyMathLab
MyMathLab®Author Biography Martin-Gay developed an acclaimed series of lecture videos to support developmental mathematics students. These highly successful videos originally served as the foundation materials for her texts. Today, the videos are specific to each book in her series. She has also created Chapter Test Prep Videos to help students during their most "teachable moment"–as they prepare for a test–along with Instructor-to-Instructor videos that provide teaching tips, hints, and suggestions for every developmental mathematics course, including basic mathematics, prealgebra, beginning algebra, and intermediate algebra.
Elayn is the author of 12 published textbooks and numerous multimedia interactive products, all specializing in developmental mathematics courses. She has also published series in Algebra 1, Algebra 2, and Geometry. She has participated as an author across a broad range of educational materials: textbooks, videos, tutorial software, and courseware. This offers an opportunity for multiple combinations for an integrated teaching and learning package, offering great consistency for the student. | 677.169 | 1 |
A: There are several courses of different levels for you to choose.
In general, if you are Secondary 3 or above, if you enjoy mathematics and usually are the better ones in your class,
there should be a course suitable for you. If you are of lower forms but already have mathematics skills of higher forms,
you You need to spend 3 intensive weeks on a course.
Besides lessons on 9 whole days, you need the other days to prepare, work on exercises, and review.
TheA: We welcome junior form student to take EPYMT course, but please note that EPYMT introduces students more advanced and sophisticated mathematics than that
they learn in school so junior form students should be mature enough to face difficulties and frustration.
It is important to consider if junior form students are able to learn mathematics
together with higher form students (both in mathematics ability and personality). The following courses are designed for the students who are advancing to Secondary 4 or 5. Yet, they are still challenging.
Geometric Perspectives of Complex Numbers (CUSA1014)
Number Theory and Cryptography (SAYT1114)
If you are fond of geometry, probably you should choose ¡§Geometric Perspectives of Complex Numbers¡¨, which is a natural extension to your secondary school coordinate geometry module. Or
if you enjoy the funniness among numbers, ¡§Number Theory and Cryptography¡¨ would be your suitable choice.
If you have learnt Calculus (differentiation and integration), you are welcome to apply ¡§Towards Differential Geometry¡¨ which suits Secondary 5 or 6 students. If you are able to complete this course at Form 4, you may have a great advantage to become a high-achiever in your Form 5 syllabus. However, without adequate knowledge of Calculus, we advise you not to apply.
Finally, the most mathematically challenging course, ¡§Understanding Non-Euclidean Geometry¡¨, is beyond the reach of the vast majority of F.4 students, and it is only suitable for senior form students who have distinguished mathematical performance or have taken any previous EPYMT course. You may consult your teacher if you are capable to take this course. you apply for the course "Towards Differential Geometry",
we do expect you to have reasonable competence with calculus.
A: Our courses are intensive and they demand tremendous efforts to master the course contents. Therefore, it is advisable not to skip any one of the lecture, even to take a leave.
Attending several dates of the course is strongly discouraged and your application may not be considered if such a condition is acknowledged.
A: A note on the application form states that ¡§recommendation is NOT required if the applicant had applied for admission to EPYMT before¡¨ so you can choose not to hand another recommendation letter to us. However,
if the teacher wants to write another stronger recommendation letters for the student this year, we will still accept.
A: If you submit the online application before deadline, we can actually allow the recommendation letter to be submitted a bit later.
Nevertheless, please inform us your situation and submit it as soon as possible so that the process of your application will not be postponed.
A: You are required to take the Admission Screening Test (if you are applying for the first time), but, under certain circumstances, we can probably provide special arrangements. Please send us an email and elaborate your situation.
A: You should receive a confirmation email listing the course(s) you applied if your online application has been successfully submitted. If you have not received a confirmation email, please contact us by () immediately.
A: The information, collected in the application form, will be used to process the application, administration and statistical purposes. Unless those admitted students request to process academic credit records through the University, we may transfer students¡¦ information to the according third party. Besides, we may also keep contact with you, in order to introduce you the latest information of EPYMT-related activities. keep checking your emails.
A: There will be ONE paper conducted in English for the Admission Screening Test. The test paper will be divided into a compulsory part and an elective part. No any past papers of the admission test can be found.
A: The course instructors prepared the Admission Screening Test to evaluate the mathematical levels the applicants have achieved,
soTo improve your mathematical ability, we advise you to concentrate on your school works and ask questions beyond what the textbook teaches.
Also, try to read ahead of your syllabus. For example, if you are a F.4 student, you should try to take a look or even finish the F.5 mathematics textbook and work out the problems by yourself.
You could also read some foreign mathematics textbooks or leisure books for enrichment. Nowadays, you can even surf the web to learn advanced mathematics.
A: We arrange alternative methods to assess your ability through a ¡§take-home¡¨ test which will be arranged in
a specific period. The admission assessment cannot be waived. Please contact us after
you have submitted your application.
A: We won't provide past papers of the Admission Screening Test to anyone. They would not help you in the coming test even if you had them and we are not giving them out. The questions are designed by our creative University teachers, so the test questions will be different each year.
A: You shall receive a notification and an attachment of a formal document from us through email. For applicants who do not receive any notification before July 1, you may consider your application unsuccessful. hasI have an outstanding student who is mathematically well beyond the average of his/her age. However, he/she is only student
"Towards Differential Geometry" and "Geometric Perspectives of Complex Number" are especially designed for F.5 students. However,
if students apply for the course "Towards Differential Geometry", we do expect students to have reasonable competence with calculus.
My student is studying in overseas / mainland high school, but he/she plans to come to Hong Kong in the period when EPYMT courses start. Can he/she still apply EPYMT?
A: We will assess his/her ability through a ¡§take-home¡¨ test that student need to spend on EPYMT?
A: Students need to spend 3 intensive weeks on a course. Besides lessons on 9 whole days, students need the other days to prepare,
work on exercises, and review. TheIf my student had applied EPYMT last year, does he/she still need me to write another recommendation letter for him this year?
A: A note on the application form states that ¡§recommendation is NOT required if the applicant had applied for admission to EPYMT before¡¨ so he/she can choose not to hand another recommendation letter to us. However, if the teacher wants to write another stronger recommendation letters for the student this year, we will still accept.
If my student student to contact us after he/she has submitted the application. ¢w "Understanding Non-Euclidean Geometry".
How can my student studentI am looking for summer classes for my child, who is good at mathematics. I wonder if EPYMT is suitable.
A: Please ask your child to visit our website. Generally, EPYMT is suitable for a child who is already among the best in his/her own school and is at least Secondary 3. The courses are not only summer activities but formal university courses. Strong commitments of the students are required.
My child is very good at mathematics and he/she may be talented. Will EPYMT have any special treatment for him/her?
A: The courses are of very high level that many gifted students had studied in the past. Usually, no special treatment is needed if the student is able to interact with others normally.
How does my student choose the courses of EPYMT? haveMy child is mathematically well beyond the average of his/her age. However, he/she is only child students apply for the course "Towards Differential Geometry", we do expect students to have reasonable competence with calculus.
My child is studying in overseas / mainland high school, but he/she plans to come to Hong Kong in the period when EPYMT courses start. Can he/she still apply EPYMT?
A: Yes. We will assess his/her ability through a ¡§take-home¡¨ test which child need to spend on EPYMT?
A: Students need to spend 3 intensive weeks on a course. Besides lessons on 9 whole days, students need the other days to prepare, work on exercises, and review. The learning pace is fast and the course schedule is tight. A lot of after class revision is needed. Students should be highly self-motivated and committed. Also, students are expected to be very curious and fond of exploring mathematics beyond his/her level. Otherwise, many of them will lag behind.
My child has participated in many other activities, can he/she be absent from a few lessons?
A: It is strongly discouraged to do so. EPYMT courses are of very high level and even the best students of Hong Kong need to pay serious effort in learning. Any student taking an EPYMT course should consider it as a strong commitment. This kind of applicant may not be considered if such a condition is acknowledged.
If my child
child to contact us after he/she has submitted the application.
When will my child know if he/she can attend the Admission Screening Test? ask your child to keep checking
his/her emails.
What is the purpose of the Admission Screening Test?How can my child childDoes my child need to pay cost on the date of Admission Screening Test?
A: No. He/She needs not to pay any cost before he/she are admitted. After we have confirmed the admitted students, we will contact them for the issue of tuition payment. | 677.169 | 1 |
EXTRAS
Differential Equations (12th Grade Mathematics)
12th Grade Mathematics
At the end of this lesson, you will be able to:
• Find the order and degree of differential
equations.
• Find the general and particular solutions of
differential equations.
• Form the differential equation when general
solution is given.
• Solve differential equation by separation of
variables method.
• Solve homogeneous diiferential equations.
• Solve linear diiferential equations. | 677.169 | 1 |
[SR: 1055473], Hardcover, [EAN: 9783642309632], Springer, Springer, Book, [PU: Springer], 226700, Geometry & Topology, 13928, Algebraic Geometry, 13930, Analytic Geometry, 13932, Differential Geometry, 13936, Non-Euclidean Geometries, 13987, Topology, 13884, Mathematics, 75, Science & Math, 1000, Subjects, 283155, Books, 13942, History, 13884, Mathematics, 75, Science & Math, 1000, Subjects, 283155, Books, 491546, Geometry, 468218, Mathematics, 468216, Science & Mathematics, 465600, New, Used & Rental Textbooks, 2349030011, Specialty Boutique, 283155, Books
[SR: 244021], Hardcover, [EAN: 9783642309632], Springer, Springer, Book, [PU: Springer], 2012-08-16, 278353, Geometry & Topology, 278320, Mathematics, 57, Science & Nature, 1025612, Subjects, 266239, Books, 278362, History of Mathematics, 278320, Mathematics, 57, Science & Nature, 1025612, Subjects, 266239, Books, 922942, Maths, 922868, Popular Science, 57, Science & Nature, 1025612, Subjects, 266239, Books, 570936, Geometry, 564352, Mathematics, 564334, Scientific, Technical & Medical, 1025612, Subjects, 266239, Books
Based on a series of lectures for adult students, this lively and entertaining book proves that, far from being a dusty, dull subject, geometry is in fact full of beauty and fascination. The author´s infectious enthusiasm is put to use in explaining many of the key concepts in the field, starting with the Golden Number and taking the reader on a geometrical journey via Shapes and Solids, through the Fourth Dimension, finishing up with Einstein´s Theories of Relativity. Equally suitable as a gift for a youngster or as a nostalgic journey back into the world of mathematics for older readers, John Barnes´. This lively book explains many of the key concepts in the field, starting with the Golden Number and taking the reader on a geometrical journey via Shapes and Solids, through the Fourth Dimension, finishing up with Einstein´s Theories of Relativity 244x156x20 mm, [GW: 608g] FixedPrice, [GW: 608g]
John Barnes
Gems of Geometry
Based
Equ
In this second edition, stimulated by recent lectures at Oxford, further material and extra illustrations have been added on many topics including Coloured Cubes, Chaos and Crystals. | 677.169 | 1 |
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1 left in stock at this price201763904
ISBN: 0201763907
Edition: 7
Publication Date: 2002
Publisher: Pearson
AUTHOR
John B. Fraleigh
SUMMARY
This is an in-depth introductory text for those on an abstract algebra course. It focuses on groups, rings, and fields, giving a firm foundation for more specialized work by emphasizing an understanding of the nature of algebraic structures.John B. Fraleigh is the author of 'A First Course in Abstract Algebra, 7th Edition', published 2002 under ISBN 9780201763904 and ISBN 02017639 | 677.169 | 1 |
Math 212: Path Integrals
In this integral instructional activity, students identify a vector field and explore line integrals. Students use the Fundamental Theorem for path integrals to solve problems. This two-page instructional activity contains three problems. | 677.169 | 1 |
Posts tagged with 'microsoft mathematics tutorial'
This is the seventh tutorial in the Microsoft Mathematics Tutorial Series. In this post, we learn how to use the Ink input of Microsoft Mathematics. Microsoft Mathematics has handwriting support that enables users to write problems and solve equations using the pen in Tablet PCs or even just a mouse when using a computer. This […] Continue reading…
This is the 6th tutorial in the Microsoft Mathematics Tutorial Series. In this post, we discuss how to use the Unit Converter of Microsoft Mathematics. Microsoft Mathematics has a built-in unit converter that allows conversion of different types of measurements. It supports measurement conversion of length, area, volume, mass, temperature, velocity, pressure, weight, energy, power, time, […] Continue reading…
This is the fourth tutorial of the Microsoft Mathematics Tutorial Series. In this tutorial, we learn how to plot 2 and 3 dimensional Cartesian graphs and 2 dimensional polar graphs. We also learn how to modify the settings of the Graphing window such as plotting range and proportional display. 1. Open Microsoft Mathematics. 2. Select […] Continue reading… | 677.169 | 1 |
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Algebra Touch is Genius!
By Cindy Downeson Sun, 08/05/2012
Algebra Touch is the perfect app for visual and kinesthetic learners. Here's how it works. There are three stages: Explain, Practice and My Problems. In the Explain stage, you can either work your way through the lessons in order or select a lesson at random. Once you have chosen your lesson, you scroll through the instructions at the bottom of the screen and tap out your answers at the top. If you make a mistake, the problem won't solve. If you're right, you see the problem solve on the screen. You really have to watch this in action to see how genius this is. Check out this video you understand how the problem works, you can go to the Practice stage to practice the concept as many times as you want. I wasn't able to determine if there was a limit to the number of practice problems or not. There were more than enough for me. The last stage is My Problems. In this stage, you can type in your own problems and solve them using the same intuitive features as in the practice problems. You can type in simple fractions and up to three unknowns, but no exponents or square roots.
Algebra Touch 1.4.4 covers:
· Addition
· Like Terms
· Negatives
· Multiplication
· Order of Operation
· Pairs
· Replacement
· Prime Numbers
· Convenience
· Substitution
· Cross Out
· Equals
· Division
· Products Only
· Variables
· Isolate
· Split
· Basic Equations
· Distribution
· Factoring Out
· Advanced (combination of above) | 677.169 | 1 |
Mathematics
Mathematics, more than the fundamental language and underlying analytical structure of science and technology, is a formal way of thinking—an art that ties together the abstract structure of reason and the formal development of the logic that defines the scientific method. From the study of just how arguments and theories are formed in language and technology to the framework of quantitative and qualitative models of the natural and social sciences, mathematics is based upon the development of precise expressions, logical arguments, and the search for and exposure of pattern and structure.
The undergraduate program in the Department of Mathematics is intended both for students interested in attaining the proper preparation for graduate study and research in pure mathematics, and for students interested in using mathematics to define, properly pose, and solve problems in the sciences, engineering, and other areas. With either purpose, the focus of the program is to help those who wish to understand further the logical content, geometric meaning, and abstract reasoning of mathematics itself. A flexible program involving a broad selection of courses is a department tradition. The program begins by introducing students to the basics of algebra and mathematical analysis and then gives them the choice of exploring topics in theoretical mathematics or studying applications to physics, economics, engineering, computer science, probability, statistics, or mechanics.
The graduate program is designed primarily to prepare students for research and teaching in mathematics. It is naturally centered around the research areas of the faculty, which include algebraic geometry, algebraic number theory, differential geometry, partial differential equations, topology, several complex variables, algebraic groups, and representation theory. The program can be supplemented in applied directions by courses in theoretical physics, computer science, mechanics, probability, and statistics offered in other departments of the Krieger School of Arts and Sciences and in the Department of Applied Mathematics in the Whiting School of Engineering.
Facilities
The Mathematics Department resides in Krieger Hall on the Keyser Quad of Homewood. Adjacent to Krieger Hall, The University's Milton S. Eisenhower Library has an unusually extensive collection of mathematics literature, including all the major research journals, almost all of which are accessible electronically. The stacks are open to students. The department also has a useful reference library, the Philip Hartman Library. Graduate students share departmental offices, and study space can also be reserved in the university library. Graduate students may access the department's Linux and Windows servers, as well as computers in graduate student offices. The department also hosts numerous research seminars, special lectures, and conferences throughout the academic year.
Math Course Placement and Sequencing for All Students
There are different versions of Calculus I and II offered by the Mathematics Departments and students should select their first course in math at JHU based on their major intentions and placement. Students intending to major in mathematics, the physical sciences, or engineering are encouraged to begin with the AS.110.108 Calculus I - AS.110.109 Calculus II (For Physical Sciences and Engineering) sequence or AS.110.113 Honors Single Variable Calculus. Students majoring in other subjects may wish to take the AS.110.106 Calculus I (Biology and Social Sciences) - AS.110.107 Calculus II (For Biological and Social Science) sequence which relates the methods of calculus to the biological and social sciences. A one-semester pre-calculus course (AS.110.105 Introduction to Calculus) is offered for students who could benefit from additional preparation in the basic tools (algebra and trigonometry) used in calculus.
After completing a Calculus II course, the courses AS.110.201 Linear Algebra, AS.110.202 Calculus III, or AS.110.302 Diff Equations/Applic may be taken in any order. The department offers honors courses in both AS.110.212 Honors Linear Algebra and AS.110.211 Honors Multivariable Calculus.
Requirements for the B.A. Degree
In addition to the Requirements for a Bachelor's Degree, a candidate for the bachelor's degree in mathematics is required to have completed the major requirements listed below. All courses used to meet these requirements must be completed with a grade of C- or better and may not be taken satisfactory/unsatisfactory.
Sample Program of Study
The following chart is one example of how a student might progress through the mathematics major. As potential math majors enter JHU with a wide range of prior math abilities, students should begin courses at their current level of knowledge.
Requirements for a Minor in Mathematics
All courses used to meet the mathematics minor requirements must be completed with a grade of C- or better and may not be taken satisfactory/unsatisfactory. One course in the Applied Mathematics and Statistics Department (at the 300-level or above) may be substituted for one of the elective courses for the minor.
Honors Program in Mathematics
As a general guideline, departmental honors are awarded to recipients of the B.A. degree who have completed AS.110.311 Methods of Complex Analysis, as well as AS.110.401 Advanced Algebra I-, AS.110.415 Honors Analysis I-AS.110.416 Honors Analysis II, and one more course at the 400-level or above with at least a 3.6 average in these six courses.
J.J. Sylvester Prize
The J.J. Sylvester Prize in Mathematics, which caries a cash award, is given each year to the one of two top-performing graduating seniors majoring in mathematics for outstanding achievement.
The B.A./M.A. Program
By applying the same courses simultaneously toward the requirements for the B.A. and M.A. degrees, an advanced student can qualify for both degrees in four years. Admission to the program is by the standard graduate application form, which should be completed in the junior year. At least a 3.0 average is required in the 400-level mathematics courses taken while resident at the university. Students may contact the graduate program assistant for further information.
Undergraduate Teaching Assistantships
The department awards many upper-level undergraduates the opportunity to act as recitation instructors to our freshman courses, enabling them to practice the art of teaching and talking mathematics and to earn a valuable credential while studying for their degree.
Admission
Admission to the Ph.D. program is based on academic records, letters of recommendation, and Graduate Record Examination scores. International applicants are required to submit a TOEFL or IELTS score if English is not their native language.
These 600-level graduate courses are preliminary to research and are built upon the foundations constituted by the 400-level courses: AS.110.401 Advanced Algebra I-AS.110.402 Advanced Algebra II, AS.110.405 Analysis I-AS.110.406 Analysis II or AS.110.311 Methods of Complex Analysis, and AS.110.413 Introduction To Topology.
The 700-level courses are designed to bring students abreast of recent developments and to prepare them for research in the area of their choice.
Requirements for the M.A. Degree
Although the Mathematics Department does not admit students seeking a terminal M.A. degree, students in the Ph.D. program may earn an M.A. degree. Advanced undergraduate students may also apply to be admitted to the accelerated B.A./M.A. program.
M.A. candidates must complete:
Four graduate courses given by the Johns Hopkins Mathematics Department;
Two additional courses at the graduate or 400-level, other than AS.110.401, AS.110.405, and AS.110.415, given by the Johns Hopkins Mathematics Department, or, with the permission of the graduate program director, graduate mathematics courses given by other departments or universities.
All courses used to satisfy the requirements must be completed with a grade of B- or better. (Advanced graduate courses completed with a grade of P can also be used to satisfy the requirements.)
Requirements for the Ph.D. Degree
The departmental requirements for the Ph.D. degree are:
Candidates must show satisfactory work in Algebra (AS.110.601-AS.110.602), Real Variables (AS.110.605), Complex Variables (AS.110.607), Algebraic Topology (AS.110.615), and one additional mathematics graduate course in their first year. The seminars and qualifying exam preparation course cannot be used to fulfill this requirement. The algebra and analysis requirements can be satisfied by passing the corresponding written qualifying exam in September of the first year; these students must complete at least two courses each semester. Students having sufficient background in topology can substitute an advanced topology course for AS.110.615, with the permission of the instructor.
Candidates must pass written qualifying exams by the beginning of their second year in Analysis (Real and Complex) and in Algebra. Exams are scheduled for September and May of each academic year.
Candidates must show satisfactory work in at least two mathematics graduate courses each semester of their second year, and, if they have not passed their oral qualifying exam, in the first semester of their third year.
Candidates must pass a departmental oral qualifying examination in the student's chosen area of research by April 8th of the third year. The topic of the exam is chosen in consultation with the faculty member who has agreed (provisionally) to be the student's thesis advisor, who will also be involved in administering the exam.
There is no longer a Mathematics Department foreign language requirement. With the vast majority of articles written in English, the importance of having the capability of reading another language has diminished. However, important earlier literature in certain areas of mathematics may be written in French, German, or Russian. Moreover, some articles are still being written in French. It is now at the discretion of the student's thesis advisor whether to impose a language requirement.
Candidates must produce a dissertation based upon independent and original research.
Candidates will gain teaching experience in mathematics as a teaching assistant for undergraduate courses. The student will be under the supervision of both the faculty member teaching the course and the director of undergraduate studies. First year students are given a reduced TA workload in the spring semester (this is related to item #2).
After completion of the thesis research the student will defend their dissertation by means of the Graduate Board Oral Exam. The exam must be held at least three weeks before the Graduate Board deadline the candidate wishes to meet.
Financial Aid
Students admitted to the Ph.D. program receive teaching assistantships and full tuition fellowships. Exceptional applicants become candidates for one of the university's George E. Owen Fellowships.
William Kelso Morrill Award
The William Kelso Morrill Award for excellence in the teaching of mathematics is awarded every spring to the graduate student who best exemplifies the traits of Kelso Morrill: a love of mathematics, a love of teaching, and a concern for students.
Excellence in Teaching Awards
Three awards are given each year to a junior faculty member and graduate student teaching assistants who have demonstrated exceptional ability and commitment to undergraduate education.
Courses
AS.110.105. Introduction to Calculus. 4.00 Credits.
This course starts from scratch and provides students with all the background necessary for the study of calculus. It includes a review of algebra, trigonometry, exponential and logarithmic functions, coordinates and graphs. Each of these tools will be introduced in its cultural and historical context. The concept of the rate of change of a function will be introduced. Not open to students who have studied calculus in high school. Instructor(s): M. Martin
Area: Quantitative and Mathematical Sciences113. Honors Single Variable Calculus. 4.00 Credits.
This is an honors alternative to the Calculus sequences AS.110.106-AS.110.107 or AS.110.108-AS.110.109 and meets the general requirement for both Calculus I and Calculus II (although the credit hours count for only one course). It is a more theoretical treatment of one variable differential and integral calculus and is based on our modern understanding of the real number system as explained by Cantor, Dedekind, and Weierstrass. Students who want to know the why's and how's of Calculus will find this course rewarding. Previous background in Calculus is not assumed. Students will learn differential Calculus (derivatives, differentiation, chain rule, optimization, related rates, etc), the theory of integration, the fundamental theorem(s) of Calculus, applications of integration, and Taylor series. Students should have a strong ability to learn mathematics quickly and on a higher level than that of the regular Calculus sequences. Instructor(s): V. Lorman
Area: Quantitative and Mathematical Sciences.
AS.110.160. Discover Hopkins: Mathematics of Infinity. 1.00 Credit.
An interdisciplinary introduction to the history of infinity in mathematics, from Zeno's paradox to the development of calculus to the crisis in the foundations of mathematics in the early 20th century. We will read about history, discuss philosophy, and learn some mathematics (including a crash course in mathematical logic and proof, building up to the rigorous definition of limits). A previous course in calculus is not required, but some mathematical maturity will be necessary. Instructor(s): V. Lorman.
AS.110.201. Linear Algebra. 4.00 Credits.
Vector spaces, matrices, and linear transformations. Solutions of systems of linear equations. Eigenvalues, eigenvectors, and diagonalization of matrices. Applications to differential equations. Prerequisites: Grade of C- or better in 110.107 or 110.109 or 110.113, or a 5 on the AP BC exam. Instructor(s): G. Di Matteo; J. Bernstein Stokes applicationsAS.110.225. Problem Solving Lab. 2.00 Credits.
An introduction to some fundamental techniques of mathematical problem solving, such as induction, invariants, inequalities and generating functions. Instructor(s): D. Savitt
Area: Quantitative and Mathematical Sciences.
AS.110.302. Diff Equations/Applic. 4.00 Credits.
This is an applied course in ordinary differential equations, which is primarily for students in the biological, physical and social sciences, and engineering. The purpose of the course is to familiarize the student with the techniques of solving ordinary differential equations. The
specific subjects to be covered include first order differential equations, second order linear differential equations, applications to electric circuits, oscillation of solutions, power series solutions, systems of linear differential equations, autonomous systems, Laplace
transforms and linear differential equations, mathematical models (e.g., in the sciences or economics). Prerequisites: Grade of C- or better in 110.107 or 110.109 or 110.113, or a 5 on the AP BC exam. Instructor(s): R. Brown; Y. Sire
Area: Quantitative and Mathematical Sciences.
AS.110.304. Elementary Number Theory. 4.00 Credits.
The student is provided with many historical examples of topics, each of which serves as an illustration of and provides a background for many years of current research in number theory. Primes and prime factorization, congruences, Euler's function, quadratic reciprocity, primitive roots, solutions to polynomial congruences (Chevalley's theorem), Diophantine equations including the Pythagorean and Pell equations, Gaussian integers, Dirichlet's theorem on primes. Prerequisites: Grade of C- or better in 110.201 or 110.212 Instructor(s): T. Wright; W. Wilson
Area: Quantitative and Mathematical Sciences.
AS.110.306. Honors Differential Equations. 4.00 Credits.
This course includes the material in 110.302 Differential Equations but with a strong emphasis on theory and proofs. Recommended only for mathematics majors or mathematically able students majoring in physical science or engineering. Prerequisites: Grade of B+ or better in AS.110.107 or AS.110.109 or AS.110.113 or a 5 on the Advanced Placement BC exam. Instructor(s): J. Gell-redman
Area: Quantitative and Mathematical Sciences.
AS.110.311. Methods of Complex Analysis. 4.00 Credits.
This course is an introduction to the theory of functions of one complex variable. Its emphasis is on techniques and applications, and it serves as a basis for more advanced courses. Functions of a complex variable and their derivatives; power series and Laurent expansions; Cauchy integral theorem and formula; calculus of residues and contour integrals; harmonic functions. Prerequisites: Grade of C- or better in 110.202 or 110.211 Instructor(s): Y. Sire
Area: Quantitative and Mathematical Sciences.
AS.110.321. Honors Complex Analysis. 4.00 Credits.
This course is an introduction to the theory of functions of one complex variable for honors students. Its emphasis is on techniques and applications, and it serves as a basis for more advanced courses. Topics will include functions of a complex variable and their derivatives; power series and Laurent expansions; Cauchy integral theorem and formula; calculus of residues and contour integrals; harmonic functions, as well as applications to number theory and harmonic analysis. Instructor(s): B. Dodson
Area: Quantitative and Mathematical Sciences.
AS.110.328. Non-Euclidean Geometry. 4.00 Credits.
For 2,000 years, Euclidean geometry was the geometry. In the 19th century, new, equally consistent but very different geometries were discovered. This course will delve into these geometries on an elementary but mathematically rigorous level. Instructor(s): M. Merling
Area: Quantitative and Mathematical Sciences.
AS.110.401. Advanced Algebra I. 4.00 Credits.
An introduction to the basic notions of modern algebra. Elements of group theory: groups, subgroups, normal subgroups, quotients, homomorphisms. Generators and relations, free groups, products, commutative (Abelian) groups, finite groups. Groups acting on sets, the Sylow theorems. Definition and examples of rings and ideals. Introduction to field theory. Linear algebra over a field. Field extensions, constructible polygons, non-trisectability. Prerequisites: Grade of C- or better in 110.201 or 110.212 Instructor(s): J. Kong
Area: Quantitative and Mathematical Sciences.
AS.110.402. Advanced Algebra II. 4.00 Credits.
Splitting field of a polynomial, algebraic closure of a field. Galois theory: correspondence between subgroups and subfields. Solvability of polynomial equations by radicals. Instructor(s): B. Smithling
Area: Quantitative and Mathematical Sciences.
AS.110.405. Analysis I. 4.00 Credits.
This course is designed to give a firm grounding in the basic tools of analysis. It is recommended as preparation (but may not be a prerequisite) for other advanced analysis courses. Real and complex number systems, topology of metric spaces, limits, continuity, infinite sequences and series, differentiation, Riemann-Stieltjes integration. Prerequisites: Grade of C- or better in 110.201 or 110.212 and 110.202 or 110.211 Instructor(s): J. Luehrmann
Area: Quantitative and Mathematical Sciences.
AS.110.406. Analysis II. 4.00 Credits.
This course continues AS.110.405 with an emphasis on the fundamental notions of modern analysis. Sequences and series of functions, Fourier series, equicontinuity and the Arzela-Ascoli theorem, the Stone-Weierstrass theorem, functions of several variables, the inverse and implicit function theorems, introduction to the Lebesgue integral. Instructor(s): J. Spruck
Area: Quantitative and Mathematical Sciences.
This highly theoretical sequence in analysis is reserved for the most able students. The sequence covers the real number system, metric spaces, basic functional analysis, the Lebesgue integral, and other topics. Instructor(s): Y. Sire
Area: Quantitative and Mathematical Sciences.
This is a course in the modern theory of Dynamical Systems. Topic include existence and uniqueness of general ODEs, nonlinear analysis and stability, including bifurcation theory and stable and center manifolds, smooth flows, limit sets, Hamiltonian mechanics, perturbation theory and structural stability. Prerequisites: Grade of C- or better in 110.201 or 110.212 OR 110.202 or 110.211 and 110.302 Instructor(s): H. Xu
Area: Quantitative and Mathematical Sciences.
AS.110.422. Representation Theory. 4.00 Credits.
This course will focus on the basic theory of representations of finite groups in characteristic zero: Schur's Lemma, Mashcke's Theorem and complete reducibility, character tables and orthogonality, direct sums and tensor products. The main examples we will try to understand are the representation theory of the symmetric group and the general linear group over a finite field. If time permits, the theory of Brauer characters and modular representations will be introduced. Prerequisites: Prereqs: ( AS.110.201 OR AS.110.212 ) AND AS.110.401 Instructor(s): M. Merling
Area: Quantitative and Mathematical Sciences.
AS.110.423. Lie Groups for Undergraduates. 4.00 Credits.
This course is an introduction to Lie Groups and their representations at the upper undergraduate level. It will cover basic Lie Groups such as SU (2), U(n) , the Euclidean Motion Group and Lorentz Group. This course is useful for students who want a working knowledge of group representations. Some aspects of the role of symmetry groups in particle physics such as some of the formal aspects of the electroweak and the strong interactions will also be discussed. Recommended Course Background: AS.110.202; prior knowledge of group theory (AS.110.401) would be helpful. Instructor(s): S. Zucker
Area: Quantitative and Mathematical Sciences.
The theory of knots and links is a facet of modern topology. The course will be mostly self-contained, but a good working knowledge of groups will be helpful. Topics include braids, knots and links, the fundamental group of a knot or link complement, spanning surfaces, and low dimensional homology groups. Instructor(s): C. McTague
Area: Quantitative and Mathematical Sciences.
AS.110.439. Introduction To Differential Geometry. 4.00 Credits.
Theory of curves and surfaces in Euclidean space: Frenet equations, fundamental forms, curvatures of a surface, theorems of Gauss and Mainardi-Codazzi, curves on a surface; introduction to tensor analysis and Riemannian geometry; theorema egregium; elementary global theorems. Prerequisites: Grade of C- or better in (AS.110.201 or AS.110.212) and (AS.110.202 or AS.100.211) Instructor(s): J. Spruck
Area: Quantitative and Mathematical Sciences.
AS.110.441. Calculus on Manifolds. 4.00 Credits.
This course provides the tools for classical three-dimensional physics and mechanics. This course extends these techniques to the general locally Euclidean spaces (manifolds) needed for an understanding of such things as Maxwell's equations or optimization in higher dimensional contexts, eg. in economics. The course will cover the theory of differential forms and integration. Specific topics include Maxwell's equations in terms of 4D Lorentz geometry, vector (in particular, tangent) bundles, an introduction to de Rham theory, and Sard's theorem on the density of regular values of smooth functions. The course is intended to be useful to mathematics students interested in analysis, differential geometry, and topology, as well as to students in physics and economics. Instructor(s): J. Morava
Area: Quantitative and Mathematical Sciences.
Topics in advanced algebra and number theory, including local fields and adeles, Iwasawa-Tate theory of zeta functions and connections with Hecke's treatment, semisimple algebras over local and number fields, adeles geometry. Instructor(s): C. Mese; G. Di Matteo
Area: Quantitative and Mathematical Sciences.
AS.110.618. Number Theory.
Topics in advanced algebra and number theory, including local fields and adeles, Iwasawa-Tate theory of zeta-functions and connections with Hecke's treatment, semi-simple algebras over local and number fields, adele geometry. Instructor(s): D. Savitt.
Microlocal analysis is the geometric study of singularities of solutions of partial differential equations. The course will begin by introducing the geometric theory of (Schwartz) distributions: Fourier transform and Sobolev spaces, pseudo-differential operators, wave front set of a distribution, elliptic operators, Lagrangean distributions, oscillatory integrals, method of stationary phase, Fourier integral operators. The second semester will develop the theory and apply it to special topics such as asymptotics of eigenvalues/eigenfunctions of the Laplace operator on a Riemann manifold, linear and non-linear wave equation asymptotics of quantum systems, Bochner-Riesz means, maximal theorems. Instructor(s): H. Lindblad.
AS.110.637. Functional Analysis.
Instructor(s): C. Mese; J. Spruck.
AS.110.643 theorem. Other topics may include Jacobian varieties, resolution of singularities, geometry on surfaces, connections with complex analytic geometry and topology, schemes. Instructor(s): V. Shokurov.
AS.110.644 Theorem. Other topics may include Jacobian varieties, resolution of singularities, geometry on surfaces, schemes, connections with complex analytic geometry and topology. Instructor(s): C. Consani
Area: Quantitative and Mathematical Sciences.
The goal is to give a self-contained course on mean curvature flow, starting with the basic linear heat equation in Euclidean space and – hopefully – getting to topics of current research. Mean curvature flow is a geometric heat equation that shares many properties with Ricci flow, harmonic map heat flow, Yang-Mills flow and the Navier-Stokes equations. Recommended Course Background: AS.110.605 and an undergraduate course in differential geometry; AS.110.645 and AS.110.631 Instructor(s): C. Mese.
AS.110.712. Topics in Mathematical Physics.
Instructor(s): J. Morava.
AS.110.722. Topics in Homotopy Theory.
The course will focus on recent developments in homotopy theory, such as Galois theory for E_n (n \geq 2) ring-spectra, and on connections with number theory; in particular, work of Bhatt, Hesselholt, Lurie, Scholze and others on topological Hochschild homology and its applications to geometry over the p-adic complex numbers. Instructor(s): J. Morava
Area: Quantitative and Mathematical Sciences.
AS.110.724. Topics in Arithmetic Geometry.
Topics around the subject of Arithmetic Geometry will be covered in this course. Instructor(s): B. Smithling
Area: Quantitative and Mathematical Sciences.
AS.110.726. Topics in Analysis.
The topics covered will involve the theory of calculus of Functors applied to Geometric problems like Embedding theory. Other related areas will be covered depending on the interest of the audience. Instructor(s): Y. Sire
Area: Quantitative and Mathematical Sciences.
AS.110.727. Topics/Algebraic Topology.
Instructor(s): C. Mese; E. Riehl.
AS.110.728. Topics in Algebraic Topology.
Instructor(s): N. Kitchloo.
AS.110.731. Topics in Geometric Analysis.
Instructor(s): Y. Wang.
AS.110.735. Topics In Hodge Theory.
Instructor(s): C. Mese; R. Brown; S. Zucker.
AS.110.737. Topics Algebraic Geometry.
Instructor(s): C. Consani.
AS.110.738. Topics Algebraic Geometry.
Introduction to toric varieties. This class is a general introduction to toric varieties. Toric varieties are special kinds of algebraic varieties which can be described by lattices and convex sets. They provide a rich source of concrete examples in complex geometry or mathematical physics. If time permits, we discuss in the end the stability of toric embeddings. Students should know basic notions of algebraic geometry (schemes, sheaves, linear systems), as covered in AS.110.643. Instructor(s): C. Mese; R. Brown; V. Shokurov
Area: Quantitative and Mathematical Sciences.
AS.110.741. Topics:Partial Differential Equations.
Instructor(s): H. Lindblad.
AS.110.742. Topics In Partial Differential Equations.
In this course we will be discussing some dispersive evolution equations, primarily the nonlinear Schrodinger equation. Topics will include well - posedness theory, conservation laws, and scattering. The course will be accessible to students who have not taken graduate partial differential equations or functional analysis. Instructor(s): B. Dodson.
AS.110.748. Topics in Geometry.
Instructor(s): J. Spruck.
AS.110.749. Topics in Differential Geometry.
In this class, we will study Aaron Naber and Jeff Cheeger's recent result on proving codimension four conjecture. We plan to talk about some early results of the structure on manifolds with lower Ricci bound by Cheeger and Colding. We will prove quantitative splitting theorem, volume convergence theorem, and the result that almost volume cone implies almost metric cone. Then we will discuss regularity of Einstein manifolds and the codimension four conjecture. Instructor(s): Y. Wang
Area: Quantitative and Mathematical Sciences. | 677.169 | 1 |
Teaching Textbooks Pre-Algebra Package
Prepare for algebra with simple equations, powers, polynomials and much more. More than 120 hours of instruction and 139 lessons on 10 interactive CDs with lectures, problems, step-by-step solutions, tests and automated grading. Includes a 750-page student workbook, answer key and 17 chapter tests. Version 2.0. Windows and Mac compatible.
Reference numbers for each problem so students and parents can see where a problem was first introduced
A digital gradebook that can manage multiple student accounts and be easily edited by a parent
Built in back up and gradebook transfer feature
Compatible with PC (Windows XP or later) and Mac (OS 10.4 or later)
Students watch video lectures on CD-ROM, do problems from the 600-page workbook. When they need help or review, you've got a printed Answer Key plus audiovisual step-by-step solutions to every homework and quiz problem. | 677.169 | 1 |
this textbook. I use the book this summer to tutor and prepare my daughter, who will be a 9th grader, for the upcoming new school year. I find the book very easy to read and each lesson is not overloaded with new material. In each exercise section, the first part of it is oral exercise that provides the new math concept practice without involving pencil and paper. There are three levles of difficulty in lesson exercise, A, B, and C. Overall, this is a great book for us. I have a strong math background so I don't need a solution book to go with it but it'll be nice to have it.
We've only used this a little bit, so far, but it looks to be similar to the first volume, which was excellent. Good, logical progression of concepts, but the best part is that there are plenty of exercises which test the concepts and force the student to show mastery of the concept. Most books we've seen, and also the hw we get at school, only test the student's ability to copy the pattern of problem that was demonstrated in the teaching section. A better test is to include problems which are different enough to require that the student really understood the underlying concept and can then apply it to a slightly different problem. We used this text to supplement whatever was used in the local school.
Relearning algebra as an adult is difficult enough. This book makes it a little easier. Thorough. As i read through all of the gaps in my understanding from previous, less thorough methods I have tried are becoming crystal clear. Tons of practice (which makes perfect right?). Answers to many in the back of the book to check yourself (although I wish a few of these perhaps one from each set had worked out solutions). Will be supplementing my daughter's education with this book.
A thorough textbook with easy to follow examples. There are numerous word problems on all topics. The exercises are divided into three categories (A, B, C) and are in increasing difficulty with section 'C' being the most challenging. I would reccommend the text to teachers who want a rigorous complete Algebra 1 course. | 677.169 | 1 |
Equations with variables give you the power to describe situations more generally than a single set of numbers allows.
If you've taught Algebra 1, do you agree with those? Am I missing some? If I ever find myself teaching this class again, I'd like to try to present those up front and highlight them repeatedly as they come up, rather than gradually discovering them for myself and trying to make them clear to my students over the course of the class.
I think one of the most problematic concepts of Algebra 1 is "Simplifying". It's so very easy to slip into "here are the rules for what constitutes simplified in this case, and this case, and this case..." (polynomials, exponents, rationals, radicals, ...) and so very difficult to access a big idea like, "We're trying to make our lives easier when we move on to the next step of solving or using a complicated equation that describes some situation in real life like particle physics or the stock market or the operation of an engine or the forces on a bridge." I can say that to students, but I have a hard time showing them or getting them to experience it...
Second, I find that I have some startlingly clear memories of my own Algebra 1 class in eighth grade in 1990, probably because some of the subject matter I learned never got used again. It seems like there's a lot of manipulation and variations on a theme that we teach students but aren't actually critical to their ongoing life as mathematical practicioners.
For example:
I've used a lot of math in my life and graphed a lot of linear functions and data. All you really need is a deep and thorough understanding of y = mx + b. You do not need the point-slope form of a line. You might need the standard form when you get around to doing linear functions at a deeper level in Linear Algebra (and all of its applications in physics and economics and ...) in college, but could it wait until then?
I've used a lot of math in my life, and while being able to factor out the greatest common factor of a polynomial is a deeply practical skill for all the math it can make easier, and factoring x^2 + bx + c offers a sort of aesthetic pleasure and can be used to ensure that students understand FOIL at a deeper, backwards operation level, no one ever tries to factor ax^2 + bx + c . You pull out the quadratic formula, because it's so much more general and so much less guess and check. Why do we then waste time teaching an essentially guess-and-check approach to solving it?
What's your perception of which parts of the standard Algebra 1 curriculum are actually essential and useful to practicing users of math in this modern, computational age, and which parts are historical cruft?
1 comment:
First, a quote: "'When are we ever going to use this in real life?' has been the cry of bored math students since time immemorial. Though the vast majority mightn't a clue what 'real life' entails, it's still a fair question that deserves an answer. Coming up with practical examples of math in real life is a reasonable approach.
But I would go further. I would deny the very implicit premises that the question is based upon: first, that the only really legitimate knowledge worth having is practical 'real life' knowledge, second that anything which lacks an immediate, direct application is by definition impractical."
From
I understand that this doesn't necessary apply to summer school, though.
I find it useful to be able to factor ax^2 + bx + c in the trivial cases: a^2 + b^2, a^2 + 2ab + b^2, etc. But, I doubt I'd recognize these as trivial without the more extensive exposure I got in school.
I have less to say on the point-slope form, except that being exposed to different views of the same information seems like a useful way to bring up that there are often multiple ways of approaching a problem. We like to chose the easiest, but we don't know which way will be the easiest unless we know which ways there are, and that requires a passing familiarity with the ways in the first place. Reinventing the wheel is fun once in a while, but giants have shoulders for a reason. :)
(Real life example: If you're writing a computer program to solve a Sudoku puzzle, there are many ways to approach it. You aren't going to realize it is trivially convertible to a graph coloring problem unless you're familiar with graph theory, though. | 677.169 | 1 |
UND Math Active Learning Lab (MALL)
Math is not a spectator sport at UND
Beginning Summer 2017, students enrolled in a math class from MATH 92 through MATH 112 (Algebra Prep II through Transition to Calculus) will learn through a student-centered emporium environment called the Math Active Learning Lab (MALL).
Spend your usual class time in the lab during any open MALL hours
Learn and work on math through interactive instructional software
Get coaching from faculty, graduate teaching assistants and undergraduate tutors
Progress at your own pace to become more comfortable and proficient with math
Register for a required weekly Focus Group meeting to connect you with an instructor | 677.169 | 1 |
ISBN-10: 1437723667
ISBN-13: 9781437723663 pharmacy technicians, and addressing the competencies developed by the American Society of Health-System Pharmacists (ASHP), Math Calculations for Pharmacy Technicians, 2nd Edition helps you learn to calculate drug dosages safely and accurately. A practical worktext format covers everything from basic math skills to reading and interpreting labels and physicians' orders, introducing key calculation and conversion concepts and then providing hundreds of problems so you can practice and master the material. Other vital topics include conversions between the various measurement systems, reconstituting liquid medications, and calculating medications based on a patient's age or body weight. Written by experienced pharmacist Robert Fulcher and educator Eugenia Fulcher, Math Calculations for Pharmacy Technicians helps you learn calculation skills and develop the competencies needed by pharmacy technicians.Learning objectives and definitions of key words begin each chapter.Pretests in each chapter allow readers to assess their current knowledge of specific topics.Step-by-step examples make it easy to learn and remember how to do equations and use formulas.Hundreds of practice problems provide practice with calculations, conversions, and measurements.Actual drug labels accompany examples and problems, for real-world experience with the information you will see in pharmacy practice.Business Math for Pharmacy Technicians chapter introduces the calculations needed in retail pharmacy settings.Body system icons appear next to medication names to help you associate different drugs with their respective disorders and body systems.Points to Remember boxes make it easy to learn and remember key information.Review of Rules sections in each chapter summarize the rules and methods for performing equations.Chapter reviews provide a quick summary of the key concepts in each chapter.Posttests in each chapter allow you to assess how well you have learned the material.A comprehensive posttest includes 50 questions that assess your knowledge of all major topics covered in the book.Helpful study tools also include an answer key for odd-numbered problems and a comprehensive glossary.Updated content meets ASHP requirements and features new topics such as powder volume and compounding problems, formulas for reducing and enlarging medications, and opportunities to write out prescription label directions.Tech Note boxes offer helpful advice on real-life situations you may encounter in the pharmacy.Tech Alert boxes warn against common pharmacy and medication errors that could impact patients' safety.Additional prescription and practice exercises give you valuable experience with translating physician directions into patient instructions | 677.169 | 1 |
Description for Discrete and Combinatorial Mathematics
Description This fifth edition of Discrete and Combinatorial Mathematics: An Applied Introductioncontinues to improve on the features that have made it themarket leader. The text is flexibly organized, enabling instructorsto adapt the book to their particular courses. Excellent exercise sets allow students to perfect skillsthrough practice. This new edition continues to feature variouscomputer science applications, making this the ideal text forpreparing students for advanced study. Historical reviews and biographies bring a human element to their assignments. Chapter summaries allow students to review what they have learned. Expanded treatment of discrete probability in Chapter 3. New material on cryptology, private-key cryptosystems in Chapter 13, public-key RSA cryptosystems in Chapter 15. Fundamental Principles of Counting Fundamentals of Logic Set theory Properties of the integers:Mathematical Induction Relations and Functions Language:Finite State Machines Relations:The Second Time Around The Principle of Inclusion and Exclusion Generating Functions Recurrence Relations An introduction to graph theory Trees Rings and modular arithmetic Boolean algebra and switching functions Algebraic structures,semigroups,monoids,groups,coding theory and polya's method of enumeration Finite fields and combinatorial designs Solved Question Papers perhaps the best reference for Discrete Math I have seen. The book is comprehensive. It gives an introduction to many topics every CS coding theory, number theory, partial orders, even automata. There are many examples and illustrations to understand the material. There are plenty of excercises. The text is well written; no errors or typos. It would be nice if the book had more proofs.
Finally! A math book which is acutally well written, has enough examples to illustrate key concepts, and has enough problems to keep the math student busy. Discrete mathematics is a fairly involved subject and books on the topic range from relatively basic to extremely difficult treatises which only a PhD or a math professor could understand. Discrete and Combinatorial Mathematics : An Applied Introduction by Ralph Grimaldi is a book which will appeal to both sides of the spectrum. The book is written so that most undergraduate students will have little difficulty understanding, but graduate students will also find it indispensable as a reference. The illustrated examples are actually relevant to the homework problems, which is often missing in mathematical texts. Finally, the book does not try to overwhelm the reader with lofty proofs or stilted language. Each chapter builds on the previous subjects learned. That's all I can ask for in a math text. I like the coverage of combinatorics in the first chapter, which does a better job than many probability textbooks. And be sure to understand Euclid's theroem and the examples given in the book. Quite a few high-tech companies will ask you about the problem Grimaldi gives as an example of Euclid's theorem in their job interviews. | 677.169 | 1 |
Deep Thinking book. Please register to get access to this book, don't worry this is free.
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Book Description
There is more than one way to think. Most people are familiar with the systematic, rule-based thinking that one finds in a mathematical proof or a computer program. But such thinking does not produce breakthroughs in mathematics and science nor is it the kind of thinking that results in significant learning. Deep thinking is a different and more basic way of using the mind. It results in the discontinuous "aha!" experience, which is the essence of creativity. It is at the heart of every paradigm shift or reframing of a problematic situation.
The identification of deep thinking as the default state of the mind has the potential to reframe our current approach to technological change, education, and the nature of mathematics and science. For example, there is an unbridgeable gap between deep thinking and computer simulations of thinking. Many people suspect that such a gap exists, but find it difficult to make this intuition precise. This book identifies the way in which the authentic intelligence of deep thinking differs from the artificial intelligence of "big data" and "analytics".
Deep thinking is the essential ingredient in every significant learning experience, which leads to a new way to think about education. It is also essential to the construction of conceptual systems that are at the heart of mathematics and science, and of the technologies that shape the modern world. Deep thinking can be found whenever one conceptual system morphs into another.
The sources of this study include the cognitive development of numbers in children, neuropsychology, the study of creativity, and the historical development of mathematics and science. The approach is unusual and original. It comes out of the author's lengthy experience as a mathematician, teacher, and writer of books about mathematics and science, such as How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics and The Blind Spot: Science and the Crisis of Uncertainty.
Contents:
What is Deep Thinking?
Conceptual Systems
Deep Thinking in Mathematics and Science
Deep Thinking in the Mind and the Brain
Deep Thinking and Creativity
Deep Learning
Good Teaching
Undergraduate Mathematics
What the Mind Can Teach Us About Mathematics
What Mathematics Can Teach Us About the Mind
References
Readership: Students, graduate students and researchers with an interest in mathematics, mathematicians, scientists, philosophers, psychologists, and readers who use mathematics in their work. Key Features:
In this book, the author, a mathematician, demonstrates the specific difference between creative mathematical thought and the analytic thought of logic and the artificial intelligence of computers
This book shows why learning is a creative activity and demonstrates how teaching and learning must undergo radical changes in this age of rapid technological change
Mathematics is a model for how people think and it reveals the essence of intelligence
The author, a mathematician, demonstrates that creativity is a basic feature of the world. The same phenomenon of creative intelligence underlies the theory of evolution, child development, learning, and scientific and mathematical research. Creativity is so natural that even babies are capable of it but so difficult that adults have great trouble with it. This book explains the reasons behind this apparent paradox | 677.169 | 1 |
Parent Category: Math
ABCs for smart people – learn how the alphabet evolved into what we use today, plus basic typography terms, and see the Greek alphabet, Chinese alphabet, sign language alphabet, Japanese alphabet and much more.
Columbia Academy learning center offers math, reading, and writing after-school academic enrichment and classroom programs, small group tutoring, and test preparation for elementary, middle, and high school students in New Jersey.
The Quadratic Equation Solver posted on this page solves for both real and imaginary (if applicable) results. Also on this site: solvers for the Cubic and Quartic equations, N Equations in N Unknowns, Eigenvalues and Eigenvectors, Cubic Spline Interpolati | 677.169 | 1 |
PDF (Acrobat) Document File
Be sure that you have an application to open this file type before downloading and/or purchasing.
9.08 MB | 48 pages
PRODUCT DESCRIPTION
Understanding linear topics is foundational for Algebra 1 and 8th Grade Math students. Yet, when I open the typical Algebra or 8th Grade Math textbook, I am overwhelmed with formulas and definitions in Key Concept boxes that DON'T give students any opportunity to interact with or discover the concepts themselves.
This bundle is a labor of love, developed over many years. The resources found here are my attempt to look at "the book", see what students need to understand, then develop a resource for them to interact with and "play" with the mathematics before being told the definitions or formulas.
I hope you find these supplemental resources to accompany your textbook and additional practice activities and handouts useful, as they help students understand linear topics more deeply. Formative assessments, discovery activities, graphic organizers, error analysis, and more, all in one place!
Please download the "Preview" PDF, which serves as a detailed Table of Contents for this bundle. Some resources are posted individually in my store already, but others are exclusive to this bundle | 677.169 | 1 |
This laboratory text and accompanying instructor's manual (available from the department) are based on an unusually effective, sophomore level course developed and taught in the Mathematics and Statistics Program of Mount Holyoke College over the last twenty-eight years.
The "Lab," as it is called, serves as a bridge between first year courses (often college geometry or number theory as well as calculus) and more sophisticated upper level mathematics courses. Students explore ideas that they will encounter later and more formally in advanced courses. They learn to experiment, to describe patterns, to generalize, to conjecture, and to argue with different degrees of certainty.
The course is central to our mathematics curriculum. The Lab is the key element in allowing us to offer students a number of alternative entries (that is, entries other than the standard calculus sequence) to the study of mathematics. It has also helped us develop an interactive, conversational mode of mathematics teaching that we have found effective in other courses. We have observed that the Lab improves the performance of students in real analysis and abstract algebra.
The student text consists of sixteen modules drawn from a wide range of mathematical and statistical contexts, and each introducing an idea or ideas that the student is likely to encounter in later courses. In a typical offering of the course, the instructor will choose six or seven modules. Each begins by placing the topic in context and providing some background. Then students respond to questions which invite them to examine examples, first by hand and then by computer. The student is encouraged to find and describe patterns, to generalize from observations, to formulate conjectures, and to support conjectures with analysis and sometimes proof. Each project requires a carefully written laboratory report describing the student's findings, conjectures and conclusions.
The course has worked far better than our initial expectations and would, we think, be easy to adapt to a variety of institutions. It is cheap to implement, could be run on calculators, and succeeds wonderfully in engaging students in doing mathematics. It is also easy to teach (although grading it is no picnic), and several sabbatical visitors have thoroughly enjoyed teaching the Lab.
With the advice of participants in our NSF-funded Undergraduate Faculty Enhancement workshops in 1997 and 1999, we have prepared some corrections and clarifications for the student text and suggestions for the instructor's manual .
Please contact the department for any questions about the materials or the course. | 677.169 | 1 |
1852331528
ISBN: 1852331526
Publication Date: 2000
Publisher: Springer Verlag
AUTHOR
Pressley, Andrew
SUMMARY
Curves and surfaces are objects that everyone can see, and many of the questions that can be asked about them are natural and easily understood. Differential geometry is concerned with the precise mathematical formulation of some of these questions, and with trying to answer them using calculus techniques. It is a subject that contains some of the most beautiful and profound results in mathematics, yet many of them are accessible to higher level undergraduates.Elementary Differential Geometry presents the main results in the differential geometry of curves and surfaces while keeping the prerequisites to an absolute minimum. Nothing more than first courses in linear algebra and multivariate calculus are required, and the most direct and straightforward approach is used at all times. Numerous diagrams illustrate both the ideas in the text and the examples of curves and surfaces discussed there.Pressley, Andrew is the author of 'Elementary Differential Geometry', published 2000 under ISBN 9781852331528 and ISBN 18523315 | 677.169 | 1 |
0 overview of our redesign
1.
Overview of our Course RedesignMontana State University BillingsChairsty Stewart, Assistant Director (cstewart@msubillings.edu)Vivian Zabrocki, Math Instructor (vzabrocki@msubillings.edu)Fall 2011Basics: We have 4 instructors and a section coordinator (who is also an instructor) that are responsiblefor a class of approximately 110 students. Students are placed into this course based on their COMPASSscore (Alg 16 – 50). During the semester, we usually have 3 classrooms available with computers, 1 or 2without computers and a lab with 40 computers. Since the inception of this course, we typically haveMOST students testing into Module A (see below) which is the beginning module, even though a lot ofthe students should be able to test out of that material based on their COMPASS score.Days 1 – 4 of Class: Students begin with an overview/explanation of the course and then they enroll intoMathXL. They begin taking a series of pre-tests (designed by us, based on our module tests) to help usfurther determine their placement (they have already been placed via COMPASS into either Introductoryor Intermediate Algebra) into one of four modules. After the 4 days, we sort them based on pre-testscores into the appropriate module. We usually have multiple groups of students testing into Module Athat are further divided based on their performance on the pre-tests (low, medium and high classes), asmall group that tests into Module B and one or two that will test into C and D.Time Block 1 (Days 5 – 19 of Class): Students begin attending the assigned module and ENROLL IN ANEW COURSE in MathXL. During this 15 day time block, instructors mix lecture with brain-based learningactivities and other strategies to teach the concepts to the students. Classes are 60 minutes long. Ifthere is time remaining at the end of the class, students are urged to use their time completing theirMathXL homework. Students have homework for each section in MathXL and quizzes after 2 -3 sections.Students MUST score 100% on their homework assignments (3 attempts) and 80% or better on theirquizzes (2 attempts). In order to take the module test (paper-pencil) on the 15th day of the time block,students MUST have completed ALL their MathXL assignments up to the required standard. Studentsmust also score 80% or better on the module test in order to progress to the next module. If they don'ttake the test or get the required score, they repeat the module.Time Block 2 (15 days): If students pass the test, they move to the next module and ENROLL IN A NEWCOURSE in MathXL. If they don't pass, they also ENROLL IN A NEW COURSE in MathXL and REPEAT ALLthe assignments. Class is as stated above.There are a total of 4 time blocks, 4 modules and a final exam. Each time a new time block starts,students need to enroll in a new MathXL course. | 677.169 | 1 |
This text aims to give undergraduate students a grasp of the basic concepts of round-off errors, stability, conditioning, and accuracy as well as an appreciation of the core numerical linear algebra algorithms, their basic properties and implementations. | 677.169 | 1 |
Monday, 23 May 2016
Essential MATLAB for Engineers and Scientists EBook by Brian D. Hahn
Third edition by Brain D. Hahn
and Danial T.Valentine on essential MATLAB for engineers and scientists. In this ebook you will learn about mathematical
tool and a programming language. There are seventeen chapters in this complete ebook
included 449 pages. This ebook is upgraded by its previous vision, now you will
learn new with this ebook.MATLAB is based on the mathematical concept of a matrix and MATLAB
computer programming. You will learn in this ebook about BASIC, Pascal, C, C++
and MATLAB. Complete solutions to many of the exercises appear in an appendix.
If you want original copy of this ebook, you can by it Amazone otherwise you
can simply download here. All the ebooks are free which are in this blog. Let me
know if you want other specific ebook for your course.
Jaivinder Singh is a SEO Executive at Two Minds Technology Pvt. Ltd. He is also a student of BCA from IGNOU. He spent most in front of computer, He is passionate about SEO. He create this blog for engineering students where students can get help about their course they can download and read Ebooks. You can connet with me on following social sites
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Computer Resources
There are many computer resources that will assist students in their math courses. Make an appointment with the Math Lab Coordinator to learn to use:
CD lecture series (for most math courses)
MyMathLab (Web resource with practice tests and assignments graded by the computer Internet Resources). MyMathLab must be purchased with a new book, or purchased as an extra with a Used Text. (Contact the Bookstore.) If your instructor does not use MyMathLab ask the Math Lab Coordinator how to use it.
CD and Video Lecture Series: VideoThere is also a video Calculus Lecture Series available for loan.
Worksheets on Difficult Subjects
Worksheets and concept aids are available in a variety of courses and to assist with understanding commonly misunderstood concepts such as
On Line Learning styles survey - Explains the four styles, lets you take a survey and gives you instructions on how to best study, given your style. Then use our brochure Learn to Study Math for Your Style
CD and Video Lecture Series:
VideoNote: PurpleMath sites have good explanations but don't have exercises; AAAmath & Gomath do have practice exercises with answers.
Ask a Math Question: Web Math
This site is a commercial site (will contain advertisements). It is both a tutorial (showing how to do a type of problem) and a place to enter a problem you can't solve. It will solve the problem, explaining all the steps. Covers Algebra and below. Also allows you to download a demo copy of their PC software. If you like it, you can order the software to have on your PC and use without going on the Web ($29.99).
Ask Dr. Math
This in non-commercial site created and maintained by Swarthmore College. You can submit a question and wait for a response. You can also search the list of questions others have asked (yours might already have been answered) and see that answer immediately.
Take a Tutorial in Math:
Khan Academy
This site has tutorials and quizzes online in all types of math, from elementary to college calculus, and many other science and social science topics.
Cool Math
This site has tutoring in all types of math, from elementary to college calculus, plus some nice explanations of algebra concepts. | 677.169 | 1 |
What is the "most uniform" way of distributing n points in the unit square? How big is the "irregularity" necessarily present in any such distribution? Such questions are treated in geometric discrepancy theory. The book is an accessible and lively introduction to this area, with numerous exercises and illustrations. In separate, more specialized parts, it also provides a comprehensive guide to recent research. Including a wide variety of mathematical techniques (from harmonic analysis, combinatorics, algebra etc.) in action on non-trivial examples, the book is suitable for a "special topic" course for early graduates in mathematics and computer science. Besides professional mathematicians, it will be of interest to specialists in fields where a large collection of objects should be "uniformly" represented by a smaller sample (such as high-dimensional numerical integration in computational physics or financial mathematics, efficient divide-and-conquer algorithms in computer science, etc.).
DOWNLOAD: Buy Premium From My Links To Get Resumable Support & Max Speed & To Support Me | 677.169 | 1 |
Math Study Guide, State Standardized Tests [Paperback]
Item description for Math Study Guide, State Standardized Tests by The Learning Hand...
Math is learned by practice. The more the practice, the better a student becomes at math. This Math Study Guide contains lots of practice exercises. Each exercise has a sample example or two which are worked out to show the student the steps needed to solve the exercise problems. The Math Study Guide is arranged methodically where each math topic leads into the next topic. Math is a step-by-step process. This Study Guide contains a complete comprehensive study and practice material for Grades 2 through 5. Ideal for home, school and standardized test preparations Math Study Guide, State Standardized Tests | 677.169 | 1 |
Showing 1 to 1 of 1
Philosophy 03
Platos Republic
PLATO(427-347 B.C)- admired Socrates
Plato is somewhat like Socrates student
He wrote novels, called dialogues: the main character in his dialogue is named
Socrates ( platos literally creation not the ACTUAL philosopher) base
AP Calculus Advice
Showing 1 to 3 of 4
I would recommend it because it is a college course and it would help a lot of college students save more money. It is a very difficult class but all it takes is time and effort, studying, and working hard to make sure you know the topic.
Course highlights:
Just everyday learning something new. Walking into class everyday I knew I would learning something new because Calculus is a very difficult subject with a lot of materials in it.
Hours per week:
6-8 hours
Advice for students:
It's going to be a grind especially if you are an student-athlete because there's going to be nights when you have to stay up studying the work & doing homework after a long practice.
I would recommend this course because it's a good class to take to prepare you for the calculus that you will receive college. It's definitely challenging though, but once you memorize all the formulas and the steps to solving the math problems it becomes a lot easier.
Course highlights:
I learned quite a lot in my AP Calculus class: such as how to solve limit problems, how to find velocity, and speed of objects, how to use graphing calculators for more than just basic math, and how to take the derivative of equations.
Hours per week:
12+ hours
Advice for students:
You can definitely pass this class, you just have to work together with study groups and keep asking for help, if necessary! | 677.169 | 1 |
...
Show More down content into manageable chunks and maximise students' concentration. An exam practice chapter at the end of the book provides detailed guidance on exam technique and makes sure students are fully prepared for the big day. This book provides: short revision sessions on every topic; worked examples to show how to tackle every aspect of Maths; hint boxes for extra guidance and support; Intermediate and Higher Level content clearly indicated; and check yourself questions to test understanding | 677.169 | 1 |
Users Community
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This math video app provides a quick and simple way for you to learn and understand the basics behind linear graphs and equations. All of the 12 computer animated videos are stored in your device. So, no internet connection required to play. These math videos are: The coordinate plane, the… more
Nullspace is a universal app that fulfills the need to solve linear systems of equations for common and advanced requirementsBrief description:Solve up to 12 equations and variablesNullspace vector solutionsSolve multiple equations at oncePresent results in fraction, decimal and scientific… more
You are stationed at a remote outpost and your mission is to guard the border. It has been quiet for several years as you have never even seen another soul. Your life is about to change. Headquarters has detected a massive enemy attack that is headed straight for you. Wave after wave of enemy… more
The lite version is ad free for a limited time. Get it while it lastsYou are stationed at a remote outpost and your mission is to guard the border. It has been quiet for several years as you have never even seen another soul. Your life is about to change. Headquarters has detected a massive… more
This is a calculator useful by various scenes depending on your conception. For instance, it is possible to use it for the calculation of the interest rate, the calorie, the experiment, and statistics etc.Anyway, you always quickly take it out of your pocket, and can use it to calculate at once. … more | 677.169 | 1 |
Literal Equations
Be sure that you have an application to open
this file type before downloading and/or purchasing.
714 KB|10 pages
Product Description
This lesson over literal equations can be used for math or science classes.It starts with a note page that students can put in their interactive notebook. The teacher will have an opportunity to talk about the process and then go through another problem. The notes take the student through a process of writing the equation in words, circling what variable is being solved for and then solving the equation. Another page takes the students through a regular equation then through a literal equation set up in the same way. The next page shows students the types of equations they will see in their science classes and math classes. | 677.169 | 1 |
Combination AB/BC Solution version of Calculus Cache of Hidden Treasures.
832 AB/BC problems/solutions in Microsoft Word Format. $189.95.
What is it? Calculus Cache of Hidden Treasures is an extensive database of multiple-choice questions in both AB and BC Calculus, intended to make it easy for teachers to create worksheets, quizzes and exams. There is a student version and a solution version.
How is it Organized? Calculus Cache of Hidden Treasures is organized by topic. There are 36 AB topics and 15 BC topics:
AB Topics
BC Topics
Limits
Absolute extrema
L'Hospital's Rule
Rates of change
Optimization
Integration by Parts
Derivative by Definition
Basic integration
Integration using Partial Fractions
Basic Derivatives
Integration w/substitution
Improper Integrals
Equations of tangent lines
Integration w trig
Euler's Method
Derivative of trig functions
Integration w/ln & e
Logistic Growth
Chain rule
Definite Integrals as Area
Arc Length
Local linear approximation
Riemann sums & trapezoid rule
Parametric Equations
Implicit differentiation
Fund. Thm. Of Calculus
Vector Functions
Derivatives of ln & exponential
Accumulation function
Polar Curves
Horizontal & Vertical tangents
Integrals of rates of change
Taylor Polynomials
Differentiability
Average value of functions
Infinite Series
Intermediate Value, Rolles', MVT
Motion using integrals
Error Bounds
Inverses
Inverse trig
Power Series
Related Rates
Area
Taylor, Manipulation of Series
Straight-line motion
Volume
Function analysis
Slope fields/DEQ's
2nd derivative test
Growth Problems
How many problems are there? 588 for AB and 244 for BC. Within any topic, there will be between 10 and 24 problems, usually 16.
What about the difficulty of the problems? Within any topic, half of the problems will be easier - usually concentrating on one aspect of the course, and half of them will be harder - the type which students can expect in the AP exam involving multiple concepts.
What prior knowledge is assumed within a topic? The approach used is that students will not be responsible for any topic prior to it being taught. For instance, problems on straight-line motion will not use integrals. So questions about distance traveled would not be asked until integrals are taught. There is a later section on straight-line motion using integrals. This allows teachers to use any question within a topic without the fear that a concept has not yet been taught.
Are calculators required? For the most part, calculators are not necessary or allowed. However there is a smattering of "calculator active" problems, especially in areas where complicated definite integrals need to be taken.
What does it cost? Calculus Cache of Hidden Treasures is absolutely free for the student version.
When will it be available? Like our Diving In and Ripped From The Headlines, Calculus Cache of Hidden Treasures is a subscription. It will begin August 20, 2015, and every other week, 3 AB topics will be available and 2 BC topics will be available for download.
How do I get the problems? If you are enrolled in the MasterMathMentor community, you get them automatically. As with Diving In, when a new issue comes out, you will receive an email inviting you to download the AB problems, BC problems, or both.
If you are not a member, complete a free registration for our Mastermathmentor community. You can do so at: Registration page. Once you do that, you will automatically receive Calculus Cache through your email address (home or school or both) every 2.5 weeks.
What is the format of the problems? The student version will be in PDF format.
What about the solutions? The most exciting part of Calculus Cache of Hidden Treasures is that teachers who purchase solutions will receive it in Microsoft Word format. This will allow teachers to create exams or worksheets that use as many or as few problems as needed, edit the problems if they wish, print out answer keys, change the size of graphs and charts, and more. While answers in PDF format will also be available upon request, having Calculus Cache of Hidden Treasures in Word format gives teachers a huge amount of flexibility. It is something that teachers will find invaluable.
I have an older version of Word. Will the solution versions work? I work on a Mac and save the files in a format that can be opened by Word: 1997 - 2004. PC's work slightly differently. So check it out. Below is a link to a sample page from Calculus Cache of Hidden Treasures in Word format. The page has all the elements that the entire database will have: text, equations, graphs, and tables. Open the page and see whether you can do the following:
Editor will be needed if you wish to change any of the problems. Teachers
purchasing solutions will receive a short tutorial.)
What do the solutions version cost? The student version is free. The AB solution version (containing the problems and solutions) is $129.95 and the BC solution version is $69.95. The cost of purchasing both is $189.95.
Can my school purchase it via a purchase order? Absolutely. Contact me at sschwartz8128@gmail.com for instructions.
I don't want to do all that downloading. Can I get it on a flash drive? Calculus Cache of Hidden Treasures is a subscription and each set of three AB topics and one set of BC are issued every two weeks. When all issues have come out, we will have an option for purchasing the entire database of problems on a flash drive. Teachers who have purchased the solutions will be able to receive a flash drive with the entire database for $17.95 for AB and $25.95 for AB and BC plus $6 shipping.
Can I place the problems and/or solutions on the Internet? Absolutely not! However, if you create your own exams by electronic cutting and pasting and modifying problems, what you do with it is up to you. Realize though that if you place your own exams on the Internet, your next year's students may very well find it and cheat. Do yourself and the rest of us a favor. These problems are your "treasures." Keep these treasures to yourself!
I hope you are as excited about Calculus Cache of Hidden Treasures as I am. Having the ability to quickly create calculus exams on the fly with full solutions is something I would have loved to have had when I taught. I am really thrilled to be able to bring this extensive collection to calculus teachers. | 677.169 | 1 |
EE math! Have topic and a simple draft! Need tips!
I have my math topic and its due soon. But the teacher is not very optimistic about the level of the math. she says its quite simple.
My essay is basically making a graphic circle on the computer screen using the coordinate system and then moving it like an animation by calculating the coordinates each time and adding a certain amount to it so it moves. and I am going to animate a projectile motion of a throwing ball so some physics will be involved. also since its the computer I can do the calculations in the binary to add a little taste to it. Question IS: HOW can i make the math more complicated so I dont fail the criteria. Any tips will be appreciated. sorry for long question. | 677.169 | 1 |
ISBN-10: 0321969928
ISBN-13: 9780321969927Mathematics in Action" series, students discover mathematical concepts through activities and applications that demonstrate how math applies to their everyday lives. Different from most math books, this series teaches through activities--encouraging students to learn by constructing, reflecting on, and applying the mathematical concepts. The user-friendly approach instills confidence in even the most reticent math students and shows them how to interpret data algebraically, numerically, symbolically, and graphically. The active style develops mathematical literacy and critical thinking skills. Updated examples, brand-new exercises, and a clearer presentation make the Fifth Edition of this text more relevant than ever to today's students 0134134427 / 9780134134420 Mathematics in Action: Algebraic, Graphical, and Trigonometric Problem Solving Plus NEW MyMathLab -- Access Card PackagePackage consists of:0321431308 / 9780321431301 MyMathLab -- Glue-in Access Card0321654064 / 9780321654069 MyMathLab Inside Star Sticker0321969928 / 9780321969927 Mathematics in Action: Algebraic, Graphical, and Trigonometric Problem Solving Students, if interested in purchasing this title with MyMathLab, ask your instructor for the correct package ISBN and Course ID. Instructors, contact your Pearson representative for more | 677.169 | 1 |
Mathematics is the common language of science, and many a theory is expressed, debunked and redeemed by the infallible truth of numbers. This programme aims to produce undergraduates with a solid foundation in Mathematics and a strong ability to solve practical as well as formulated problems.
The programme aims to produce undergraduates with a strong background in the Mathematical sciences and to produce graduates with problem-solving abilities to meet the challenges of an ever-changing world. | 677.169 | 1 |
Abstract
We know to multiply the first input by the first value, the second input by the second value, etc., and add the result. The ICCA conferences have a strong interdisciplinary character. Information posted prior to the beginning of the semester is frequently tentative, or based on previous semesters. It felt more like a mathematical discussion rather than a lesson. Rubber bands or string can be used to form various shapes around the raised notches or rods.
This book is on abstract algebra (abstract algebraic systems), an advanced set of topics related to. The personnel mix of the department has also shifted. Some features of this site may not work without it. Number and equations are actually used in almost anywhere in the world. Description: Advanced topics in abstract algebra, including ring theory and field theory; introduction to Galois theory. Prentice hall pre calculus answers, 9th grade english textbook, how to solve difficult exponential equations, is otto bretscher linear algebra bad, algebra 2 mcdougal answer.
Where volume one helped establish the basics and form the reader's understanding, volume two expands that knowledge in a way that demands full immersion into the text. This undergraduate-level course is 5 weeks. At my school, I took linear algebra and abstract algebra in the same semester, and I wasn't skirting any prerequisites. Let me now address a second audience: those willing to try a new approach. Isomorphism theorems for groups and rings. Below are the most common reasons: You have cookies disabled in your browser.
Under multiplication, G and H aren't even groups, being that there are no 6.42 Prove that Q, the group of rational numbers under addition, is not isomorphic to a Suppose that H is a proper subgroup of Q. We should separate the inputs into groups: And how could we run the same input through several operations? Discrete mathematics is the foundation of computing. Be sure when either multiplying or dividing to carry out the action upon all separate parts of the equation, which will isolate and solve one of the variables.
Because x appears to the first power, we call that a linear equation. This is not a replacement for coming to class and taking notes yourself. I summarized those talks in this blog post. You can access Fields and Galois Theory, by John Howie as a Columbia library ebook, with pdf available via SpringerLink ebook link on the right side on the page. You can also get the problems, and their solutions that I assigned for take home work when I taught the class in the Fall of 2014.
An archetypical example of this evolution can be seen in the theory of groups. Why does the fact that they are even make it an "alternating" group? It is harder than "Elements of Modern Algebra" by Gilbert and Gilbert, and finally it is harder than "Abstract Algebra" by Blair and Beachy. 2. The only exceptions to the faculty member�s reporting obligation are when incidents of sexual violence are communicated by a student during a classroom discussion, in a writing assignment for a class, or as part of a University-approved research project.
I tried looking it up and all I see is a ton of weird symbols and terms. Many groups can be defined in terms of symmetry of geometrical objects; in fact, group theory has been referred to as the mathematical study of symmetry. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras. This site has automated quiz and word problem generators for these types of common algebra word problems: two-number problems (sum & product known), consecutive integers, distance/time/rate, average/count, sum, markup, markdown, percent, percentage, two coins, and work word problems.
Comments are included throughout the text dealing with the historical development of abstract algebra as well as profiles of notable mathematicians. So part of algebra is learning the language of how to write down consistently precisely what you mean when you are trying to express or represent something numerically; it is about learning a precise language. You will surely want to buy an additional book on how to write proofs if your school is using this book for a intro course to abstract algebra. ...
Real nxn invertible matrices form a group. It is intended for undergraduate students taking an abstract algebra class at the junior/senior level, as well as for students taking their first graduate algebra course. An integer d is a divisor of an integer n if n = da for some integer a. You will be presented with a variety of links for pdf files associated with the page you are on. In its English usage in the 14th century, algeber meant "bone-setting," close to its original meaning. | 677.169 | 1 |
If your expression is incomplete or if you make a syntax error the = sign changes to ? mark. When you touch it, this artificial intelligent graphing calculator gives you a feedback by precisely indicating the error.
The graphing calculator graphs functions on an interval. When no interval is specified, the graphing calculator automatically appends a default interval to the function or parametric expression and draws the graph on that interval. You can change the endpoints of the interval, of course.
Touch Polar on the upper right of the graphing area to draw the graphs using the polar coordinate system.
Touch and drag the graphs to see the required region. To center the graph on the screen, touch the coordinate indicator on the top left of the graphing area.
Touch ++ or -- on the lower right corner of the graphing area to zoom in or zoom out.
Touch the lower left corner of the graphing area (where the caption is) to reset zoom.
On this graphing calculate most keys are assigned a second functionality by long pressing. For example
* Long press of the back-space button x←: Clears the input box. Immediately after pressing it will display back the deleted expression.
* Long press of ^ (power) sign: Appends the 'intvl=(' to the expression if it is not already appended or deleted. Long press of decimal point inserts comma to separate the end-points.
All of the above with more details are included in the built-in instruction menu of the graphing calculator.
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Comic Bible 漫畫聖經 FULL version
Original comic bible hand drawings. This book included 90 topics of comic
drawings with reference to the Bible. Let our family and children learning
God's words through a relaxing and happy method. The whole book was
written in Chinese-English bilingual format.
根據聖經繪畫,原創聖經漫畫,傳統四格漫畫,以輕鬆活潑的全新方式表達上帝的話語及聖經重 | 677.169 | 1 |
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