Elementary Linear Algebra: Applications Version

1276 Pages·2010·32.38 MB·English

by  Howard Anton

About The Author

Howard Anton obtained his B.A. from Lehigh University, his M.A. from the University of Illinois, and his
Ph.D. from the Polytechnic University of Brooklyn, all in mathematics. In the early 1960s he worked for
Burroughs Corporation and Avco Corporation at Cape Canaveral, Florida, where he was involved with the
manned space program. In 1968 he joined the Mathematics Department at Drexel University, where he taught
full time until 1983. Since then he has devoted the majority of his time to textbook writing and activities for
mathematical associations. Dr. Anton was president of the EPADEL Section of the Mathematical Association
of America (MAA), served on the Board of Governors of that organization, and guided the creation of the
Student Chapters of the MAA. In addition to various pedagogical articles, he has published numerous
research papers in functional analysis, approximation theory, and topology. He is best known for his textbooks
in mathematics, which are among the most widely used in the world. There are currently more than 150
versions of his books, including translations into Spanish, Arabic, Portuguese, Italian, Indonesian, French,
Japanese, Chinese, Hebrew, and German. For relaxation, Dr. Anton enjoys travel and photography.
Copyright © 2010 John Wiley & Sons, Inc. All rights reserved.
Preface

This edition of Elementary Linear Algebra gives an introductory treatment of linear algebra that is suitable for
a first undergraduate course. Its aim is to present the fundamentals of linear algebra in the clearest possible
way—sound pedagogy is the main consideration. Although calculus is not a prerequisite, there is some
optional material that is clearly marked for students with a calculus background. If desired, that material can
be omitted without loss of continuity.
Technology is not required to use this text, but for instructors who would like to use MATLAB, Mathematica,
Maple, or calculators with linear algebra capabilities, we have posted some supporting material that can be
accessed at either of the following Web sites:
www.howardanton.com
www.wiley.com/college/anton
Summary of Changes in this Edition
This edition is a major revision of its predecessor. In addition to including some new material, some of the old
material has been streamlined to ensure that the major topics can all be covered in a standard course. These
are the most significant changes:
• Vectors in 2-space, 3-space, and n-space Chapters 3 and 4 of the previous edition have been combined
into a single chapter. This has enabled us to eliminate some duplicate exposition and to juxtapose concepts
in n-space with those in 2-space and 3-space, thereby conveying more clearly how n-space ideas generalize
those already familiar to the student.
• New Pedagogical Elements Each section now ends with a Concept Review and a Skills mastery that
provide the student a convenient reference to the main ideas in that section.
• New Exercises Many new exercises have been added, including a set of True/False exercises at the end of
most sections.
• Earlier Coverage of Eigenvalues and Eigenvectors The chapter on eigenvalues and eigenvectors, which
was Chapter 7 in the previous edition, is Chapter 5 in this edition.
• Complex Vector Spaces The chapter entitled Complex Vector Spaces in the previous edition has been
completely revised. The most important ideas are now covered in Section 5.3 and Section 7.5 in the context
of matrix diagonalization. A brief review of complex numbers is included in the Appendix.
• Quadratic Forms This material has been extensively rewritten to focus more precisely on the most
important ideas.
• New Chapter on Numerical Methods In the previous edition an assortment of topics appeared in the last
chapter. That chapter has been replaced by a new chapter that focuses exclusively on numerical methods of
linear algebra. We achieved this by moving those topics not concerned with numerical methods elsewhere
in the text.
• Singular-Value Decomposition In recognition of its growing importance, a new section on Singular-Value
Decomposition has been added to the chapter on numerical methods.
• Internet Search and the Power Method A new section on the Power Method and its application to
Internet search engines has been added to the chapter on numerical methods.
• Applications There is an expanded version of this text by Howard Anton and Chris Rorres entitled
th
Elementary Linear Algebra: Applications Version, 10 (ISBN 9780470432051), whose purpose is to
supplement this version with an extensive body of applications. However, to accommodate instructors who
asked us to include some applications in this version of the text, we have done so. These are generally less
detailed than those appearing in the Anton/Rorres text and can be omitted without loss of continuity.
Hallmark Features
• Relationships Among Concepts One of our main pedagogical goals is to convey to the student that linear
algebra is a cohesive subject and not simply a collection of isolated definitions and techniques. One way in
which we do this is by using a crescendo of Equivalent Statements theorems that continually revisit
relationships among systems of equations, matrices, determinants, vectors, linear transformations, and
eigenvalues. To get a general sense of how we use this technique see Theorems 1.5.3, 1.6.4, 2.3.8, 4.8.10,
4.10.4 and then Theorem 5.1.6, for example.
• n
Smooth Transition to Abstraction Because the transition from R to general vector spaces is difficult for
many students, considerable effort is devoted to explaining the purpose of abstraction and helping the
student to “visualize” abstract ideas by drawing analogies to familiar geometric ideas.
• Mathematical Precision When reasonable, we try to be mathematically precise. In keeping with the level
of student audience, proofs are presented in a patient style that is tailored for beginners. There is a brief
section in the Appendix on how to read proof statements, and there are various exercises in which students
are guided through the steps of a proof and asked for justification.
• Suitability for a Diverse Audience This text is designed to serve the needs of students in engineering,
computer science, biology, physics, business, and economics as well as those majoring in mathematics.
• Historical Notes To give the students a sense of mathematical history and to convey that real people
created the mathematical theorems and equations they are studying, we have included numerous Historical
Notes that put the topic being studied in historical perspective.
About the Exercises
• Graded Exercise Sets Each exercise set begins with routine drill problems and progresses to problems
with more substance.
• True/False Exercises Most exercise sets end with a set of True/False exercises that are designed to check
conceptual understanding and logical reasoning. To avoid pure guessing, the students are required to justify
their responses in some way.
• Supplementary Exercise Sets Most chapters end with a set of supplementary exercises that tend to be
more challenging and force the student to draw on ideas from the entire chapter rather than a specific
section.
Supplementary Materials for Students
• Student Solutions Manual This supplement provides detailed solutions to most theoretical exercises and
to at least one nonroutine exercise of every type (ISBN 9780470458228).
• Technology Exercises and Data Files The technology exercises that appeared in the previous edition have
been moved to the Web site that accompanies this text. Those exercises are designed to be solved using
MATLAB, Mathematica, or Maple and are accompanied by data files in all three formats. The exercises and
data can be downloaded from either of the following Web sites.
www.howardanton.com
www.wiley.com/college/anton
Supplementary Materials for Instructors
• Instructor's Solutions Manual This supplement provides worked-out solutions to most exercises in the
text (ISBN 9780470458235).
• WileyPLUS™ This is Wiley's proprietary online teaching and learning environment that integrates a
digital version of this textbook with instructor and student resources to fit a variety of teaching and learning
styles. WileyPLUS will help your students master concepts in a rich and structured environment that is
available to them 24/7. It will also help you to personalize and manage your course more effectively with
student assessments, assignments, grade tracking, and other useful tools.
• Your students will receive timely access to resources that address their individual needs and will
receive immediate feedback and remediation resources when needed.
• There are also self-assessment tools that are linked to the relevant portions of the text that will enable
your students to take control of their own learning and practice.
• WileyPLUS will help you to identify those students who are falling behind and to intervene in a
timely manner without waiting for scheduled office hours.
More information about WileyPLUS can be obtained from your Wiley representative.
A Guide for the Instructor
Although linear algebra courses vary widely in content and philosophy, most courses fall into two categories
—those with about 35–40 lectures and those with about 25–30 lectures. Accordingly, we have created long
and short templates as possible starting points for constructing a course outline. Of course, these are just
guides, and you will certainly want to customize them to fit your local interests and requirements. Neither of
these sample templates includes applications. Those can be added, if desired, as time permits.
Long Template Short Template
Chapter 1: Systems of Linear Equations and Matrices 7 lectures 6 lectures
Chapter 2: Determinants 3 lectures 2 lectures
Long Template Short Template
Chapter 3: Euclidean Vector Spaces 4 lectures 3 lectures
Chapter 4: General Vector Spaces 10 lectures 10 lectures
Chapter 5: Eigenvalues and Eigenvectors 3 lectures 3 lectures
Chapter 6: Inner Product Spaces 3 lectures 1 lecture
Chapter 7: Diagonalization and Quadratic Forms 4 lectures 3 lectures
Chapter 8: Linear Transformations 3 lectures 2 lectures
Total: 37 lectures 30 lectures
Acknowledgements
I would like to express my appreciation to the following people whose helpful guidance has greatly improved
the text.
Reviewers and Contributors
Don Allen, Texas A&M University
John Alongi, Northwestern University
John Beachy, Northern Illinois University
Przemslaw Bogacki, Old Dominion University
Robert Buchanan, Millersville University of Pennsylvania
Ralph Byers, University of Kansas
Evangelos A. Coutsias, University of New Mexico
Joshua Du, Kennesaw State University
Fatemeh Emdad, Michigan Technological University
Vincent Ervin, Clemson University
Anda Gadidov, Kennesaw State University
Guillermo Goldsztein, Georgia Institute of Technology
Tracy Hamilton, California State University, Sacramento
Amanda Hattway, Wentworth Institute of Technology
Heather Hulett, University of Wisconsin—La Crosse
David Hyeon, Northern Illinois University
Matt Insall, Missouri University of Science and Technology
Mic Jackson, Earlham College
Anton Kaul, California Polytechnic Institute, San Luis Obispo
Harihar Khanal, Embry-Riddle University
Hendrik Kuiper, Arizona State University
Kouok Law, Georgia Perimeter College
James McKinney, California State University, Pomona
Eric Schmutz, Drexel University
Qin Sheng, Baylor University
Adam Sikora, State University of New York at Buffalo
Allan Silberger, Cleveland State University
Dana Williams, Dartmouth College
Mathematical Advisors
Special thanks are due to a number of talented teachers and mathematicians who provided pedagogical
guidance, provided help with answers and exercises, or provided detailed checking or proofreading:
John Alongi, Northwestern University
Scott Annin, California State University, Fullerton
Anton Kaul, California Polytechnic State University
Sarah Streett
Cindy Trimble, C Trimble and Associates
Brad Davis, C Trimble and Associates
The Wiley Support Team
David Dietz, Senior Acquisitions Editor
Jeff Benson, Assistant Editor
Pamela Lashbrook, Senior Editorial Assistant
Janet Foxman, Production Editor
Maddy Lesure, Senior Designer
Laurie Rosatone, Vice President and Publisher
Sarah Davis, Senior Marketing Manager
Diana Smith, Marketing Assistant
Melissa Edwards, Media Editor
Lisa Sabatini, Media Project Manager
Sheena Goldstein, Photo Editor
Carol Sawyer, Production Manager
Lilian Brady, Copyeditor
Special Contributions
The talents and dedication of many individuals are required to produce a book such as this, and I am fortunate
to have benefited from the expertise of the following people:
David Dietz — my editor, for his attention to detail, his sound judgment, and his unwavering faith in me.
Jeff Benson — my assistant editor, who did an unbelievable job in organizing and coordinating the many
threads required to make this edition a reality.
Carol Sawyer — of The Perfect Proof, who coordinated the myriad of details in the production process. It
will be a pleasure to finally delete from my computer the hundreds of emails we exchanged in the course of
working together on this book.
Scott Annin — California State University at Fullerton, who critiqued the previous edition and provided
valuable ideas on how to improve the text. I feel fortunate to have had the benefit of Prof. Annin's teaching
expertise and insights.
Dan Kirschenbaum — of The Art of Arlene and Dan Kirschenbaum, whose artistic and technical expertise
resolved some difficult and critical illustration issues.
Bill Tuohy — who read parts of the manuscript and whose critical eye for detail had an important influence
on the evolution of the text.
Pat Anton — who proofread manuscript, when needed, and shouldered the burden of household chores to
free up time for me to work on this edition.
Maddy Lesure — our text and cover designer whose unerring sense of elegant design is apparent in the
pages of this book.
Rena Lam — of Techsetters, Inc., who did an absolutely amazing job of wading through a nightmare of
author edits, scribbles, and last-minute changes to produce a beautiful book.
John Rogosich — of Techsetters, Inc., who skillfully programmed the design elements of the book and
resolved numerous thorny typesetting issues.
Lilian Brady — my copyeditor of many years, whose eye for typography and whose knowledge of language
never ceases to amaze me.
The Wiley Team — There are many other people at Wiley who worked behind the scenes and to whom I owe
a debt of gratitude: Laurie Rosatone, Ann Berlin, Dorothy Sinclair, Janet Foxman, Sarah Davis, Harry Nolan,
Sheena Goldstein, Melissa Edwards, and Norm Christiansen. Thanks to you all.
Copyright © 2010 John Wiley & Sons, Inc. All rights reserved.
CHAPTER
1 Systems of Linear
Equations and Matrices

CHAPTER CONTENTS
1.1. Introduction to Systems of Linear Equations
1.2. Gaussian Elimination
1.3. Matrices and Matrix Operations
1.4. Inverses; Algebraic Properties of Matrices
1.5. Elementary Matrices and a Method for Finding
1.6. More on Linear Systems and Invertible Matrices
1.7. Diagonal, Triangular, and Symmetric Matrices
1.8. Applications of Linear Systems
• Network Analysis (Traffic Flow)
• Electrical Circuits
• Balancing Chemical Equations
• Polynomial Interpolation
1.9. Leontief Input-Output Models
INTRODUCTION
Information in science, business, and mathematics is often organized into rows and
columns to form rectangular arrays called “matrices” (plural of “matrix”). Matrices often
appear as tables of numerical data that arise from physical observations, but they occur in
various mathematical contexts as well. For example, we will see in this chapter that all of
the information required to solve a system of equations such as
is embodied in the matrix
and that the solution of the system can be obtained by performing appropriate operations
on this matrix. This is particularly important in developing computer programs for solving
systems of equations because computers are well suited for manipulating arrays of
numerical information. However, matrices are not simply a notational tool for solving
systems of equations; they can be viewed as mathematical objects in their own right, and
there is a rich and important theory associated with them that has a multitude of practical
applications. It is the study of matrices and related topics that forms the mathematical field
that we call “linear algebra.” In this chapter we will begin our study of matrices.
Copyright © 2010 John Wiley & Sons, Inc. All rights reserved.