Elementary linear algebra 10th edition

1276 Pages·2011·32.56 MB·English

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

Elementary Linear Algebra: Applications Version, 10th (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.

• Smooth Transition to Abstraction Because the transition from Rn 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.

1.1 Introduction to Systems of Linear Equations

Systems of linear equations and their solutions constitute one of the major topics that we will study in this
course. In this first section we will introduce some basic terminology and discuss a method for solving such
systems.

Linear Equations

Recall that in two dimensions a line in a rectangular xy-coordinate system can be represented by an equation of
the form

and in three dimensions a plane in a rectangular xyz-coordinate system can be represented by an equation of the
form

These are examples of “linear equations,” the first being a linear equation in the variables x and y and the second
a linear equation in the variables x, y, and z. More generally, we define a linear equation in the n variables

to be one that can be expressed in the form

(1)

where
and b are constants, and the a's are not all zero. In the special cases where
or
,

we will often use variables without subscripts and write linear equations as

(2)

(3)

In the special case where
, Equation 1 has the form

(4)

which is called a homogeneous linear equation in the variables
.

EXAMPLE 1 Linear Equations

Observe that a linear equation does not involve any products or roots of variables. All variables
occur only to the first power and do not appear, for example, as arguments of trigonometric,
logarithmic, or exponential functions. The following are linear equations:

The following are not linear equations:

A finite set of linear equations is called a system of linear equations or, more briefly, a linear system. The
variables are called unknowns. For example, system 5 that follows has unknowns x and y, and system 6 has
unknowns
,
, and
.

(5)

(6)

The double subscripting on the coefficients
of the unknowns gives their location in the
system—the first subscript indicates the equation
in which the coefficient occurs, and the second
indicates which unknown it multplies. Thus,
is in the first equation and multiplies
.

A general linear system of m equations in the n unknowns
can be written as

(7)

A solution of a linear system in n unknowns
is a sequence of n numbers
for which

the substitution

makes each equation a true statement. For example, the system in 5 has the solution

and the system in 6 has the solution

These solutions can be written more succinctly as

in which the names of the variables are omitted. This notation allows us to interpret these solutions geometrically
as points in two-dimensional and three-dimensional space. More generally, a solution

of a linear system in n unknowns can be written as

which is called an ordered n-tuple. With this notation it is understood that all variables appear in the same order

in each equation. If
, then the n-tuple is called an ordered pair, and if
, then it is called an ordered

triple.

Linear Systems with Two and Three Unknowns

Linear systems in two unknowns arise in connection with intersections of lines. For example, consider the linear
system

in which the graphs of the equations are lines in the xy-plane. Each solution (x, y) of this system corresponds to a
point of intersection of the lines, so there are three possibilities (Figure 1.1.1):

1. The lines may be parallel and distinct, in which case there is no intersection and consequently no solution.

2. The lines may intersect at only one point, in which case the system has exactly one solution.

3. The lines may coincide, in which case there are infinitely many points of intersection (the points on the

common line) and consequently infinitely many solutions.

Figure 1.1.1

In general, we say that a linear system is consistent if it has at least one solution and inconsistent if it has no
solutions. Thus, a consistent linear system of two equations in two unknowns has either one solution or infinitely
many solutions—there are no other possibilities. The same is true for a linear system of three equations in three
unknowns

in which the graphs of the equations are planes. The solutions of the system, if any, correspond to points where
all three planes intersect, so again we see that there are only three possibilities—no solutions, one solution, or
infinitely many solutions (Figure 1.1.2).

Figure 1.1.2

We will prove later that our observations about the number of solutions of linear systems of two equations in two
unknowns and linear systems of three equations in three unknowns actually hold for all linear systems. That is:

Every system of linear equations has zero, one, or infinitely many solutions. There are no other
possibilities.

EXAMPLE 2 A Linear System with One Solution

Solve the linear system

Solution We can eliminate x from the second equation by adding −2 times the first equation to
the second. This yields the simplified system

From the second equation we obtain
, and on substituting this value in the first equation we

obtain
. Thus, the system has the unique solution

Geometrically, this means that the lines represented by the equations in the system intersect at the

single point
. We leave it for you to check this by graphing the lines.

EXAMPLE 3 A Linear System with No Solutions

Solve the linear system

Solution We can eliminate x from the second equation by adding −3 times the first equation to
the second equation. This yields the simplified system

The second equation is contradictory, so the given system has no solution. Geometrically, this
means that the lines corresponding to the equations in the original system are parallel and distinct.
We leave it for you to check this by graphing the lines or by showing that they have the same slope
but different y-intercepts.

EXAMPLE 4 A Linear System with Infinitely Many Solutions

Solve the linear system

In Example 4 we could have also obtained
parametric equations for the solutions by
solving 8 for y in terms of x, and letting

be the parameter. The resulting

parametric equations would look different
but would define the same solution set.

Solution We can eliminate x from the second equation by adding −4 times the first equation to
the second. This yields the simplified system

The second equation does not impose any restrictions on x and y and hence can be omitted. Thus,
the solutions of the system are those values of x and y that satisfy the single equation