Linear Algebra Done Right

Undergraduate Texts in Mathematics

Linear Algebra

Done Right

Sheldon Axler

Third Edition

Undergraduate Texts in Mathematics

Undergraduate Texts in Mathematics

Series Editors:

Sheldon Axler

San Francisco State University, San Francisco, CA, USA

Kenneth Ribet

University of California, Berkeley, CA, USA

Advisory Board:

Colin Adams, Williams College, Williamstown, MA, USA

Alejandro Adem, University of British Columbia, Vancouver, BC, Canada

Ruth Charney, Brandeis University, Waltham, MA, USA

Irene M. Gamba, The University of Texas at Austin, Austin, TX, USA

Roger E. Howe, Yale University, New Haven, CT, USA

David Jerison, Massachusetts Institute of Technology, Cambridge, MA, USA

Jeffrey C. Lagarias, University of Michigan, Ann Arbor, MI, USA

Jill Pipher, Brown University, Providence, RI, USA

Fadil Santosa, University of Minnesota, Minneapolis, MN, USA

Amie Wilkinson, University of Chicago, Chicago, IL, USA

Undergraduate Texts in Mathematics are generally aimed at third- and fourth￾year undergraduate mathematics students at North American universities. These

texts strive to provide students and teachers with new perspectives and novel

approaches. The books include motivation that guides the reader to an appreciation

of interrelations among different aspects of the subject. They feature examples that

illustrate key concepts as well as exercises that strengthen understanding.

http://www.springer.com/series/666

For further volumes:

Sheldon Axler

Linear Algebra Done Right

Third edition

123

ISSN 0172-6056 ISSN 2197-5604 (electronic)

ISBN 978-3-319-11079-0 ISBN 978-3-319-11080-6 (eBook)

DOI 10.1007/978-3-319-11080-6

Springer Cham Heidelberg New York Dordrecht London

Library of Congress Control Number: 2014954079

Mathematics Subject Classification (2010): 15-01, 15A03, 15A04, 15A15, 15A18, 15A21

c Springer International Publishing 2015

This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part

of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,

recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or

information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar

methodology now known or hereafter developed. Exempted from this legal reservation are brief

excerpts in connection with reviews or scholarly analysis or material supplied specifically for the

purpose of being entered and executed on a computer system, for exclusive use by the purchaser of

the work. Duplication of this publication or parts thereof is permitted only under the provisions of the

Copyright Law of the Publisher’s location, in its current version, and permission for use must always be

obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright

Clearance Center. Violations are liable to prosecution under the respective Copyright Law.

The use of general descriptive names, registered names, trademarks, service marks, etc. in this

publication does not imply, even in the absence of a specific statement, that such names are exempt

from the relevant protective laws and regulations and therefore free for general use.

While the advice and information in this book are believed to be true and accurate at the date of

publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for

any errors or omissions that may be made. The publisher makes no warranty, express or implied, with

respect to the material contained herein.

Typeset by the author in LaTeX

Printed on acid-free paper

Springer is part of Springer Science+Business Media (www.springer.com)

Sheldon Axler

San Francisco State University

Department of Mathematics

San Francisco, CA, USA

Cover figure: For a statement of Apollonius

exercise in Section 6.A.

’s Identity and its connection to linear algebra, see the last

Contents

Preface for the Instructor xi

Preface for the Student xv

Acknowledgments xvii

1 Vector Spaces 1

1.A Rn and Cn 2

Complex Numbers 2

Lists 5

Fn 6

Digression on Fields 10

Exercises 1.A 11

1.B Definition of Vector Space 12

Exercises 1.B 17

1.C Subspaces 18

Sums of Subspaces 20

Direct Sums 21

Exercises 1.C 24

2 Finite-Dimensional Vector Spaces 27

2.A Span and Linear Independence 28

Linear Combinations and Span 28

Linear Independence 32

Exercises 2.A 37

v

vi Contents

2.B Bases 39

Exercises 2.B 43

2.C Dimension 44

Exercises 2.C 48

3 Linear Maps 51

3.A The Vector Space of Linear Maps 52

Definition and Examples of Linear Maps 52

Algebraic Operations on L.V; W / 55

Exercises 3.A 57

3.B Null Spaces and Ranges 59

Null Space and Injectivity 59

Range and Surjectivity 61

Fundamental Theorem of Linear Maps 63

Exercises 3.B 67

3.C Matrices 70

Representing a Linear Map by a Matrix 70

Addition and Scalar Multiplication of Matrices 72

Matrix Multiplication 74

Exercises 3.C 78

3.D Invertibility and Isomorphic Vector Spaces 80

Invertible Linear Maps 80

Isomorphic Vector Spaces 82

Linear Maps Thought of as Matrix Multiplication 84

Operators 86

Exercises 3.D 88

3.E Products and Quotients of Vector Spaces 91

Products of Vector Spaces 91

Products and Direct Sums 93

Quotients of Vector Spaces 94

Exercises 3.E 98

Contents vii

3.F Duality 101

The Dual Space and the Dual Map 101

The Null Space and Range of the Dual of a Linear Map 104

The Matrix of the Dual of a Linear Map 109

The Rank of a Matrix 111

Exercises 3.F 113

4 Polynomials 117

Complex Conjugate and Absolute Value 118

Uniqueness of Coefficients for Polynomials 120

The Division Algorithm for Polynomials 121

Zeros of Polynomials 122

Factorization of Polynomials over C 123

Factorization of Polynomials over R 126

Exercises 4 129

5 Eigenvalues, Eigenvectors, and Invariant Subspaces 131

5.A Invariant Subspaces 132

Eigenvalues and Eigenvectors 133

Restriction and Quotient Operators 137

Exercises 5.A 138

5.B Eigenvectors and Upper-Triangular Matrices 143

Polynomials Applied to Operators 143

Existence of Eigenvalues 145

Upper-Triangular Matrices 146

Exercises 5.B 153

5.C Eigenspaces and Diagonal Matrices 155

Exercises 5.C 160

6 Inner Product Spaces 163

6.A Inner Products and Norms 164

Inner Products 164

Norms 168

Exercises 6.A 175

viii Contents

6.B Orthonormal Bases 180

Linear Functionals on Inner Product Spaces 187

Exercises 6.B 189

6.C Orthogonal Complements and Minimization Problems 193

Orthogonal Complements 193

Minimization Problems 198

Exercises 6.C 201

7 Operators on Inner Product Spaces 203

7.A Self-Adjoint and Normal Operators 204

Adjoints 204

Self-Adjoint Operators 209

Normal Operators 212

Exercises 7.A 214

7.B The Spectral Theorem 217

The Complex Spectral Theorem 217

The Real Spectral Theorem 219

Exercises 7.B 223

7.C Positive Operators and Isometries 225

Positive Operators 225

Isometries 228

Exercises 7.C 231

7.D Polar Decomposition and Singular Value Decomposition 233

Polar Decomposition 233

Singular Value Decomposition 236

Exercises 7.D 238

8 Operators on Complex Vector Spaces 241

8.A Generalized Eigenvectors and Nilpotent Operators 242

Null Spaces of Powers of an Operator 242

Generalized Eigenvectors 244

Nilpotent Operators 248

Exercises 8.A 249

Contents ix

8.B Decomposition of an Operator 252

Description of Operators on Complex Vector Spaces 252

Multiplicity of an Eigenvalue 254

Block Diagonal Matrices 255

Square Roots 258

Exercises 8.B 259

8.C Characteristic and Minimal Polynomials 261

The Cayley–Hamilton Theorem 261

The Minimal Polynomial 262

Exercises 8.C 267

8.D Jordan Form 270

Exercises 8.D 274

9 Operators on Real Vector Spaces 275

9.A Complexification 276

Complexification of a Vector Space 276

Complexification of an Operator 277

The Minimal Polynomial of the Complexification 279

Eigenvalues of the Complexification 280

Characteristic Polynomial of the Complexification 283

Exercises 9.A 285

9.B Operators on Real Inner Product Spaces 287

Normal Operators on Real Inner Product Spaces 287

Isometries on Real Inner Product Spaces 292

Exercises 9.B 294

10 Trace and Determinant 295

10.A Trace 296

Change of Basis 296

Trace: A Connection Between Operators and Matrices 299

Exercises 10.A 304

x Contents

10.B Determinant 307

Determinant of an Operator 307

Determinant of a Matrix 309

The Sign of the Determinant 320

Volume 323

Exercises 10.B 330

Photo Credits 333

Symbol Index 335

Index 337

Preface for the Instructor

You are about to teach a course that will probably give students their second

exposure to linear algebra. During their first brush with the subject, your

students probably worked with Euclidean spaces and matrices. In contrast,

this course will emphasize abstract vector spaces and linear maps.

The audacious title of this book deserves an explanation. Almost all

linear algebra books use determinants to prove that every linear operator on

a finite-dimensional complex vector space has an eigenvalue. Determinants

are difficult, nonintuitive, and often defined without motivation. To prove the

theorem about existence of eigenvalues on complex vector spaces, most books

must define determinants, prove that a linear map is not invertible if and only

if its determinant equals 0, and then define the characteristic polynomial. This

tortuous (torturous?) path gives students little feeling for why eigenvalues

exist.

In contrast, the simple determinant-free proofs presented here (for example,

see 5.21) offer more insight. Once determinants have been banished to the

end of the book, a new route opens to the main goal of linear algebra—

understanding the structure of linear operators.

This book starts at the beginning of the subject, with no prerequisites

other than the usual demand for suitable mathematical maturity. Even if your

students have already seen some of the material in the first few chapters, they

may be unaccustomed to working exercises of the type presented here, most

of which require an understanding of proofs.

Here is a chapter-by-chapter summary of the highlights of the book:

Chapter 1: Vector spaces are defined in this chapter, and their basic proper￾ties are developed.

Chapter 2: Linear independence, span, basis, and dimension are defined in

this chapter, which presents the basic theory of finite-dimensional vector

spaces.

xi

xii Preface for the Instructor

Chapter 3: Linear maps are introduced in this chapter. The key result here

is the Fundamental Theorem of Linear Maps (3.22): if T is a linear map

on V, then dim V D dim null T C dim range T. Quotient spaces and duality

are topics in this chapter at a higher level of abstraction than other parts

of the book; these topics can be skipped without running into problems

elsewhere in the book.

Chapter 4: The part of the theory of polynomials that will be needed

to understand linear operators is presented in this chapter. This chapter

contains no linear algebra. It can be covered quickly, especially if your

students are already familiar with these results.

Chapter 5: The idea of studying a linear operator by restricting it to small

subspaces leads to eigenvectors in the early part of this chapter. The

highlight of this chapter is a simple proof that on complex vector spaces,

eigenvalues always exist. This result is then used to show that each linear

operator on a complex vector space has an upper-triangular matrix with

respect to some basis. All this is done without defining determinants or

characteristic polynomials!

Chapter 6: Inner product spaces are defined in this chapter, and their basic

properties are developed along with standard tools such as orthonormal

bases and the Gram–Schmidt Procedure. This chapter also shows how

orthogonal projections can be used to solve certain minimization problems.

Chapter 7: The Spectral Theorem, which characterizes the linear operators

for which there exists an orthonormal basis consisting of eigenvectors,

is the highlight of this chapter. The work in earlier chapters pays off

here with especially simple proofs. This chapter also deals with positive

operators, isometries, the Polar Decomposition, and the Singular Value

Decomposition.

Chapter 8: Minimal polynomials, characteristic polynomials, and gener￾alized eigenvectors are introduced in this chapter. The main achievement

of this chapter is the description of a linear operator on a complex vector

space in terms of its generalized eigenvectors. This description enables

one to prove many of the results usually proved using Jordan Form. For

example, these tools are used to prove that every invertible linear operator

on a complex vector space has a square root. The chapter concludes with a

proof that every linear operator on a complex vector space can be put into

Jordan Form.

Preface for the Instructor xiii

Chapter 9: Linear operators on real vector spaces occupy center stage in

this chapter. Here the main technique is complexification, which is a natural

extension of an operator on a real vector space to an operator on a complex

vector space. Complexification allows our results about complex vector

spaces to be transferred easily to real vector spaces. For example, this

technique is used to show that every linear operator on a real vector space

has an invariant subspace of dimension 1 or 2. As another example, we

show that that every linear operator on an odd-dimensional real vector space

has an eigenvalue.

Chapter 10: The trace and determinant (on complex vector spaces) are

defined in this chapter as the sum of the eigenvalues and the product of the

eigenvalues, both counting multiplicity. These easy-to-remember defini￾tions would not be possible with the traditional approach to eigenvalues,

because the traditional method uses determinants to prove that sufficient

eigenvalues exist. The standard theorems about determinants now become

much clearer. The Polar Decomposition and the Real Spectral Theorem are

used to derive the change of variables formula for multivariable integrals in

a fashion that makes the appearance of the determinant there seem natural.

This book usually develops linear algebra simultaneously for real and

complex vector spaces by letting F denote either the real or the complex

numbers. If you and your students prefer to think of F as an arbitrary field,

then see the comments at the end of Section 1.A. I prefer avoiding arbitrary

fields at this level because they introduce extra abstraction without leading

to any new linear algebra. Also, students are more comfortable thinking

of polynomials as functions instead of the more formal objects needed for

polynomials with coefficients in finite fields. Finally, even if the beginning

part of the theory were developed with arbitrary fields, inner product spaces

would push consideration back to just real and complex vector spaces.

You probably cannot cover everything in this book in one semester. Going

through the first eight chapters is a good goal for a one-semester course. If

you must reach Chapter 10, then consider covering Chapters 4 and 9 in fifteen

minutes each, as well as skipping the material on quotient spaces and duality

in Chapter 3.

A goal more important than teaching any particular theorem is to develop in

students the ability to understand and manipulate the objects of linear algebra.

Mathematics can be learned only by doing. Fortunately, linear algebra has

many good homework exercises. When teaching this course, during each

class I usually assign as homework several of the exercises, due the next class.

Going over the homework might take up a third or even half of a typical class.

xiv Preface for the Instructor

Major changes from the previous edition:

This edition contains 561 exercises, including 337 new exercises that were

not in the previous edition. Exercises now appear at the end of each section,

rather than at the end of each chapter.

Many new examples have been added to illustrate the key ideas of linear

algebra.

Beautiful new formatting, including the use of color, creates pages with an

unusually pleasant appearance in both print and electronic versions. As a

visual aid, definitions are in beige boxes and theorems are in blue boxes (in

color versions of the book).

Each theorem now has a descriptive name.

New topics covered in the book include product spaces, quotient spaces,

and duality.

Chapter 9 (Operators on Real Vector Spaces) has been completely rewritten

to take advantage of simplifications via complexification. This approach

allows for more streamlined presentations in Chapters 5 and 7 because

those chapters now focus mostly on complex vector spaces.

Hundreds of improvements have been made throughout the book. For

example, the proof of Jordan Form (Section 8.D) has been simplified.

Please check the website below for additional information about the book. I

may occasionally write new sections on additional topics. These new sections

will be posted on the website. Your suggestions, comments, and corrections

are most welcome.

Best wishes for teaching a successful linear algebra class!

Sheldon Axler

Mathematics Department

San Francisco State University

San Francisco, CA 94132, USA

website: linear.axler.net

e-mail: linear@axler.net

Twitter: @AxlerLinear