This best-selling textbook for a second course in linear algebra is aimed at undergrad math majors and graduate students. The novel approach taken here banishes determinants to the end of the book. The text focuses on the central goal of linear algebra: understanding the structure of linear operators on finite-dimensional vector spaces. The author has taken unusual care to motivate concepts and to simplify proofs. A variety of interesting exercises in each chapter helps students understand and manipulate the objects of linear algebra.

The **third edition** contains major improvements and revisions throughout the book. More than 300 new exercises have been added since the previous edition. Many new examples have been added to illustrate the key ideas of linear algebra. New topics covered in the book include product spaces, quotient spaces, and dual spaces. Beautiful new formatting creates pages with an unusually pleasant appearance in both print and electronic versions.

No prerequisites are assumed other than the usual demand for suitable mathematical maturity. Thus the text starts by discussing vector spaces, linear independence, span, basis, and dimension. The book then deals with linear maps, eigenvalues, and eigenvectors. Inner-product spaces are introduced, leading to the finite-dimensional spectral theorem and its consequences. Generalized eigenvectors are then used to provide insight into the structure of a linear operator.

**Table of 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

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

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

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

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

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

**Sheldon Axler** was valedictorian of his high school in Miami, Florida. He received his AB from **Princeton University** with highest honors, followed by a PhD in Mathematics from the **University of California at Berkeley**.

As a Moore Instructor at MIT, Axler received a university-wide teaching award. He was then an assistant professor, associate professor, and professor in the Mathematics Department at Michigan State University, where he received the first J. Sutherland Frame Teaching Award and the Distinguished Faculty Award.

Axler came to San Francisco State University as Chair of the **Mathematics** Department in 1997. In 2002, he became Dean of the College of Science & Engineering at SF State, a position he held until returning full-time to mathematics in 2015.

Axler received the Lester R. Ford Award for expository writing from the Mathematical Association of America in 1996. In addition to publishing numerous research papers, Axler is the author of five mathematics textbooks, ranging from freshman to graduate level. His book **Linear Algebra Done Right** **has been adopted as a textbook at over 300 universities.**

Axler has served as Editor-in-Chief of the Mathematical Intelligencer and as Associate Editor of the American Mathematical Monthly. He has been a member of the Council of the American Mathematical Society and a member of the Board of Trustees of the Mathematical Sciences Research Institute. Axler currently serves on the editorial board of Springer’s series Undergraduate Texts in Mathematics, Graduate Texts in Mathematics, and Universitext.

The American Mathematical Society honored Axler by selecting him as a member of its inaugural group of Fellows in 2013.

## Reviews

There are no reviews yet.