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.
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