基本信息

---

下载

   源码介绍

---

Contents
Preface to the Instructor ix
Preface to the Student xiii
Acknowledgments xv
Chapter 1
Vector Spaces 1
Complex Numbers .......................... 2
Definition of Vector Space ...................... 4
Properties of Vector Spaces ..................... 11
Subspaces ............................... 13
Sums and Direct Sums ........................ 14
Exercises ................................ 19
Chapter 2
Finite-Dimensional Vector Spaces 21
Span and Linear Independence ................... 22
Bases .................................. 27
Dimension ............................... 31
Exercises ................................ 35
Chapter 3
Linear Maps 37
Definitions and Examples ...................... 38
Null Spaces and Ranges ....................... 41
The Matrix of a Linear Map ..................... 48
Invertibility .............................. 53
Exercises ................................ 59
v
vi Contents
Chapter 4
Polynomials 63
Degree ................................. 64
Complex Coefficients ........................ 67
Real Coefficients ........................... 69
Exercises ................................ 73
Chapter 5
Eigenvalues and Eigenvectors 75
Invariant Subspaces ......................... 76
Polynomials Applied to Operators ................. 80
Upper-Triangular Matrices ..................... 81
Diagonal Matrices ........................... 87
Invariant Subspaces on Real Vector Spaces ........... 91
Exercises ................................ 94
Chapter 6
Inner-Product Spaces 97
Inner Products ............................. 98
Norms ................................. 102
Orthonormal Bases .......................... 106
Orthogonal Projections and Minimization Problems ...... 111
Linear Functionals and Adjoints .................. 117
Exercises ................................ 122
Chapter 7
Operators on Inner-Product Spaces 127
Self-Adjoint and Normal Operators ................ 128
The Spectral Theorem ........................ 132
Normal Operators on Real Inner-Product Spaces ........ 138
Positive Operators .......................... 144
Isometries ............................... 147
Polar and Singular-Value Decompositions ............ 152
Exercises ................................ 158
Chapter 8
Operators on Complex Vector Spaces 163
Generalized Eigenvectors ...................... 164
The Characteristic Polynomial ................... 168
Decomposition of an Operator ................... 173
Contents vii
Square Roots .............................. 177
The Minimal Polynomial ....................... 179
Jordan Form .............................. 183
Exercises ................................ 188
Chapter 9
Operators on Real Vector Spaces 193
Eigenvalues of Square Matrices ................... 194
Block Upper-Triangular Matrices .................. 195
The Characteristic Polynomial ................... 198
Exercises ................................ 210
Chapter 10
Trace and Determinant 213
Change of Basis ............................ 214
Trace .................................. 216
Determinant of an Operator .................... 222
Determinant of a Matrix ....................... 225
Volume ................................. 236
Exercises ................................ 244
Symbol Index 247
Index 249