实例介绍
【实例简介】
【实例截图】
【实例截图】
【核心代码】
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
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