Yitzhak Katznelson, Stanford University, Stanford, CA,
Yonatan R. Katznelson, University of California, Santa Cruz, Santa Cruz, CA
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Preface 10
1. Vector Spaces 12
1.1 Groups and fields 12
1.2 Vector spaces 15
1.3 Linear dependence, bases, and dimension 25
1.4 Systems of linear equations 33
*1.5 Normed finite-dimensional linear spaces 43
2. Linear Operators and Matrices 46
2.1 Linear operators 46
2.2 Operator multiplication 50
2.3 Matrix multiplication 52
2.4 Matrices and operators 57
2.5 Kernel, range, nullity, and rank 62
*2.6 Operator norms 67
3. Duality of Vector Spaces 68
3.1 Linear functionals 68
3.2 The adjoint 73
4. Determinants 76
4.1 Permutations 76
4.2 Multilinear maps 80
4.3 Alternating n-forms 85
4.4 Determinant of an operator 87
4.5 Determinant of a matrix 90
5. Invariant Subspaces 96
5.1 The characteristic polynomial 96
5.2 Invariant subspaces 99
5.3 The minimal polynomial 104
6. Inner-Product Spaces 114
6.1 Inner products 114
6.2 Duality and the adjoint 122
6.3 Self-adjoint operators 124
6.4 Normal operators 130
6.5 Unitary and orthogonal operators 132
*6.6 Positive definite operators 138
*6.7 Polar decomposition 139
*6.8 Contractions and unitary dilations 143
7. Structure Theorems 146
7.1 Reducing subspaces 146
7.2 Semisimple systems 153
7.3 Nilpotent operators 158
7.4 The Jordan canonical form 162
*7.5 The cyclic decomposition, general case 163
*7.6 The Jordan canonical form, general case 167
8. Additional Topics 170
8.1 Functions of an operator 170
8.2 Quadratic forms 173
8.3 Perron-Frobenius theory 177
8.4 Stochastic matrices 189
8.5 Representation of finite groups 191
A Appendix 198
A.1 Equivalence relations-partitions 198
A.2 Maps 199
A.3 Groups 200
*A.4 Group actions 205
A.5 Rings and algebras 207
A.6 Polynomials 212
• Index 222
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