Table of contents for A (terse) introduction to linear algebra / Yitzhak Katznelson, Yonatan R. Katznelson.


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1    Vector Spaces                                           1
1.1  Groups and fields . ................. ..             1
1.2  Vector spaces      ..............   ........         4
1.3  Linear dependence, bases, and dimension  . .....    14
1.4  Systems of linear equations  . .............       22
*1.5  Normed finite-dimensional linear spaces ....... .  32
2     Linear Operators and Matrices                         35
2.1  Linear operators ................... ..             35
2.2  Operator multiplication . ................ 39
2.3  Matrix multiplication . .................           41
2.4  Matrices and operators . ................ 46
2.5   Kernel, range, nullity, and rank ............. 51
*2.6  Operator norms ................... ..              56
3     Duality of Vector Spaces                              57
3.1  Linear functionals  . .................. 57
3.2  The adjoint .......................                 62
4     Determinants                                          65
4.1  Permutations ................... ...                65
4.2  Multilinear maps . .................. . 69
4.3  Alternating n-forms . ..................            74
4.4  Determinant of an operator . ..............         76
4.5  Determinant of a matrix  . ...............          79
5    Invariant Subspaces                                 85
5.1  The characteristic polynomial . .............   85
5.2 Invariant subspaces . ..................         88
5.3  The minimal polynomial . ...............        93
6    Inner-Product Spaces                               103
6.1 Inner products . .................. ... 103
6.2  Duality and the adjoint  . ............... 111
6.3  Self-adjoint operators  . .................     113
6.4  Normal operators . .................. . 119
6.5  Unitary and orthogonal operators . ........... 121
*6.6 Positive definite operators ........... . . . .  127
*6.7 Polar decomposition  . ................. 128
*6.8 Contractions and unitary dilations ........... 132
7    Structure Theorems                                 135
7.1  Reducing subspaces . .................. 135
7.2  Semisimple systems . .................. 142
7.3  Nilpotent operators . .................. 147
7.4  The Jordan canonical form  ................ 151
*7.5 The cyclic decomposition, general case  . ....... 152
*7.6 The Jordan canonical form, general case ........ 156
8    Additional Topics                                  159
8.1  Functions of an operator . ............... 159
8.2  Quadratic forms . .................. .. 162
8.3  Perron-Frobenius theory  ................ . 166
8.4  Stochastic matrices .. ..................       178
8.5  Representation of finite groups . ............ 180
A    Appendix                                           187
A.1 Equivalence relations-partitions . ........... 187
A.2 Maps ..       .   ...........    ..........      188
A.3 Groups ...............          ............189
*A.4 Group actions ................... ... 194
A.5 Rings and algebras .   .................. 196
A.6  Polynomials ......... .............. 201



Library of Congress subject headings for this publication: Algebras, Linear