Page 12 - Advanced Linear Algebra

xiv Contents

Topological Vector Spaces, 79
Linear Operators on = d , 82
Exercises, 83
3 The Isomorphism Theorems, 87
Quotient Spaces, 87
The Universal Property of Quotients and
the First Isomorphism Theorem, 90
Quotient Spaces, Complements and Codimension, 92
Additional Isomorphism Theorems, 93
Linear Functionals, 94
Dual Bases, 96
Reflexivity, 100
Annihilators, 101
Operator Adjoints, 104
Exercises, 106
4 Modules I: Basic Properties, 109
Motivation, 109
Modules, 109
Submodules, 111
Spanning Sets, 112
Linear Independence, 114
Torsion Elements, 115
Annihilators, 115
Free Modules, 116
Homomorphisms, 117
Quotient Modules, 117
The Correspondence and Isomorphism Theorems, 118
Direct Sums and Direct Summands, 119
Modules Are Not as Nice as Vector Spaces, 124
Exercises, 125
5 Modules II: Free and Noetherian Modules, 127
The Rank of a Free Module, 127
Free Modules and Epimorphisms, 132
Noetherian Modules, 132
The Hilbert Basis Theorem, 136
Exercises, 137
6 Modules over a Principal Ideal Domain, 139
Annihilators and Orders, 139
Cyclic Modules, 140
Free Modules over a Principal Ideal Domain, 142
Torsion-Free and Free Modules, 145
The Primary Cyclic Decomposition Theorem, 146
The Invariant Factor Decomposition, 156
Characterizing Cyclic Modules, 158