Preface ........................................................ ix
Notation and convention ...................................... xiii
1 Vector spaces ................................................ 1
1.1 Vector spaces ........................................... 1
1.2 Subspaces, span, and linear dependence .................. 8
1.3 Bases, dimensionality, and coordinates ................. 13
1.4 Dual spaces ............................................ 16
1.5 Constructions of vector spaces ......................... 20
1.6 Quotient spaces ........................................ 25
1.7 Normed spaces .......................................... 28
2 Linear mappings ............................................. 34
2.1 Linear mappings ........................................ 34
2.2 Change of basis ........................................ 45
2.3 Adjoint mappings ....................................... 50
2.4 Quotient mappings ...................................... 53
2.5 Linear mappings from a vector space into itself ........ 55
2.6 Norms of linear mappings ............................... 70
3 Determinants ................................................ 78
3.1 Motivational examples .................................. 78
3.2 Definition and properties of determinants .............. 88
3.3 Adjugate matrices and Cramer's rule ................... 102
3.4 Characteristic polynomials and Cayley-Hamilton
theorem ............................................... 107
4 Scalar products ............................................ 115
4.1 Scalar products and basic properties .................. 115
4.2 Non-degenerate scalar products ........................ 120
4.3 Positive definite scalar products ..................... 127
4.4 Orthogonal resolutions of vectors ..................... 137
4.5 Orthogonal and unitary versus isometric mappings ...... 142
5 Real quadratic forms and self-adjoint mappings ............. 147
5.1 Bilinear and quadratic forms .......................... 147
5.2 Self-adjoint mappings ................................. 151
5.3 Positive definite quadratic forms, mappings, and
matrices .............................................. 157
5.4 Alternative characterizations of positive definite
matrices .............................................. 164
5.5 Commutativity of self-adjoint mappings ................ 170
5.6 Mappings between two spaces ........................... 172
6 Complex quadratic forms and self-adjoint mappings .......... 180
6.1 Complex sesquilinear and associated quadratic forms ... 180
6.2 Complex self-adjoint mappings ......................... 184
6.3 Positive definiteness ................................. 188
6.4 Commutative self-adjoint mappings and consequences .... 194
6.5 Mappings between two spaces via self-adjoint
mappings .............................................. 199
7 Jordan decomposition ....................................... 205
7.1 Some useful facts about polynomials ................... 205
7.2 Invariant subspaces of linear mappings ................ 208
7.3 Generalized eigenspaces as invariant subspaces ........ 211
7.4 Jordan decomposition theorem .......................... 218
8 Selected topics ............................................ 226
8.1 Schur decomposition ................................... 226
8.2 Classification of skewsymmetric bilinear forms ........ 230
8.3 Perron-Frobenius theorem for positive matrices ........ 237
8.4 Markov matrices ....................................... 242
9 Excursion: Quantum mechanics in a nutshell ................. 248
9.1 Vectors in C" and Dirac bracket ....................... 248
9.2 Quantum mechanical postulates ......................... 252
9.3 Non-commutativity and uncertainty principle ........... 257
9.4 Heisenberg picture for quantum mechanics .............. 262
Solutions to selected exercises ............................... 267
Bibliographic notes ........................................... 311
References .................................................... 313
Index ......................................................... 315
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