Advanced linear algebra

Cover......Page 1
Title Page......Page 5
Preface to the Third Edition......Page 9
Preface to the Second Edition......Page 11
Preface to the First Edition......Page 13
Contents......Page 15
Part 1: Preliminaries......Page 21
Part 2: Algebraic Structures......Page 37
Part I—Basic Linear Algebra......Page 53
Vector Spaces......Page 55
Subspaces......Page 57
Direct Sums......Page 60
Spanning Sets and Linear Independence......Page 64
The Dimension of a Vector Space......Page 68
Ordered Bases and Coordinate Matrices......Page 71
The Row and Column Spaces of a Matrix......Page 72
The Complexification of a Real Vector Space......Page 73
Exercises......Page 75
Linear Transformations......Page 79
The Kernel and Image of a Linear Transformation......Page 81
Isomorphisms......Page 82
The Rank Plus Nullity Theorem......Page 83
Linear Transformations from F^n to F^m......Page 84
Change of Basis Matrices......Page 85
The Matrix of a Linear Transformation......Page 86
Equivalence of Matrices......Page 88
Similarity of Matrices......Page 90
Similarity of Operators......Page 91
Invariant Subspaces and Reducing Pairs......Page 92
Projection Operators......Page 93
Topological Vector Spaces......Page 99
Linear Operators on V^ℂ......Page 102
Exercises......Page 103
Quotient Spaces......Page 107
The Universal Property of Quotients and the First Isomorphism Theorem......Page 110
Quotient Spaces, Complements and Codimension......Page 112
Additional Isomorphism Theorems......Page 113
Linear Functionals......Page 114
Dual Bases......Page 116
Reflexivity......Page 120
Annihilators......Page 121
Operator Adjoints......Page 124
Exercises......Page 126
Modules......Page 129
Submodules......Page 131
Spanning Sets......Page 132
Linear Independence......Page 134
Annihilators......Page 135
Free Modules......Page 136
Quotient Modules......Page 137
The Correspondence and Isomorphism Theorems......Page 138
Direct Sums and Direct Summands......Page 139
Modules Are Not as Nice as Vector Spaces......Page 144
Exercises......Page 145
The Rank of a Free Module......Page 147
Noetherian Modules......Page 152
The Hilbert Basis Theorem......Page 156
Exercises......Page 157
Annihilators and Orders......Page 159
Cyclic Modules......Page 160
Free Modules over a Principal Ideal Domain......Page 162
Torsion-Free and Free Modules......Page 165
The Primary Cyclic Decomposition Theorem......Page 166
The Invariant Factor Decomposition......Page 176
Indecomposable Modules......Page 178
Exercises......Page 179
7 The Structure of a Linear Operator......Page 183
The Module Associated with a Linear Operator......Page 184
The Primary Cyclic Decomposition of V_τ......Page 187
The Characteristic Polynomial......Page 190
Cyclic and Indecomposable Modules......Page 191
The Big Picture......Page 194
The Rational Canonical Form......Page 196
Exercises......Page 202
Eigenvalues and Eigenvectors......Page 205
Geometric and Algebraic Multiplicities......Page 209
The Jordan Canonical Form......Page 210
Triangularizability and Schur\'s Lemma......Page 212
Diagonalizable Operators......Page 216
Exercises......Page 218
9 Real and Complex Inner Product Spaces......Page 225
Norm and Distance......Page 228
Isometries......Page 230
Orthogonality......Page 231
Orthogonal and Orthonormal Sets......Page 232
The Projection Theorem and Best Approximations......Page 239
The Riesz Representation Theorem......Page 241
Exercises......Page 243
The Adjoint of a Linear Operator......Page 247
Orthogonal Projections......Page 251
Unitary Diagonalizability......Page 253
Normal Operators......Page 254
Special Types of Normal Operators......Page 258
Self-Adjoint Operators......Page 259
Unitary Operators and Isometries......Page 260
The Structure of Normal Operators......Page 265
Functional Calculus......Page 267
Positive Operators......Page 270
The Polar Decomposition of an Operator......Page 272
Exercises......Page 274
Part II—Topics......Page 277
Symmetric, Skew-Symmetric and Alternate Forms......Page 279
The Matrix of a Bilinear Form......Page 281
Quadratic Forms......Page 284
Orthogonality......Page 285
Linear Functionals......Page 288
Orthogonal Complements and Orthogonal Direct Sums......Page 289
Isometries......Page 291
Hyperbolic Spaces......Page 292
Nonsingular Completions of a Subspace......Page 293
The Witt Theorems: A Preview......Page 295
The Classification Problem for Metric Vector Spaces......Page 296
Symplectic Geometry......Page 297
The Structure of Orthogonal Geometries: Orthogonal Bases......Page 302
The Classification of Orthogonal Geometries: Canonical Forms......Page 305
The Orthogonal Group......Page 311
The Witt Theorems for Orthogonal Geometries......Page 314
Maximal Hyperbolic Subspaces of an Orthogonal Geometry......Page 315
Exercises......Page 317
The Definition......Page 321
Open and Closed Sets......Page 324
Convergence in a Metric Space......Page 325
The Closure of a Set......Page 326
Dense Subsets......Page 328
Continuity......Page 330
Completeness......Page 331
Isometries......Page 335
The Completion of a Metric Space......Page 336
Exercises......Page 341
A Brief Review......Page 345
Hilbert Spaces......Page 346
Infinite Series......Page 350
An Approximation Problem......Page 351
Hilbert Bases......Page 355
Fourier Expansions......Page 356
Hilbert Dimension......Page 366
A Characterization of Hilbert Spaces......Page 367
The Riesz Representation Theorem......Page 369
Exercises......Page 372
Universality......Page 375
Bilinear Maps......Page 379
Tensor Products......Page 381
When Is a Tensor Product Zero?......Page 387
Coordinate Matrices and Rank......Page 388
Characterizing Vectors in a Tensor Product......Page 391
Defining Linear Transformations on a Tensor Product......Page 394
The Tensor Product of Linear Transformations......Page 395
Change of Base Field......Page 399
Multilinear Maps and Iterated Tensor Products......Page 402
Tensor Spaces......Page 405
Special Multilinear Maps......Page 410
The Symmetric and Antisymmetric Tensor Algebras......Page 412
The Determinant......Page 423
Exercises......Page 426
15 Positive Solutions to Linear Systems: Convexity and Separation......Page 431
Convex, Closed and Compact Sets......Page 433
Convex Hulls......Page 434
Linear and Affine Hyperplanes......Page 436
Separation......Page 438
Exercises......Page 443
Affine Geometry......Page 447
Affine Combinations......Page 448
Affine Hulls......Page 450
The Lattice of Flats......Page 451
Affine Independence......Page 453
Affine Transformations......Page 455
Projective Geometry......Page 457
Exercises......Page 460
Singular Values......Page 463
The Moore–Penrose Generalized Inverse......Page 466
Least Squares Approximation......Page 468
Exercises......Page 469
Associative Algebras......Page 471
Division Algebras......Page 482
Exercises......Page 489
Formal Power Series......Page 491
The Umbral Algebra......Page 493
Formal Power Series as Linear Operators......Page 497
Sheffer Sequences......Page 500
Examples of Sheffer Sequences......Page 508
Umbral Operators and Umbral Shifts......Page 510
Continuous Operators on the Umbral Algebra......Page 512
Operator Adjoints......Page 513
Umbral Operators and Automorphisms of the Umbral Algebra......Page 514
Umbral Shifts and Derivations of the Umbral Algebra......Page 519
The Transfer Formulas......Page 524
A Final Remark......Page 525
Exercises......Page 526
References......Page 527
Index of Symbols......Page 533
Index......Page 535