Advanced Linear Algebra

Cover
......Page 1
Graduate Texts in Mathematics......Page 3
Title......Page 4
Preface to the Third Edition......Page 7
Preface to the Second Edition......Page 8
Preface to the First Edition......Page 9
Part I—Basic Linear Algebra......Page 11
Part II—Topics......Page 13
Part 1 Preliminaries......Page 17
Part 2 Algebraic Structures......Page 33
Part I—Basic Linear Algebra......Page 48
Vector Spaces......Page 49
Subspaces......Page 51
Direct Sums......Page 54
Spanning Sets and Linear Independence......Page 58
The Dimension of a Vector Space......Page 62
Ordered Bases and Coordinate Matrices......Page 65
The Row and Column Spaces of a Matrix......Page 66
The Complexification of a Real Vector Space......Page 67
Exercises......Page 69
Linear Transformations......Page 73
The Kernel and Image of a Linear Transformation......Page 75
Isomorphisms......Page 76
The Rank Plus Nullity Theorem......Page 77
Linear Transformations from Fn to Fm......Page 78
Change of Basis Matrices......Page 79
The Matrix of a Linear Transformation......Page 80
Equivalence of Matrices......Page 82
Similarity of Matrices......Page 84
Similarity of Operators......Page 85
Invariant Subspaces and Reducing Pairs......Page 86
Projection Operators......Page 87
Topological Vector Spaces......Page 93
Linear Operators on Vc......Page 96
Exercises......Page 97
Quotient Spaces......Page 100
The Universal Property of Quotients and the First Isomorphism Theorem......Page 103
Quotient Spaces, Complements and Codimension......Page 105
Additional Isomorphism Theorems......Page 106
Linear Functionals......Page 107
Dual Bases......Page 109
Reflexivity......Page 113
Annihilators......Page 114
Operator Adjoints......Page 117
Exercises......Page 119
Modules......Page 121
Submodules......Page 123
Spanning Sets......Page 124
Linear Independence......Page 126
Annihilators......Page 127
Free Modules......Page 128
Quotient Modules......Page 129
The Correspondence and Isomorphism Theorems......Page 130
Direct Sums and Direct Summands......Page 131
Modules Are Not as Nice as Vector Spaces......Page 136
Exercises......Page 137
The Rank of a Free Module......Page 139
Noetherian Modules......Page 144
The Hilbert Basis Theorem......Page 148
Exercises......Page 149
Annihilators and Orders......Page 151
Cyclic Modules......Page 152
Free Modules over a Principal Ideal Domain......Page 154
Torsion-Free and Free Modules......Page 157
The Primary Cyclic Decomposition Theorem......Page 158
The Invariant Factor Decomposition......Page 168
Indecomposable Modules......Page 170
Exercises......Page 171
7.
The Structure of a Linear Operator......Page 174
The Module Associated with a Linear Operator......Page 175
The Primary Cyclic Decomposition of Vtau......Page 178
The Characteristic Polynomial......Page 181
Cyclic and Indecomposable Modules......Page 182
The Big Picture......Page 185
The Rational Canonical Form......Page 187
Exercises......Page 193
Eigenvalues and Eigenvectors......Page 196
Geometric and Algebraic Multiplicities......Page 200
The Jordan Canonical Form......Page 201
Triangularizability and Schur\'s Lemma......Page 203
Diagonalizable Operators......Page 207
Exercises......Page 209
9.
Real and Complex Inner Product Spaces......Page 215
Norm and Distance......Page 218
Isometries......Page 220
Orthogonality......Page 221
Orthogonal and Orthonormal Sets......Page 222
The Projection Theorem and Best Approximations......Page 229
The Riesz Representation Theorem......Page 231
Exercises......Page 233
The Adjoint of a Linear Operator......Page 237
Orthogonal Projections......Page 241
Unitary Diagonalizability......Page 243
Normal Operators......Page 244
Special Types of Normal Operators......Page 248
Self-Adjoint Operators......Page 249
Unitary Operators and Isometries......Page 250
The Structure of Normal Operators......Page 255
Functional Calculus......Page 257
Positive Operators......Page 260
The Polar Decomposition of an Operator......Page 262
Exercises......Page 264
Part II—Topics......Page 266
Symmetric, Skew-Symmetric and Alternate Forms......Page 267
The Matrix of a Bilinear Form......Page 269
Quadratic Forms......Page 272
Orthogonality......Page 273
Linear Functionals......Page 276
Orthogonal Complements and Orthogonal Direct Sums......Page 277
Isometries......Page 279
Hyperbolic Spaces......Page 280
Nonsingular Completions of a Subspace......Page 281
The Witt Theorems: A Preview......Page 283
The Classification Problem for Metric Vector Spaces......Page 284
Symplectic Geometry......Page 285
The Structure of Orthogonal Geometries: Orthogonal Bases......Page 290
The Classification of Orthogonal Geometries: Canonical Forms......Page 293
The Orthogonal Group......Page 299
The Witt Theorems for Orthogonal Geometries......Page 302
Maximal Hyperbolic Subspaces of an Orthogonal Geometry......Page 303
Exercises......Page 305
The Definition......Page 309
Open and Closed Sets......Page 312
Convergence in a Metric Space......Page 313
The Closure of a Set......Page 314
Dense Subsets......Page 316
Continuity......Page 318
Completeness......Page 319
Isometries......Page 323
The Completion of a Metric Space......Page 324
Exercises......Page 329
A Brief Review......Page 333
Hilbert Spaces......Page 334
Infinite Series......Page 338
An Approximation Problem......Page 339
Hilbert Bases......Page 343
Fourier Expansions......Page 344
Hilbert Dimension......Page 354
A Characterization of Hilbert Spaces......Page 355
The Riesz Representation Theorem......Page 357
Exercises......Page 360
Universality......Page 362
Bilinear Maps......Page 366
Tensor Products......Page 368
When Is a Tensor Product Zero?......Page 374
Coordinate Matrices and Rank......Page 375
Characterizing Vectors in a Tensor Product......Page 378
Defining Linear Transformations on a Tensor Product......Page 381
The Tensor Product of Linear Transformations......Page 382
Change of Base Field......Page 386
Multilinear Maps and Iterated Tensor Products......Page 389
Tensor Spaces......Page 392
Special Multilinear Maps......Page 397
The Symmetric and Antisymmetric Tensor Algebras......Page 399
The Determinant......Page 410
Exercises......Page 413
15.
Positive Solutions to Linear Systems: Convexity and Separation......Page 417
Convex, Closed and Compact Sets......Page 419
Convex Hulls......Page 420
Linear and Affine Hyperplanes......Page 422
Separation......Page 424
Exercises......Page 429
Affine Geometry......Page 432
Affine Combinations......Page 433
Affine Hulls......Page 435
The Lattice of Flats......Page 436
Affine Independence......Page 438
Affine Transformations......Page 440
Projective Geometry......Page 442
Exercises......Page 445
Singular Values......Page 447
The Moore–Penrose Generalized Inverse......Page 450
Least Squares Approximation......Page 452
Exercises......Page 453
Associative Algebras......Page 455
Division Algebras......Page 466
Exercises......Page 473
Formal Power Series......Page 475
The Umbral Algebra......Page 477
Formal Power Series as Linear Operators......Page 481
Sheffer Sequences......Page 484
Examples of Sheffer Sequences......Page 492
Umbral Operators and Umbral Shifts......Page 494
Continuous Operators on the Umbral Algebra......Page 496
Operator Adjoints......Page 497
Umbral Operators and Automorphisms of the Umbral Algebra......Page 498
Umbral Shifts and Derivations of the Umbral Algebra......Page 503
The Transfer Formulas......Page 508
A Final Remark......Page 509
Exercises......Page 510
References......Page 511
Index of Symbols......Page 516
Index......Page 518
Graduate Texts in Mathematics (continud from page ii)......Page 526