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[资源] Advanced Linear Algebra - Steven Roman .pdf

Advanced Linear Algebra - Steven Roman .pdf

Preface to the Second Edition, vii
Preface to the First Edition, ix
Preliminaries, 1
Part 1 Preliminaries, 1
Part 2 Algebraic Structures, 16
Part I—Basic Linear Algebra, 31
1 Vector Spaces, 33
Vector Spaces, 33
Subspaces, 35
Direct Sums, 38
Spanning Sets and Linear Independence, 41
The Dimension of a Vector Space, 44
Ordered Bases and Coordinate Matrices, 47
The Row and Column Spaces of a Matrix, 48
The Complexification of a Real Vector Space, 49
Exercises, 51
2 Linear Transformations, 55
Linear Transformations, 55
Isomorphisms, 57
The Kernel and Image of a Linear Transformation, 57
Linear Transformations from 􀀭􀀛 to 􀀭􀀚, 59
The Rank Plus Nullity Theorem, 59
Change of Basis Matrices, 60
The Matrix of a Linear Transformation, 61
Change of Bases for Linear Transformations, 63
Equivalence of Matrices, 64
Similarity of Matrices, 65
Similarity of Operators, 66
Invariant Subspaces and Reducing Pairs, 68
xii Contents
Topological Vector Spaces, 68
Linear Operators on 􀀽 􀁤, 71
Exercises, 72
3 The Isomorphism Theorems, 75
Quotient Spaces, 75
The Universal Property of Quotients and
the First Isomorphism Theorem, 77
Quotient Spaces, Complements and Codimension, 79
Additional Isomorphism Theorems, 80
Linear Functionals, 82
Dual Bases, 83
Reflexivity, 84
Annihilators, 86
Operator Adjoints, 88
Exercises, 90
4 Modules I: Basic Properties, 93
Modules, 93
Motivation, 93
Submodules, 95
Spanning Sets, 96
Linear Independence, 98
Torsion Elements, 99
Annihilators, 99
Free Modules, 99
Homomorphisms, 100
Quotient Modules, 101
The Correspondence and Isomorphism Theorems, 102
Direct Sums and Direct Summands, 102
Modules Are Not As Nice As Vector Spaces, 106
Exercises, 106
5 Modules II: Free and Noetherian Modules, 109
The Rank of a Free Module, 109
Free Modules and Epimorphisms, 114
Noetherian Modules, 115
The Hilbert Basis Theorem, 118
Exercises, 119
6 Modules over a Principal Ideal Domain, 121
Annihilators and Orders, 121
Cyclic Modules, 122
Free Modules over a Principal Ideal Domain, 123
Torsion-Free and Free Modules, 125
Contents xiii
Prelude to Decomposition: Cyclic Modules, 126
The First Decomposition, 127
A Look Ahead, 127
The Primary Decomposition, 128
The Cyclic Decomposition of a Primary Module, 130
The Primary Cyclic Decomposition Theorem, 134
The Invariant Factor Decomposition, 135
Exercises, 138
7 The Structure of a Linear Operator, 141
A Brief Review, 141
The Module Associated with a Linear Operator, 142
Orders and the Minimal Polynomial, 144
Cyclic Submodules and Cyclic Subspaces, 145
Summary, 147
The Decomposition of 􀀽􀀚 , 147
The Rational Canonical Form, 148
Exercises, 151
8 Eigenvalues and Eigenvectors, 153
The Characteristic Polynomial of an Operator, 153
Eigenvalues and Eigenvectors, 155
Geometric and Algebraic Multiplicities, 157
The Jordan Canonical Form, 158
Triangularizability and Schur's Lemma, 160
Diagonalizable Operators, 165
Projections, 166
The Algebra of Projections, 167
Resolutions of the Identity, 170
Spectral Resolutions, 172
Projections and Invariance, 173
Exercises, 174
9 Real and Complex Inner Product Spaces , 181
Norm and Distance, 183
Isometries, 186
Orthogonality, 187
Orthogonal and Orthonormal Sets, 188
The Projection Theorem and Best Approximations, 192
Orthogonal Direct Sums, 194
The Riesz Representation Theorem, 195
Exercises, 196
10 Structure Theory for Normal Operators, 201
The Adjoint of a Linear Operator, 201
xiv Contents
Unitary Diagonalizability, 204
Normal Operators, 205
Special Types of Normal Operators, 207
Self-Adjoint Operators, 208
Unitary Operators and Isometries, 210
The Structure of Normal Operators, 215
Matrix Versions, 222
Orthogonal Projections, 223
Orthogonal Resolutions of the Identity, 226
The Spectral Theorem, 227
Spectral Resolutions and Functional Calculus, 228
Positive Operators, 230
The Polar Decomposition of an Operator, 232
Exercises, 234
Part II—Topics, 235
11 Metric Vector Spaces: The Theory of Bilinear Forms, 239
Symmetric, Skew-Symmetric and Alternate Forms, 239
The Matrix of a Bilinear Form, 242
Quadratic Forms, 244
Orthogonality, 245
Linear Functionals, 248
Orthogonal Complements and Orthogonal Direct Sums, 249
Isometries, 252
Hyperbolic Spaces, 253
Nonsingular Completions of a Subspace, 254
The Witt Theorems: A Preview, 256
The Classification Problem for Metric Vector Spaces, 257
Symplectic Geometry, 258
The Structure of Orthogonal Geometries: Orthogonal Bases, 264
The Classification of Orthogonal Geometries:
Canonical Forms, 266
The Orthogonal Group, 272
The Witt's Theorems for Orthogonal Geometries, 275
Maximal Hyperbolic Subspaces of an Orthogonal Geometry, 277
Exercises, 279
12 Metric Spaces, 283
The Definition, 283
Open and Closed Sets, 286
Convergence in a Metric Space, 287
The Closure of a Set, 288
Contents xv
Dense Subsets, 290
Continuity, 292
Completeness, 293
Isometries, 297
The Completion of a Metric Space, 298
Exercises, 303
13 Hilbert Spaces, 307
A Brief Review, 307
Hilbert Spaces, 308
Infinite Series, 312
An Approximation Problem, 313
Hilbert Bases, 317
Fourier Expansions, 318
A Characterization of Hilbert Bases, 328
Hilbert Dimension, 328
A Characterization of Hilbert Spaces, 329
The Riesz Representation Theorem, 331
Exercises, 334
14 Tensor Products, 337
Universality, 337
Bilinear Maps, 341
Tensor Products, 343
When Is a Tensor Product Zero? 348
Coordinate Matrices and Rank, 350
Characterizing Vectors in a Tensor Product, 354
Defining Linear Transformations on a Tensor Product, 355
The Tensor Product of Linear Transformations, 357
Change of Base Field, 359
Multilinear Maps and Iterated Tensor Products, 363
Tensor Spaces, 366
Special Multilinear Maps, 371
Graded Algebras, 372
The Symmetric Tensor Algebra, 374
The Antisymmetric Tensor Algebra:
The Exterior Product Space, 380
The Determinant, 387
Exercises, 391
15 Positive Solutions to Linear Systems:
Convexity and Separation 395
Convex, Closed and Compact Sets, 398
Convex Hulls, 399
xvi Contents
Linear and Affine Hyperplanes, 400
Separation, 402
Exercises, 407
16 Affine Geometry, 409
Affine Geometry, 409
Affine Combinations, 41
Affine Hulls, 412
The Lattice of Flats, 413
Affine Independence, 416
Affine Transformations, 417
Projective Geometry, 419
Exercises, 423
17 Operator Factorizations: QR and Singular Value, 425
The QR Decomposition, 425
Singular Values, 428
The Moore–Penrose Generalized Inverse, 430
Least Squares Approximation, 433
Exercises, 434
18 The Umbral Calculus, 437
Formal Power Series, 437
The Umbral Algebra, 439
Formal Power Series as Linear Operators, 443
Sheffer Sequences, 446
Examples of Sheffer Sequences, 454
Umbral Operators and Umbral Shifts, 456
Continuous Operators on the Umbral Algebra, 458
Operator Adjoints, 459
Umbral Operators and Automorphisms
of the Umbral Algebra, 460
Umbral Shifts and Derivations of the Umbral Algebra, 465
The Transfer Formulas, 470
A Final Remark, 471
Exercises, 472
References, 473
Index, 475

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2015-02-01 21:12:22, 4.34 M