Advanced Linear Algebra, Second Edition 2005
Springer-Verlag Graduate Texts in Mathematics, Second Edition, Volume 135
Preface to Second Edition
The text has been completely rewritten. I hope that an additional 12 years and roughly 20 books worth of
experience has enabled me to improve the quality of my exposition. Also, the exercise sets have been completely rewritten.
The second edition contains two new chapters: a chapter on convexity, separation and positive solutions
to linear systems (Chapter 15) and a chapter on the QR decomposition, singular values and pseudoinverses (Chapter 17).
The treatments of tensor products and the umbral calculus have been greatly expanded and I have included discussions
of determinants (in the chapter on tensor products), the complexification of a real vector space,
Schur's lemma and Geršgorin disks.
Preface to First Edition
This is a graduate level textbook covering an especially broad range of
topics. The first part of the book contains a careful but rapid discussion of
the basics of linear algebra, including vector spaces, linear transformations,
quotient spaces, and isomorphism theorems. The author then proceeds to a
discussion of modules, emphasizing a comparison with vector spaces. A thorough
discussion of inner product spaces, eigenvalues, eigenvectors, and finite
dimensional spectral theory follows, culminating in the finite dimensional
spectral theorem for normal operators.
The second part of the book is a collection of topics, including metric
vector spaces, metric spaces, Hilbert spaces, tensor products, and affine
geometry. The final chapter contains a discussion of the umbral calculus, a
relatively new area of modern algebra that is important in applications. This is
the first time that this topic has appeared in a textbook. The book contains
numerous exercises, and is suitable both as a textbook and as a reference for
students and instructors.
2005, ISBN 0-387-24766-1, 482 pp., Hardcover
Contents
Preliminaries, 1
Vector Spaces, 33
Vector Spaces, 33
Subspaces, 35
Direct Sums, 37
Spanning Sets and Linear Independence, 41
The Dimension of a Vector Space, 44
Ordered Bases and Coordinate Matrices, 47
The Row and Column Space of a Matrix, 47
The Complexification of a Real Vector Space, 48
Exercises, 51
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
Topological Vector Spaces, 68
Linear Operators on , 71
Exercises, 72
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
Modules I: Basic Properties, 93
Modules, 93
Motivation , 93
Submodules, 95
Spanning Sets, 96
Linear Independence, 98
Annihilators, 99
Free Modules, 99
Torsion Elements, 99
Homomorphisms, 100
Quotient Modules, 101
Direct Sums and Direct Summands, 102
The Correspondence and Isomorphism Theorems, 102
Exercises, 106
Modules are Not as Nice as Vector Spaces, 106
Modules II: Free and Noetherian Modules, 109
The Rank of a Free Module, 109
Free Modules and Epimorphisms, 114
Noetherian Modules, 114
The Hilbert Basis Theorem, 118
Exercises, 119
Modules Over a Principal Ideal Domain, 121
Annihilators and Orders, 121
Cyclic Modules, 121
Free Modules over a Principal Ideal Domain, 123
Torsion-Free and Free Modules, 125
Prelude to Decomposition: Cyclic Modules, 126
The First Decomposition, 126
A Look Ahead, 127
The Primary Decomposition, 128
The Cyclic Decomposition of a Primary Module, 130
The Primary Cyclic Decomposition Theorem, 133
The Invariant Factor Decomposition, 135
Exercises, 137
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
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, 171
Projections and Invariance, 173
Exercises, 174
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, 193
The Riesz Representation Theorem, 194
Exercises, 196
Structure Theory for Normal Operators, 199
The Adjoint of a Linear Operator, 199
Unitary Diagonalizability, 202
Normal Operators, 203
Special Types of Normal Operators, 205
Self-Adjoint Operators, 206
Unitary Operators and Isometries, 208
The Structure of Normal Operators, 213
Matrix Versions, 220
Orthogonal Projections, 221
Orthogonal Resolutions of the Identity, 224
The Spectral Theorem, 225
Spectral Resolutions and Functional Calculus, 226
Positive Operators, 229
The Polar Decomposition of an Operator, 230
Exercises, 232
Metric Vector Spaces: The Theory of Bilinear Forms, 237
Symmetric, Skew-Symmetric and Alternate Forms, 237
The Matrix of a Bilinear Form, 240
Quadratic Forms, 242
Orthogonality, 243
Linear Functionals, 246
Orthogonal Complements and Orthogonal Direct Sums, 247
Isometries, 250
Hyperbolic Spaces, 251
Nonsingular Completions of a Subspace, 252
The Witt Theorems: A Preview, 254
The Classification Problem for Metric Vector Spaces, 255
Symplectic Geometry, 256
The Structure of Orthogonal Geometries: Orthogonal Bases, 261
The Classification of Orthogonal Geometries: Canonical Forms, 264
The Orthogonal Group, 270
The Witt's Theorems for Orthogonal Geometries, 273
Maximal Hyperbolic Subspaces of an Orthogonal Geometry, 275
Exercises, 277
Metric Spaces, 281
The Definition, 281
Open and Closed Sets, 284
Convergence in a Metric Space, 285
The Closure of a Set, 286
Dense Subsets, 288
Continuity, 290
Completeness, 291
Isometries, 295
The Completion of a Metric Space, 296
Exercises, 301
Hilbert Spaces, 305
A Brief Review, 305
Hilbert Spaces, 306
Infinite Series, 310
An Approximation Problem, 311
Hilbert Bases, 315
Fourier Expansions, 316
A Characterization of Hilbert Bases, 326
Hilbert Dimension, 326
A Characterization of Hilbert Spaces, 327
The Riesz Representation Theorem, 329
Exercises, 332
Tensor Products, 335
Universality, 335
Bilinear Maps, 339
Tensor Products, 341
When is a Tensor Product Zero?, 346
Coordinate Matrices and Rank, 348
Characterizing Vectors in a Tensor Product, 352
Defining Linear Transformations on a Tensor Product, 353
The Tensor Product of Linear Transformations, 355
Change of Base Field, 357
Multilinear Maps and Iterated Tensor Products, 361
Tensor Spaces, 364
Special Multilinear Maps, 369
Graded Algebras, 370
The Symmetric Tensor Algebra, 372
The AntiSymmetric Tensor Algebra: The Exterior Product Space, 378
The Determinant, 384
Exercises, 388
Positive Solutions to Linear Systems: Convexity and Separation , 393
Convex, Closed and Compact Sets, 396
Convex Hulls, 397
Linear and Affine Hyperplanes, 398
Separation, 400
Exercises, 405
Affine Geometry, 407
Affine Geometry, 407
Affine Combinations, 409
Affine Hulls, 410
The Lattice of Flats, 411
Affine Independence, 414
Affine Transformations, 415
Projective Geometry, 417
Exercises, 421
Operator Factorizations: QR and Singular Value, 423
The QR Decomposition, 423
Singular Values, 426
The Moore-Penrose Generalized Inverse, 428
Least Squares Approximation, 431
Exercises, 432
The Umbral Calculus, 435
Formal Power Series, 435
The Umbral Algebra, 437
Formal Power Series as Linear Operators, 441
Sheffer Sequences, 444
Examples of Sheffer Sequences, 451
Umbral Operators and Umbral Shifts, 454
Continuous Operators on the Umbral Algebra, 455
Operator Adjoints, 457
Umbral Operators and Automorphisms of the Umbral Algebra, 458
Umbral Shifts and Derivations of the Umbral Algebra, 462
The Transfer Formulas, 467
A Final Remark, 468
Exercises, 469
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