Advanced linear algebra:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York
Springer
2008
|
| Edition: | Third edition |
| Series: | Graduate texts in mathematics
135 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVIII, 522 Seiten |
| ISBN: | 9780387728285 9781441924988 |
Staff View
MARC
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Contents
Preface to the Third Edition, vii
Preface to the Second Edition, ix
Preface to the First Edition, xi
Preliminaries, 1
Part 1: Preliminaries, 1
Part 2: Algebraic Structures, 17
Part I—Basic Linear Algebra, 33
1 Vector Spaces, 35
Vector Spaces, 35
Subspaces, 37
Direct Sums, 40
Spanning Sets and Linear Independence, 44
The Dimension of a Vector Space, 48
Ordered Bases and Coordinate Matrices, 51
The Row and Column Spaces of a Matrix, 52
The Complexification of a Real Vector Space, 53
Exercises, 55
2 Linear Transformations, 59
Linear Transformations, 59
The Kernel and Image of a Linear Transformation, 61
Isomorphisms, 62
The Rank Plus Nullity Theorem, 63
Linear Transformations from F" to F"\ 64
Change of Basis Matrices, 65
The Matrix of a Linear Transformation, 66
Change of Bases for Linear Transformations, 68
Equivalence of Matrices, 68
Similarity of Matrices, 70
Similarity of Operators, 71
Invariant Subspaces and Reducing Pairs, 72
Projection Operators, 73
xiv Contents
Topological Vector Spaces, 79
Linear Operators on Vc, 82
Exercises, 83
3 The Isomorphism Theorems, 87
Quotient Spaces, 87
The Universal Property of Quotients and
the First Isomorphism Theorem, 90
Quotient Spaces, Complements and Codimension, 92
Additional Isomorphism Theorems, 93
Linear Functionals, 94
Dual Bases, 96
Reflexivity, 100
Annihilators, 101
Operator Adjoints, 104
Exercises, 106
4 Modules I: Basic Properties, 109
Motivation, 109
Modules, 109
Submodules, 111
Spanning Sets, 112
Linear Independence, 114
Torsion Elements, 115
Annihilators, 115
Free Modules, 116
Homomorphisms, 117
Quotient Modules, 117
The Correspondence and Isomorphism Theorems, 118
Direct Sums and Direct Summands, 119
Modules Are Not as Nice as Vector Spaces, 124
Exercises, 125
5 Modules II: Free and Noetherian Modules, 127
The Rank of a Free Module, 127
Free Modules and Epimorphisms, 132
Noetherian Modules, 132
The Hilbert Basis Theorem, 136
Exercises, 137
6 Modules over a Principal Ideal Domain, 139
Annihilators and Orders, 139
Cyclic Modules, 140
Free Modules over a Principal Ideal Domain, 142
Torsion-Free and Free Modules, 145
The Primary Cyclic Decomposition Theorem, 146
The Invariant Factor Decomposition, 156
Characterizing Cyclic Modules, 158
Contents xv
Indecomposable Modules, 158
Exercises, 159
7 The Structure of a Linear Operator, 163
The Module Associated with a Linear Operator, 164
The Primary Cyclic Decomposition of VT, 167
The Characteristic Polynomial, 170
Cyclic and Indecomposable Modules, 171
The Big Picture, 174
The Rational Canonical Form, 176
Exercises, 182
8 Eigenvalues and Eigenvectors, 185
Eigenvalues and Eigenvectors, 185
Geometric and Algebraic Multiplicities, 189
The Jordan Canonical Form, 190
Triangularizability and Schur's Theorem, 192
Diagonalizable Operators, 196
Exercises, 198
9 Real and Complex Inner Product Spaces, 205
Norm and Distance, 208
Isometries, 210
Orthogonality, 211
Orthogonal and Orthonormal Sets, 212
The Projection Theorem and Best Approximations, 219
The Riesz Representation Theorem, 221
Exercises, 223
10 Structure Theory for Normal Operators, 227
The Adjoint of a Linear Operator, 227
Orthogonal Projections, 231
Unitary Diagonalizability, 233
Normal Operators, 234
Special Types of Normal Operators, 238
Self-Adjoint Operators, 239
Unitary Operators and Isometries, 240
The Structure of Normal Operators, 245
Functional Calculus, 247
Positive Operators, 250
The Polar Decomposition of an Operator, 252
Exercises, 254
Part II—Topics, 257
11 Metric Vector Spaces: The Theory of Bilinear Forms, 259
Symmetric, Skew-Symmetric and Alternate Forms, 259
The Matrix of a Bilinear Form, 261
xvi Contents
Quadratic Forms, 264
Orthogonality, 265
Linear Functionals, 268
Orthogonal Complements and Orthogonal Direct Sums, 269
Isometries, 271
Hyperbolic Spaces, 272
Nonsingular Completions of a Subspace, 273
The Witt Theorems: A Preview, 275
The Classification Problem for Metric Vector Spaces, 276
Symplectic Geometry, 277
The Structure of Orthogonal Geometries: Orthogonal Bases, 282
The Classification of Orthogonal Geometries:
Canonical Forms, 285
The Orthogonal Group, 291
The Witt Theorems for Orthogonal Geometries, 294
Maximal Hyperbolic Subspaces of an Orthogonal Geometry, 295
Exercises, 297
12 Metric Spaces, 301
The Definition, 301
Open and Closed Sets, 304
Convergence in a Metric Space, 305
The Closure of a Set, 306
Dense Subsets, 308
Continuity, 310
Completeness, 311
Isometries, 315
The Completion of a Metric Space, 316
Exercises, 321
13 Hilbert Spaces, 325
A Brief Review, 325
Hilbert Spaces, 326
Infinite Series, 330
An Approximation Problem, 331
Hilbert Bases, 335
Fourier Expansions, 336
A Characterization of Hilbert Bases, 346
Hilbert Dimension, 346
A Characterization of Hilbert Spaces, 347
The Riesz Representation Theorem, 349
Exercises, 352
14 Tensor Products, 355
Universality, 355
Bilinear Maps, 359
Tensor Products, 361
Contents xvii
When Is a Tensor Product Zero?, 367
Coordinate Matrices and Rank, 368
Characterizing Vectors in a Tensor Product, 371
Defining Linear Transformations on a Tensor Product, 374
The Tensor Product of Linear Transformations, 375
Change of Base Field, 379
Multilinear Maps and Iterated Tensor Products, 382
Tensor Spaces, 385
Special Multilinear Maps, 390
Graded Algebras, 392
The Symmetric and Antisymmetric
Tensor Algebras, 392
The Determinant, 403
Exercises, 406
15 Positive Solutions to Linear Systems:
Convexity and Separation, 411
Convex, Closed and Compact Sets, 413
Convex Hulls, 414
Linear and Affine Hyperplanes, 416
Separation, 418
Exercises, 423
16 Affine Geometry, 427
Affine Geometry, 427
Affine Combinations, 428
Affine Hulls, 430
The Lattice of Flats, 431
Affine Independence, 433
Affine Transformations, 435
Projective Geometry, 437
Exercises, 440
17 Singular Values and the Moore-Penrose Inverse, 443
Singular Values, 443
The Moore-Penrose Generalized Inverse, 446
Least Squares Approximation, 448
Exercises, 449
18 An Introduction to Algebras, 451
Motivation, 451
Associative Algebras, 451
Division Algebras, 462
Exercises, 469
19 The Umbral Calculus, 471
Formal Power Series, 471
The Umbral Algebra, 473
xviii Contents
Formal Power Series as Linear Operators, 477
Sheffer Sequences, 480
Examples of Sheffer Sequences, 488
Umbral Operators and Umbral Shifts, 490
Continuous Operators on the Umbral Algebra, 492
Operator Adjoints, 493
Umbral Operators and Automorphisms
of the Umbral Algebra, 494
Umbral Shifts and Derivations of the Umbral Algebra, 499
The Transfer Formulas, 504
A Final Remark, 505
Exercises, 506
References, 507
Index of Symbols, 513
Index, 515 |
| adam_txt |
Contents
Preface to the Third Edition, vii
Preface to the Second Edition, ix
Preface to the First Edition, xi
Preliminaries, 1
Part 1: Preliminaries, 1
Part 2: Algebraic Structures, 17
Part I—Basic Linear Algebra, 33
1 Vector Spaces, 35
Vector Spaces, 35
Subspaces, 37
Direct Sums, 40
Spanning Sets and Linear Independence, 44
The Dimension of a Vector Space, 48
Ordered Bases and Coordinate Matrices, 51
The Row and Column Spaces of a Matrix, 52
The Complexification of a Real Vector Space, 53
Exercises, 55
2 Linear Transformations, 59
Linear Transformations, 59
The Kernel and Image of a Linear Transformation, 61
Isomorphisms, 62
The Rank Plus Nullity Theorem, 63
Linear Transformations from F" to F"\ 64
Change of Basis Matrices, 65
The Matrix of a Linear Transformation, 66
Change of Bases for Linear Transformations, 68
Equivalence of Matrices, 68
Similarity of Matrices, 70
Similarity of Operators, 71
Invariant Subspaces and Reducing Pairs, 72
Projection Operators, 73
xiv Contents
Topological Vector Spaces, 79
Linear Operators on Vc, 82
Exercises, 83
3 The Isomorphism Theorems, 87
Quotient Spaces, 87
The Universal Property of Quotients and
the First Isomorphism Theorem, 90
Quotient Spaces, Complements and Codimension, 92
Additional Isomorphism Theorems, 93
Linear Functionals, 94
Dual Bases, 96
Reflexivity, 100
Annihilators, 101
Operator Adjoints, 104
Exercises, 106
4 Modules I: Basic Properties, 109
Motivation, 109
Modules, 109
Submodules, 111
Spanning Sets, 112
Linear Independence, 114
Torsion Elements, 115
Annihilators, 115
Free Modules, 116
Homomorphisms, 117
Quotient Modules, 117
The Correspondence and Isomorphism Theorems, 118
Direct Sums and Direct Summands, 119
Modules Are Not as Nice as Vector Spaces, 124
Exercises, 125
5 Modules II: Free and Noetherian Modules, 127
The Rank of a Free Module, 127
Free Modules and Epimorphisms, 132
Noetherian Modules, 132
The Hilbert Basis Theorem, 136
Exercises, 137
6 Modules over a Principal Ideal Domain, 139
Annihilators and Orders, 139
Cyclic Modules, 140
Free Modules over a Principal Ideal Domain, 142
Torsion-Free and Free Modules, 145
The Primary Cyclic Decomposition Theorem, 146
The Invariant Factor Decomposition, 156
Characterizing Cyclic Modules, 158
Contents xv
Indecomposable Modules, 158
Exercises, 159
7 The Structure of a Linear Operator, 163
The Module Associated with a Linear Operator, 164
The Primary Cyclic Decomposition of VT, 167
The Characteristic Polynomial, 170
Cyclic and Indecomposable Modules, 171
The Big Picture, 174
The Rational Canonical Form, 176
Exercises, 182
8 Eigenvalues and Eigenvectors, 185
Eigenvalues and Eigenvectors, 185
Geometric and Algebraic Multiplicities, 189
The Jordan Canonical Form, 190
Triangularizability and Schur's Theorem, 192
Diagonalizable Operators, 196
Exercises, 198
9 Real and Complex Inner Product Spaces, 205
Norm and Distance, 208
Isometries, 210
Orthogonality, 211
Orthogonal and Orthonormal Sets, 212
The Projection Theorem and Best Approximations, 219
The Riesz Representation Theorem, 221
Exercises, 223
10 Structure Theory for Normal Operators, 227
The Adjoint of a Linear Operator, 227
Orthogonal Projections, 231
Unitary Diagonalizability, 233
Normal Operators, 234
Special Types of Normal Operators, 238
Self-Adjoint Operators, 239
Unitary Operators and Isometries, 240
The Structure of Normal Operators, 245
Functional Calculus, 247
Positive Operators, 250
The Polar Decomposition of an Operator, 252
Exercises, 254
Part II—Topics, 257
11 Metric Vector Spaces: The Theory of Bilinear Forms, 259
Symmetric, Skew-Symmetric and Alternate Forms, 259
The Matrix of a Bilinear Form, 261
xvi Contents
Quadratic Forms, 264
Orthogonality, 265
Linear Functionals, 268
Orthogonal Complements and Orthogonal Direct Sums, 269
Isometries, 271
Hyperbolic Spaces, 272
Nonsingular Completions of a Subspace, 273
The Witt Theorems: A Preview, 275
The Classification Problem for Metric Vector Spaces, 276
Symplectic Geometry, 277
The Structure of Orthogonal Geometries: Orthogonal Bases, 282
The Classification of Orthogonal Geometries:
Canonical Forms, 285
The Orthogonal Group, 291
The Witt Theorems for Orthogonal Geometries, 294
Maximal Hyperbolic Subspaces of an Orthogonal Geometry, 295
Exercises, 297
12 Metric Spaces, 301
The Definition, 301
Open and Closed Sets, 304
Convergence in a Metric Space, 305
The Closure of a Set, 306
Dense Subsets, 308
Continuity, 310
Completeness, 311
Isometries, 315
The Completion of a Metric Space, 316
Exercises, 321
13 Hilbert Spaces, 325
A Brief Review, 325
Hilbert Spaces, 326
Infinite Series, 330
An Approximation Problem, 331
Hilbert Bases, 335
Fourier Expansions, 336
A Characterization of Hilbert Bases, 346
Hilbert Dimension, 346
A Characterization of Hilbert Spaces, 347
The Riesz Representation Theorem, 349
Exercises, 352
14 Tensor Products, 355
Universality, 355
Bilinear Maps, 359
Tensor Products, 361
Contents xvii
When Is a Tensor Product Zero?, 367
Coordinate Matrices and Rank, 368
Characterizing Vectors in a Tensor Product, 371
Defining Linear Transformations on a Tensor Product, 374
The Tensor Product of Linear Transformations, 375
Change of Base Field, 379
Multilinear Maps and Iterated Tensor Products, 382
Tensor Spaces, 385
Special Multilinear Maps, 390
Graded Algebras, 392
The Symmetric and Antisymmetric
Tensor Algebras, 392
The Determinant, 403
Exercises, 406
15 Positive Solutions to Linear Systems:
Convexity and Separation, 411
Convex, Closed and Compact Sets, 413
Convex Hulls, 414
Linear and Affine Hyperplanes, 416
Separation, 418
Exercises, 423
16 Affine Geometry, 427
Affine Geometry, 427
Affine Combinations, 428
Affine Hulls, 430
The Lattice of Flats, 431
Affine Independence, 433
Affine Transformations, 435
Projective Geometry, 437
Exercises, 440
17 Singular Values and the Moore-Penrose Inverse, 443
Singular Values, 443
The Moore-Penrose Generalized Inverse, 446
Least Squares Approximation, 448
Exercises, 449
18 An Introduction to Algebras, 451
Motivation, 451
Associative Algebras, 451
Division Algebras, 462
Exercises, 469
19 The Umbral Calculus, 471
Formal Power Series, 471
The Umbral Algebra, 473
xviii Contents
Formal Power Series as Linear Operators, 477
Sheffer Sequences, 480
Examples of Sheffer Sequences, 488
Umbral Operators and Umbral Shifts, 490
Continuous Operators on the Umbral Algebra, 492
Operator Adjoints, 493
Umbral Operators and Automorphisms
of the Umbral Algebra, 494
Umbral Shifts and Derivations of the Umbral Algebra, 499
The Transfer Formulas, 504
A Final Remark, 505
Exercises, 506
References, 507
Index of Symbols, 513
Index, 515 |
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| series | Graduate texts in mathematics |
| series2 | Graduate texts in mathematics |
| spelling | Roman, Steven Verfasser (DE-588)1085648982 aut Advanced linear algebra Steven Roman Third edition New York Springer 2008 XVIII, 522 Seiten txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 135 Algèbre linéaire Algèbre linéaire ram Algebras, Linear Lineare Algebra (DE-588)4035811-2 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Lineare Algebra (DE-588)4035811-2 s DE-604 Erscheint auch als Online-Ausgabe 978-0-387-72831-5 Graduate texts in mathematics 135 (DE-604)BV000000067 135 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016318624&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Roman, Steven Advanced linear algebra Graduate texts in mathematics Algèbre linéaire Algèbre linéaire ram Algebras, Linear Lineare Algebra (DE-588)4035811-2 gnd |
| subject_GND | (DE-588)4035811-2 (DE-588)4123623-3 |
| title | Advanced linear algebra |
| title_auth | Advanced linear algebra |
| title_exact_search | Advanced linear algebra |
| title_exact_search_txtP | Advanced linear algebra |
| title_full | Advanced linear algebra Steven Roman |
| title_fullStr | Advanced linear algebra Steven Roman |
| title_full_unstemmed | Advanced linear algebra Steven Roman |
| title_short | Advanced linear algebra |
| title_sort | advanced linear algebra |
| topic | Algèbre linéaire Algèbre linéaire ram Algebras, Linear Lineare Algebra (DE-588)4035811-2 gnd |
| topic_facet | Algèbre linéaire Algebras, Linear Lineare Algebra Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016318624&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000067 |
| work_keys_str_mv | AT romansteven advancedlinearalgebra |