Advanced Linear Algebra

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Springer Science & Business Media, Dec 31, 2007 - Mathematics - 502 pages
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This is a graduate 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 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 last chapter discusses the umbral calculus, an area of modern algebra with important applications. The second edition contains two new chapters: a chapter on convexity, separation and positive solutions to linear systems and a chapter on the QR decomposition, singular values and pseudoinverses. The treatments of tensor products and the umbral calculus have been greatly expanded and there is now a discussion of determinants (in the chapter on tensor products), the complexification of a real vector space, Schur's lemma and Gersgorin disks.
 

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Contents

Spectral Resolutions and Functional Calculus
228
Positive Operators
230
The Polar Decomposition of an Operator
232
Exercises
234
Part IITopics
236
Metric Vector Spaces
237
The Matrix of a Bilinear Form
242
Quadratic Forms
244

Exercises
51
Linear Transformations
55
The Kernel and Image of a Linear Transformation
57
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
Linear Operators on
71
Exercises
72
The Isomorphism Theorems
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
92
Submodules
95
Spanning Sets
96
Linear Independence
98
Torsion Elements
99
Homomorphisms
100
Quotient Modules
101
The Correspondence and Isomorphism Theorems
102
Modules Are Not As Nice As Vector Spaces
106
Exercises
107
Modules II Free and Noetherian Modules
109
Free Modules and Epimorphisms
114
Noetherian Modules
115
The Hilbert Basis Theorem
118
Exercises
119
Modules over a Principal Ideal Domain
121
Cyclic Modules
122
Free Modules over a Principal Ideal Domain
123
TorsionFree and Free Modules
125
Cyclic Modules
126
The First Decomposition
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
The Structure of a Linear Operator
141
The Module Associated with a Linear Operator
142
Orders and the Minimal Polynomial
144
Cyclic Submodules and Cyclic Subspaces
145
Summary
147
The Rational Canonical Form
148
Exercises
151
Eigenvalues and Eigenvectors
153
Eigenvalues and Eigenvectors
155
Geometric and Algebraic Multiplicities
157
The Jordan Canonical Form
158
Triangularizability and Schurs 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
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
Structure Theory for Normal Operators
200
Unitary Diagonalizability
204
Normal Operators
205
Special Types of Normal Operators
207
SelfAdjoint 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
Orthogonality
245
Linear Functionals
248
Orthogonal Complements and Orthogonal Direct Sums
249
Isometries
252
Hyperbolic Spaces
253
Nonsingular Completions of a Subspace
254
A Preview
256
The Classification Problem for Metric Vector Spaces
257
Symplectic Geometry
258
Orthogonal Bases
264
Canonical Forms
266
The Orthogonal Group
272
The Witts Theorems for Orthogonal Geometries
275
Maximal Hyperbolic Subspaces of an Orthogonal Geometry
277
Exercises
279
Metric Spaces
283
Open and Closed Sets
286
Convergence in a Metric Space
287
The Closure of a Set
288
Dense Subsets
290
Continuity
292
Completeness
293
Isometries
297
The Completion of a Metric Space
298
Exercises
303
Hilbert Spaces
307
Hilbert Spaces
308
Infinite Series
312
An Approximation Problem
313
Hilbert Bases
317
Fourier Expansions
318
A Characterization of Hilbert Bases
328
A Characterization of Hilbert Spaces
329
The Riesz Representation Theorem
331
Exercises
334
Tensor Products
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 Exterior Product Space
380
The Determinant
387
Exercises
391
Positive Solutions to Linear Systems Convexity and Separation
394
Convex Closed and Compact Sets
398
Convex Hulls
399
Linear and Affine Hyperplanes
400
Separation
402
Exercises
407
Affine Geometry
409
Affine Combinations
411
Affine Hulls
412
The Lattice of Flats
413
Affine Independence
416
Affine Transformations
417
Projective Geometry
419
Exercises
423
Operator Factorizations QR and Singular Value
425
Singular Values
428
The MoorePenrose Generalized Inverse
430
Least Squares Approximation
433
Exercises QRFactorization
434
The Umbral Calculus
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
Copyright

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Page 14 - S. Then if xe X, we have by the definition of X that x ^ X. On the other hand, if x X, we have again by the definition of X that x X. This contradiction implies that X ^ im(/) and so / is not surjective. D Cardinal Arithmetic Now let us define addition, multiplication and exponentiation of cardinal numbers. If S and T are sets, the cartesian product S x T is the set of all ordered pairs The set of all functions from T to S is denoted by ST. Definition Let K and A denote cardinal numbers. Let...
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References to this book

Field Theory
Steven Roman
Limited preview - 2005
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About the author (2007)

Dr. Roman has authored 32 books, including a number of books on mathematics, such as Introduction to the Finance of Mathematics, Coding and Information Theory, and Field Theory, published by Springer-Verlag. He has also written Modules in Mathematics, a series of 15 small books designed for the general college-level liberal arts student. Besides his books for O'Reilly, Dr. Roman has written two other computer books, both published by Springer-Verlag.

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