Advanced Linear Algebra, Second Edition

Steven Roman0387247661, 9780387247663

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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Table of contents :
Front Matter….Pages i-xii
Preliminaries….Pages 1-24
Front Matter….Pages 25-25
Vector Spaces….Pages 27-43
Linear Transformations….Pages 45-62
The Isomorphism Theorems….Pages 63-81
Modules I….Pages 83-95
Modules II….Pages 97-106
Modules over Principal Ideal Domains….Pages 107-119
The Structure of a Linear Operator….Pages 121-133
Eigenvalues and Eigenvectors….Pages 135-156
Real and Complex Inner Product Spaces….Pages 157-174
The Spectral Theorem for Normal Operators….Pages 175-202
Front Matter….Pages 203-203
Metric Vector Spaces….Pages 205-237
Metric Spaces….Pages 239-261
Hilbert Spaces….Pages 263-290
Tensor Products….Pages 291-314
Affine Geometry….Pages 315-328
The Umbral Calculus….Pages 329-352
Back Matter….Pages 353-366