Advanced Linear Algebra

Graduate Texts in Mathematics #135

by Steven Roman

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 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.

Genres: Mathematics, Algebra, Textbooks

504 pages, Hardcover

First published October 20, 1995

Reviews

[Goo]

I bought this as a supplement for my first linear algebra course. Needless to say, I was surprised when I opened it.

You need a basic algebra background before tackling this book.

[Alberto]

The doctoral qualifying algebra exam at my school included chapters 1-3, 7-10 & 17 of this book. It's one of the best linear algebra presentations I've ever seen. There was the odd error (referencing a term before defining it, things like that), but not enough to detract from a 5-star rating.