Advanced Linear Algebra

Let me begin by thanking the readers of the first edition for their many helpful comments and suggestions. The second edition represents a major change from the first edition. Indeed, one might say that it is a totally new book, with the exception of the general range of topics covered. 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 Gersgorin disks. Steven Roman Irvine, California February 2005 Preface to the First Edition This book is a thorough introduction to linear algebra, for the graduate or advanced undergraduate student. Prerequisites are limited to a knowledge of the basic properties of matrices and determinants. However, since we cover the basics of vector spaces and linear transformations rather rapidly, a prior course in linear algebra (even at the sophomore level), along with a certain measure of "mathematical maturity," is highly desirable.