Advanced Linear Algebra

1. Front Matter

   Pages i-xvi

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2. Preliminaries

   1. Preliminaries

      Pages 1-30

3. Basic Linear Algebra

   1. Front Matter

      Pages 31-31

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   2. Vector Spaces

      Pages 33-54

   3. Linear Transformations

      Pages 55-74

   4. The Isomorphism Theorems

      Pages 75-91

   5. Modules I: Basic Properties

      Pages 93-108

   6. Modules II: Free and Noetherian Modules

      Pages 109-120

   7. Modules over a Principal Ideal Domain

      Pages 121-140

   8. The Structure of a Linear Operator

      Pages 141-152

   9. Eigenvalues and Eigenvectors

      Pages 153-179

   10. Real and Complex Inner Product Spaces

       Pages 181-199

   11. Structure Theory for Normal Operators

       Pages 201-235

4. Topics

   1. Front Matter

      Pages 237-237

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   2. Metric Vector Spaces: The Theory of Bilinear Forms

      Pages 239-282

   3. Metric Spaces

      Pages 283-306

   4. Hilbert Spaces

      Pages 307-335

   5. Tensor Products

      Pages 337-393

   6. Positive Solutions to Linear Systems: Convexity and Separation

      Pages 395-408

   7. Affine Geometry

      Pages 409-423

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 Geršgorin 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.

• Eigenvalue
• Eigenvector
• Finite
• Morphism
• algebra
• calculus
• geometry
• linear algebra
• theorem
• matrix theory

From the reviews of the first edition:

"… The book is very well written and has a good set of exercises. It is a suitable choice as a graduate textbook as well as a reference book." A.A. Jafarian for ZentralblattMATH

From the reviews of the second edition:

"In this 2nd edition, the author has rewritten the entire book and has added more than 100 pages of new materials. … As in the previous edition, the text is well written and gives a thorough discussion of many topics of linear algebra and related fields. … the exercises are rewritten and expanded. … Overall, I found the book a very useful one. … It is a suitable choice as a graduate text or as a reference book." (Ali-Akbar Jafarian, Zentralblatt MATH, Vol. 1085, 2006)

"This is a formidable volume, a compendium of linear algebra theory, classical and modern … . The development of the subject is elegant … . The proofs are neat … . The exercise sets are good, with occasional hints given for the solution of trickier problems. … It represents linear algebra and does so comprehensively." (Henry Ricardo, MathDL, May, 2005)

• University of California, Irvine, Irvine, USA

  Steven Roman

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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