* Vector Spaces * Linear Transformations * The Isomorphism Theorems * Modules I: Basic Properties * Modules II: Free and Noetherian Modules * Modules over a Principal Ideal Domain * The Structure of a Linear Operator * Eigenvalues and Eigenvectors * Real and Complex Inner Product Spaces * Structure Theory for Normal Operators * Metric Vector Spaces: The Theory of Bilinear Forms * Metric Spaces * Hilbert Spaces * Tensor Products * Positive Solutions to Linear Systems: Convexity and Separation * Affine Geometry * Operator Factorizations: QR and Singular Value * The Umbral Calculus * References * Index

For the third edition, the author has added a new chapter on associative algebras that includes the well known characterizations of the finite-dimensional division algebras over the real field (a theorem of Frobenius) and over a finite field (Wedderburn's theorem); polished and refined some arguments (such as the discussion of reflexivity, the rational canonical form, best approximations and the definitions of tensor products); upgraded some proofs that were originally done only for finite-dimensional/rank cases; added new theorems, including the spectral mapping theorem; corrected all known errors; the reference section has been enlarged considerably, with over a hundred references to books on linear algebra.
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, ZentralblattMATH
¿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

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.