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