Mathematics books are often considerably more difficult to read than their authors prepare their audiences to believe; this book is a happy exception. It is written for an audience of readers at a specific place in their studies (ones who know linear algebra but want to take their understanding of it to a deeper level), and it reaches this audience very well. The emphasis of this book is on linear algebra in abstract mathematics; it is less useful for people interested in numerical linear algebra.

As the name suggests, this book requires a fair amount of background. The introductory chapter moves very fast, but is thorough, and exciting to read. The rest of the book presents advanced topics at a more leisurely pace, while still remaining fairly concise. Some difficult concepts, such as the universal property, are introduced several times at several different places in the book, so that someone working through the book will be more familiar with them when it is finally necessary to understand them on a deeper level.

I find the material on modules outstanding; the author explores the analogies between modules and vector spaces, rigorously exploring which analogies hold, and giving examples of cases in which other analogies fail. The presentation of modules in this book differs greatly from that encountered in most abstract algebra texts: while most books focus on modules' similarities to rings and applications in commutative algebra, this text focuses on their similarities to vector spaces and applications to the study linear operators on vector spaces.

One should not be scared by the word "advanced" in the book's title. Although the book covers advanced topics, it is very clear. When proofs are omitted, it is usually because they are very easy for the reader to supply. The exercises are very valuable (some are critical for understanding the material), but they're not diabolically difficult.

I think this book would make an outstanding textbook for an introductory graduate-level course in linear algebra, or perhaps a senior-level undergraduate course for students with a strong background. It is also very well-suited to self-study. A student with prior background in abstract algebra (group theory, ring theory, etc.) will find this book much more manageable than a student who has not covered such material. People wanting a more introductory text might want to look to the book by Axler, or the old classic by Shilov.

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