This book combines a very rigorous treatment (with a flavoring of abstract algebra) and interesting applications. The presentation is very clear and straightforward. You get theorems, a proof of each one, and curious exercises. Some exercises also challenge you to develop and prove results about some side topics. As you go through the chapters and learn more, you prove further results. Also, this book is the first which presented Jordan forms lucidly and thoroughly. Other texts shove it into the appendix, which is a mistake, since this topic is important.

Finally, the applications are plenty. Standard ones like Markov chains, plus a few fascinating applications, like an entire section devoted to the development of the basics of Special Relativity.

This should be the standard text on linear algebra, instead of that drivel by Strang.

One of the best linear algebra expositions, however I would advice some familiarity with matrix algebra and modern algebra before attempting to read this book. The book emphasizes abstract vector spaces, giving special attention to the relationship between linear transformations and matrices. It covers all main subjects in linear algebra from invariant spaces to canonical forms and includes applications to numerical analysis (condition number),statistics(markov chains), and linear differential equations.

In my experience thus far, I have not come read another introductory algebra that is as comprehensive and thorough as this one. It does not sacrific clarity for the mathematics and similarly it does not sacrifice mathematics for clarity. From the beginning it builds and expects you to keep up as it introduces new topics. There is a definate succession and continuity in this volume which does not exists in many other introductory algebra texts. Furthermore, it presents good proofs and asks for the reader's help where appropriate.

The only aspect of the book that I would critique is its problems. Even though they have somewhat challenging ones, there are none which truly test the depths of ones thinking on the material presented. For example, Spivak does this well in his "Calculus".

Nonetheless, this is a great book. It covers standard topics with a few applications thrown in for good measure. Even so, it is unmistakebly a math book, not a science/engineering text on mathematics. I would recommend this to anyone who want a solid start in linear algebra.