Linear Algebra Done Right

This book is very easy to read, especially compared to other books on abstract linear algebra. The proofs are easy to follow and give intuitive insight into the results. Most importantly, this book makes linear algebra fun, unlike most of the typical introductory undergrad texts. The largest weakness of this book is that it does not cover much material. It covers the basics of vector spaces, defines and proves a few basic theorems about eigenvectors and eigenvalues, and then ends. A lot of the discussion in the more advanced chapters (the chapter on inner products comes to mind) is inadequate for anyone who intends to actually use the material. The discussion of determinants is an afterthought, and the book doesn't even touch bilinear forms, doesn't explore geometry very much, nor does it really provide any glimpse into any of the vast applications of linear algebra that are out there. This book is minimalist; it is excellent in that respect, but it is not even close to comprehensive. I think that if you use this book for a class, it should be supplemented by one or more other books. This book would be excellent for self-study because it is so clear, but it is not very useful as a reference; 90% of the time I try to look something up in this book, I find it's simply not there. I find that this book can be complemented very well by Shilov's book (which starts from determinants, something this book does not focus on).

This is probably the best book on linear algebra I have seen. It approaches linear algebra from a theoretical point of view (i.e. linear maps instead of matrices), it is not "watered-down", and yet it is accessible to undergrads. The key reason for this is because Dr. Axler has kept the book well-focused, has put in only the necessary results needed, and has tied these elements together clearly. I obtained the book, because like the author, I never liked some of the more standard proofs of "classical" theorems. However, even the material at the beginning is superbly organized and well thought out, and a joy to read. His use of a single lemma in chapter 2 for example, (the so-called "linear dependence lemma") makes many of the results given later on in the chapter (and the book), trivial to prove. One of the reasons Axler can use this lemma so effectively, is because he is careful about his notation, and uses ordered tuples instead of sets of vectors. This is just one example of where his care in such matters pays off immensely for the reader. I agree with one of the earlier reveiwer that his use of side-comments, though uncommon in texts, is very helpful and enjoyable. Also, even though the book is not application oriented, Axler does give a lot of examples of abstract defintions, which for someone learning linear algebra, is essential to have. He ties new abstract notions, like say linear maps, to things that an undergrad with modest math background would understand (like derivatives, etc). These examples I think are also crucial to a good abstract math book. Too often, an abstract math book will go from theorem to theorem. In this book, I felt like I was pacing myself. There were a lot of theorems that followed sequentially, but there were also "breather sections", where Axler will stop and take a look at what he is doing. This, I think, gives the student time to stop, reflect on what he is doing, and get a better, deeper, fuller understanding of the material. If you can purchase one book on linear algebra, this is the one book I would suggest!

This is a short, elegant presentation of linear algebra appropriate for upper level undergraduate math majors with a theoretical bent. The student has perhaps taken a linear algebra course designed for engineers and scientists. Such a student is comfortable reading mathematics and writing proofs. It is meant to be read and re-read until the ideas are absorbed. The exercises are relatively easy and no answers are provided. With exercises of this sort you generally know if you are on the right track and they require you to understand the presentation in the text and process the ideas in a straight forward way.Of course, there is nothing in this book about applications or the computational aspects of applying linear algebra. The price is right. This could be a very useful purchase even if it's not assigned as a text.

I was very much the typical person in the target audience of this. I was a computer science major and I had a semester of linear algebra where all I learned how to solve Ax = b. Then, I happened to pick up Axler one winter evening because the title looked intriguiging. That day changed my life.Now, I'm a pure math major and Axler is the reason. The exposition is clean and very elegant. By minimizing the use of matrices in his proofs, he presents the subject of linear algebra as an elegant piece of mathematics rather than a subject "spoilt" by applications. He starts with a study of vector spaces and then moves onto transformations, eigenvalues, inner product spaces, etc. all the way upto the jordan form. All along, the use of matrices in minimal. In fact, he introduces them quite late in the book just as a convenient notation and nothing else. This is an admirable aspect because it simplies a lot of the proofs. The proof that every linear operator over a finite-dimensional vector space has an eigenvalue is breathtakingly short and simple. He uses determinants in the last chapter of the book and there too, does an excellent job. (although the point of writing this book was NOT to use determinants, his exposition about determinants is itself one of the best ones I've seen).Get this book if you wish to understand the theory. It's a typical higher level math text - definition, theorem, proof, exercises (most of which are theorems). If you like math, you won't regret this.

I've seen many linear algebra books and this is by far the best treatment of them all. After going through this book one wonder why most linear algebra presentations don't follow Axler's sound and more reasonable approach. It leaves Hoffman & Kunze in the dust (although you may still want to hang on to Hoffman since it contains some material not found in Axler).Not only is Axler's approach sound, but his writing is very lucid and clear as well. You will never leave a proof feeling unsatisfied or confused; it almost reads like a book. I wish all math books were written this way. My only gripe with the book is the lack of solutions to the problems. Those who use the book for self-study will feel particular frustrated in this regard. I hope some effort is taken to assuage this problem in future editions. Also, more material on linear functionals and multilinear mappings (tensors) would be nice.In summary, this is an outstanding book; I highly recommend it.