Linear Algebra Done Right

Rating:
Summary: One of the best books on the subject.
Review: An well written book with an elegant approach to linear algebra, by a famous author.
The book has good exercises, thou not very hard. I only wish Prof Axler had talked a little bit more about applied linear algebra.

Rating:
Summary: good book
Review: I had this for an intermediate course on linear algebra (after the 1st course on abstract algebra) & I thought it was good for the level I was at. There's just enough stuff in this book to fill up a 1-term course, but no more so you'll have to get another book to find more applications, or other stuff to look at. The texts by Hoffman/Kunze or Finite-Dimensional Vector Spaces by Halmos are good references. As for this Axler book, I like how it's written in a relatively informal style, including comments in the margins by the author. I also like how he emphasizes the concepts of vector spaces, inner product spaces, etc rather than matrices (although they do appear but they're not emphasized) while determinants are done last. This is the only book I know of that does it this way & I think I liked it better like that.

Rating:
Summary: Simply Amazing
Review: 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.

Rating:
Summary: Simply Amazing
Review: I've only loooked through this book a bit, but I found the proofs to be very enlightening. It presents the ``correct'' view of linear algebra a the study of vectors spaces, not the study of R^n, and n x m matrices. The book introduces matrices towards the end for a very good reason: matrices aren't that important. The real substance of linear algebra: linear operators and vector spaces. Introducing linear operators as matrices would be like defining a homomorphism on a group by giving what the homomorphism does to the presentation for the group. An idiotic and counterintuitive method of defining homomorphisms. Yet in combinatorial group theory, it is helpful sometimes to do this. Much as it is sometimes helpful to work with matrices--but certainly not from the start.

Rating:
Summary: Superb. The best book on the subjet.
Review: 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.

Rating:
Summary: A Nice Approach for Die-hard Math fans
Review: Sheldon Axler's "Linear Algebra Done Right" is an excellent book for the strong of heart. I am an undergraduate student majoring in mathematics, and my professors are obsessed with this book. I, however, am not.

First of all, there are no solutions to the exercises at the end of each chapter, so students are left frustrated when they cannot arrive at the next step of a proof.

Second, what the author sees and deems "obvious" as far as steps and recollections are concerned is not necessarily obvious to the reader.

Axler tries to motivate readers for the proofs by offering little exercises for them to "verify." That's overkill, but to many professors and analysts, overkill in the abstract is probably necessary in order to ensure a given student's success in an advanced linear algebra course.

I'm taking one such course to fulfill my requirements for a math major, and must say these abstract/proof courses get very monotonous and, thus, ridiculously boring. The text, itself, for this book does not particularly motivate me, and I expect to consult the chapters to learn and understand concepts, not to verify info from the chaper. Essentially, the flow of the concepts is ruined by the lack of examples; how are we supposed to verify ideas when the author hasn't really even exemplified the components of them yet?

I find myself falling asleep before I even complete one or two pages in this book. The layout is dull and the propositions and theorems seem endless.

My point is the following: That which is good for the instructor is not necessarily good for the student. Students need motivation, and it is difficult to achieve this goal without offering students detailed examples, interest-catchers and solutions to the time-consuming and overwhelming exercises and concepts.

In the realm of the college curriculum, this book is average. I understand the difficulty of making abstract algebra interesting, but this notion is precisely what students need. To the student math geniuses and professors that live and breathe math, this book is a gift from the gods.

To your average math student, however, that lacks patience and the desire to give up their free time to submerge himself/herself into this book, this book, like mine, will just end up sitting on the shelf and collecting dust.

Rating:
Summary: Elegant theoretical presentation of linear algebra
Review: 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.

Rating:
Summary: Elegant theoretical presentation of linear algebra
Review: 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.