Customer Reviews

Linear Algebra Done Right (Undergraduate Texts in Mathematics)

5 star:		(35)
4 star:		(10)
3 star:		(9)
2 star:		(7)
1 star:		(4)

 

 

I have used this text for a beginning graduate course in linear algebra, mostly because I prefer its treatment of eigenvalues and eigenvectors over Hoffman and Kunze, and it sticks to the basics: complex scalars. It also has a good treatment of inner product spaces. The basic concepts and theorems are indeed presented cleanly and elegantly. Its use of linearly independent sequences (rather than sets) is a little nonstandard (what if the set of vectors is infinite?) but the adjustment is minor. Two things though I found treated in a less than desirable fashion: He pretends that we don't know about matrices, doesn't want to develop the machinery, and the treatment of coordinate vectors and matrix representations suffers. Students also get no sense of how to compute the solution of concrete vector space problems, which is easily done once the theory is established, and which is an essential skill to have after a second course in linear algebra. I have to give them supplementary notes. Second, the treatment of determinants suffers, apparently for ideological/political reasons. I think students deserve a straightforward development of determinants simply because that theory is widely used in applications, in engineering, and in discrete mathematics, and it has its own beauty. It is not hard to do, and I do it myself from notes, adapted from the treatment of Hoffman and Kunze. Now that undergraduate linear algebra courses have in many places dropped any substantial theorem-proving component, students need a serious course in linear algebra which can take them, e.g. all the way into Jordan form. There are not many good books for this, and this text does a good job with the basics without overkill on the abstraction, so I use it despite the drawbacks mentioned above.

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.

I have no doubt that this is one of the most thought provoking math books that I have come across. I used this book for a linear algebra course last fall '07 and I learned a ton. Specifically about the structure of vector spaces and linear operators. However, the most important function that this book serves is to move students towards the methodology of mathematics, which means proof construction and counter examples. It also trains students to let go of their intuitions. But you can not self-study this book, there are no answers and more importantly the structure of the course begs for instruction. I would recommend before taking this course doing what i didn't do and have had to do since, make sure you have your first course of linear algebra solidly under your belt, and that doesn't mean having gotten an A in the prior class is sufficient. Go through the most difficult proof driven exercises in your first text, that should serve as practice for easiest homework problems in this book.

All that said, there are serious limitations to this book. It would be nice if the author worked out 1 comprehensive semi-difficult exercise in each chapter of the text. While struggling to solve the problems can be enlightening, there is only so many times I can read the same sections over and over again, looking for some insight from the kiddie exercises provided by the author. It would also help if some of the kiddie exercises were accompanied with graphs, especially when describing the sums of vector spaces. Sometimes a picture is worth a thousand words - sometimes!

Last but not least, the author has a copyright on the solutions to the book. Where he does not allow professors to post homework solutions to exercises. This had a devastating effect on the class I was in, because there were many students who were lost in the first couple of homework sets and basically were never given a chance to figure out what was going on. Pedagogically, this is unacceptable. Furthermore it sets a dangerous trend, math problems simply stated should not be copyrighted. For this reason I suggest that people not purchase this book, but I still strongly recommend that they get a hold of it.

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.

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).

Did I just say that a book that has no examples of how to use a concept in applied situations is great for applied mathematics?

Yes. And I mean it.

I'm currently taking a graduate course in numerical analysis, which is about as applied as you can get. Here's how I read my textbook: I read through a section on say, the QR-algorithm for finding eigenvalues. After a long, brutal period of intense focus, I go "what did that just say?", and then I read it again. I think, "what in the hell's an invariant subspace? What's this weird projection operator doing, what's this . . ."

And then I remember how to make the pain stop: Pick up Sheldon Axler's book, and read the appropriate section in his book. Then go back to numerical analysis, and breeze right through the exposition of the algorithm.

Dr. Axler doesn't claim that his book is for applied mathematicians or engineers. He wants you to understand the structure of linear operators. But guess what? Applied mathematics exploits many-no, all-of the deep concepts relating to the structure of linear operators. So if you want to be a good applied mathematician, someday you're going to have learn linear operator theory. If Maple does all of the math you need for your job, then you won't need to learn linear operator theory. And so don't use this book. But realize that someday, you will hit a wall.

Now onto some specifics of the book:

Some have complained about the lack of use of determinants in this book. Think about a mathematical concept that you use over and over again, that you can use as an analogy to other things and helps you go about your day. For me, I understand these concepts down to the very core. Things I don't understand down to the core, I can't extend to novel situations. And if we can't extend a concept to a novel situation, why learn it? Certainly we can tell Maple to compute it if we really need. For me, the determinant is a conceptual dead end. Not once in my life has anyone been able to explain why det(AB) = det(A)det(B). And I doubt that the person who does know finds that the proof very enlightening. Similarly, why should the determinant have anything to do with how a matrix scales it's eigenvectors? No one knows! Ok, you say, but we do need to know if a matrix is invertible, and so we should teach people that the determinant can tell them that. Fair enough. I'll give your students a 100x100 matrix, and then I'll come back next year to find out whether or not it's singular. They'll say, well we got 10^-12. And I'll say, "does that mean all that matrices eigenvalues are 3/4, or one of them is zero and you just got round-off error?" They'll have a blank stare, and then I'll estimate the matrix's condition number in a few seconds and tell them the answer. So for engineers and applied mathematics, the determinant is truly useless. Dr. Axler is correct to scrap it.


Next:

Some have complained about the lack of solutions to the problems. This is false. There are many solved problems in this book. They just have the solution immediately after the statement of the problem. I find it baffling that if the author had just moved the proofs to the back of the book, and labeled the theorems "Problem", then this complaint would disappear. This is a math book, not a Charles Dickens novel. You read a Charles Dickens novel, you battle a math book. When you read the word "Theorem" in a math book, you try to prove it yourself. Then, after failing, you read the author's proof. I don't see how this is at all a controversial statement, but if people read math books like math books, then no one would complain about the lack of solved problems. The fact that Dr. Axler has to copyright the solutions (provided this is true, I don't know) is a sad testament to the fact that people don't want to battle math, they want to regurgitate it. They would rather sleuth the internet for 12 hours trying to find posted solutions rather than spend one hour thinking. I've done it myself. But once these solutions are available, and people know that sleuthing will prevent thinking, then your book has shot its wad.


This book has saved me countless hours. If I value my time at $10 per hour, then this book has already saved me well over $1000. The fact that it is only $30 is a steal.

This book is very well written and a pleasure to read. I used this book for my second linear algebra course. It was a wonderful account on finite-dimensional vectorspaces and finite-dimensional operators. Axler's approach of not using determinants is most efficient and very helpful to "see what's going on." This book isn't meant for those that want an applied course in linear algebra; only abstract material here! It's meant for junior/senior math majors who have some "math maturity." While Axler's very careful with the presentation, those with little experience with reading and writing proofs may find it challenging. If you're into math, pick it up and have fun!!!

A brief book on linear algebra that develops the theory by emphasizing vector spaces and linear maps; this leads to clearer, more elegant proofs than the traditional, matrix-based approach. This approach manages to be both more lucid and more abstract.

Among the many fine features of this book are the author's marginal notes highlighting important points, commenting on strategy, and mentioning other names that a concept may go by (e.g., an injective mapping is also known as one-to-one, this is quite useful for beginning students).