Linear Algebra Done Right (Undergraduate Texts in Mathematics)

What an awful title. I guess Sheldon Axler decided on the ungrammatical "Linear Algebra Done Right" to avoid "Linear Algebra Done Properly" or something similar, which would have sounded intolerably arrogant. He justifies this title repeatedly by rather obnoxiously flaunting his determinant-free proof that operators on complex (or real, odd-dimensional) vector spaces have an eigenvalue. (It's pretty cool, but I've seen cooler constructions, and I'm not even a mathematician.) He also often makes other snarky jabs at the unnamed body of traditional linear algebra texts. Read the book, and you will forgive him on all counts. Other reviewers have already been thorough in their praise/criticism of Axler's elegant exposition that deprecates matrices and determinants. The highlight in my view is how Axler cleans up proofs by simplifying notation and carefully abstracting common algorithms into lemmas (like 2.4, his Linear Dependence Lemma) that are used over and over. This greatly improves readability and promotes the development of intuition. Some of his nonstandard choices of notation are used to such great pedagogical effect that they seem to threaten to redefine what is standard. The prose is correspondingly clear, concise, and full of useful motivation for difficult points. The formatting is impeccable - definitions, equations, inequalities, and theorems/lemmas are all given a uniform numbering system, making them easy and unambiguous to cite. Subsidiary comments are relegated to the margins of the book, keeping the main line of exposition free of digressions. The text is quite shockingly free of errors. Finally, the layout has a clean but cheerful flower-power look that reminds the reader that math is about beauty and fun - not just intimidating formalism. Axler's refusal to refer directly to others for inspiration (he seemingly proudly omits a bibliography) does cause some warts. For instance, when looking at orthogonal projections for optimization, he asks the reader to do inner-product gymnastics in polynomial space on [0,1] instead of on [-1,1]. The latter choice gives rise to the all-important Legendre polynomials, whose symmetry properties are much clearer. Also, while the pristine algebraic presentation was remarkable, I'd have liked to see more geometric insight in places. I got into this book because my undergraduate linear algebra experience, with Apostol Vol. 2, was so frustrating - all of the sweeping and magical structure theorems of self-adjoint operators and so forth seemed to reduce to incomprehensible index-pushing. For me, what finally cleared up these notions to me was drawing, on graph paper, the fate of vectors in R^2 under various linear operators. This was not in the book, but Axler's inclusion of the theory of polar and singular-value decompositions did give some important tools to help unravel these beautiful but elusive issues. Finally, the crystal clarity of the exposition rolls off in Chapters 8 and 9 when getting into the structure theory of general operators on real and complex vector spaces. The symbols get more abstruse, and the arguments get more murky. But I've never seen another author make anything but a mess of, say, the proof of Jordan form. It is hard stuff, and it is not fair to be too hard on authors for failing to make it look easy. The end-of-chapter problems are abundant enough to give a good feel for the material, with an appropriate range of difficulties for an advanced undergraduate book. There are enough of the routine computations and simple proofs that familiarize readers with the new machinery they are learning, but at least a proof or two in each chapter require creative constructions to complete. I just finished the last of the 224 problems, a task that took me five years' worth of sporadic effort in my free time and vacations as a high school math teacher and then as a graduate student in chemistry. A few problems took me the better part of a year to figure out, though this was without the benefit of collaboration. I found the equivalent of at one sequence of problems (problems 6-8 in Chapter 6) as a starred problem in a graduate functional analysis text. I consider myself a good but not award-winning math student, so this indicates that the problems are consistently tractable but can get pretty tough in places. Axler does not mark his most difficult problems as such; for the teacher assigning Axler's problems for a course, then, it is imperative to work through the problems beforehand. All told, this is quite a remarkable book. I now feel like I understand linear algebra, something I couldn't say when I first studied the subject eight years ago. The title does not do it justice.

OK, I admit that I have never much liked matrices. Standard textbooks for undergraduates typically seem to devote 200-250 pages to telling one in an informal and concrete way what a vector is, what a matrix is, what a determinant is and how to do all sorts of highly boring / tedious matrix operations, etc., i.e., there is no real algebra in these so-called linear algebra books until one is pretty much bored to tears. And even when they do introduce some algebra, they deal almost exclusively with the reals (with a tiny nod here and there to complex numbers) and the presentation does not provide a unified coherent picture nor prepare one for more advanced mathematics. So if one wants to learn about the algebra in linear algebra, one needs to look elsewhere and Linear Algebra Done Right not only does it right but also makes it fun. The exposition is generally very clear (this does not mean one does not need to work a bit to understand the material!) and since it sticks to essentials, it moves along at a nice clip. What a breath of fresh air! Despite my lavish praise, I would not recommend this book as one's first or only book on linear algebra. But I recommend this book very highly to anyone who wants to appreciate the beauty of (actual) linear algebra or needs a stepping stone to more advanced books such as Roman's Advanced Linear Algebra (Graduate Texts in Mathematics) or books on linear functional analysis and Hilbert spaces such as Linear Functional Analysis (Springer Undergraduate Mathematics Series) , Kreyszig Introductory Functional Analysis with Applications or Introduction to Spectral Theory in Hilbert Space (Dover Books on Mathematics) .

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!

First things first: this is intended for a second course in the subject. This could not be made more clear, being written on the back cover and the foreword. Still, about half of the one and two star reviewers somehow missed this and (surprise) didn't enjoy the book. Anyway, this is one of several sources from which I learned Linear Algebra, the others being Lay and Leon for my first exposure, and Shilov as a companion to this book for my second exposure. Top that off with Professor Strang's recorded lectures and I really should know much more than I actually do! Although the word is used a lot, this book really is elegant. The first three or four chapters cover the basics and are done well, but I think the book really takes off at the start of chapter 5 (I know a lot about determinants from Shilov, so I haven't looked much at chapter 10). Sure, there isn't an abundance of worked out examples, but there are plenty of sources for that. If you want a clean, efficient, and helpful presentation of the theory of Linear Algebra, this is your book. I also appreciate the fact that Axler gives the reader a taste of the main idea and the motivation for a new topic in a few simple sentences. This, along with his ability to make the subject seem deceptively simple, with straightforward proofs that don't rely on any "tricks", led to a number of aha! moments while reading. I don't have any complaints, but there is a lot of material in Shilov that isn't in this book (Shilov's is about twice as long), so it may be possible for a good student to go right to that after a first course. It is harder, provides less motivation, and relies heavily on determinants throughout, especially early on, so beware of that. Skipping between Shilov and Axler simultaneously worked well for me.

This is an excellent book but it's not for everyone. Math majors in particular will benefit from this book the most, while those who simply need a quick, intuitive introduction to the principle theorems and matrix operations of linear algebra would be served best elsewhere. Unfortunately you're probably not going to realize which group you belong to until either 1) you take your first upper level math class and realize that your understanding of finite-dimensional vector spaces provided to you by your standard undergraduate LA class is grossly inadequate for dealing with subjects like real and numerical analysis or 2) you buy this book in order to help you learn basic LA calculations that depend on algorithmic matrix manipulation and determinants and are incredibly frustrated and disappointed with it. If you're an undergraduate who planning on taking any higher level math courses in subjects like numerical analysis, mathematical statistics, or abstract algebra, do yourself a favor and get this book! It will totally help you to understand the underlying principles of concepts like continuous functions on multidimensional spaces, hypothesis testing, and abstract fields. This book will also be extremely helpful if you are going into any variation of computational mathematics where you have to use programming languages like MatLab or R that already assume you have a good working knowledge of finite-dimensional vector spaces (not to be mistaken with standard matrix operations). On the other hand, if you are just having trouble following along in your standard undergraduate LA class or are looking for a good study aid or textbook supplement I would recommend something more along the lines of Linear Algebra by Strange. A quick way to know if this is you, just ask yourself, "Do the problems I'm trying to solve require the use of determinants?" If the answer is yes, then this is probably not the book for you.

I read this book a few years ago and it's one of the most enlightening books I've read. What makes it unique is it's way of very simply communicating the basic concepts of vector spaces and linear maps between them. It's one of the few books where you usually immediately realize how to prove each lemma, proposition and theorem before even looking at the proof. Everything just comes out "naturally". Some people here seem to complain about the lack of breadth in the exposition. However, I see this book as a way of teaching the reader to understand the concepts and proof techniques of abstract linear algebra (of course you won't see how to use determinants in proofs) and it prepares a student very well for a course in functional analysis. Probably a natural continuation would be to read a book focusing on modules. The new edition of Roman's Advanced Linear Algebra seems very good (compared to the 1st edition which was bad) and is probably a good choice. Those who want formulas and applications should probably take a look at a book titled "engineering mathematics" or maybe Strang's Linear Algebra which is quite practically oriented (and has corresponding video lectures at MIT OCW). It's true that Axler lacks solutions to exercises, but most are very straightforward. When I read the book I typed up solutions to every exercise in the book. These seem to have spread online last spring through some of the people who I gave them to for self-studying, so I've put my solutions out online. They should be found by googling "solutions axler linear algebra done right" or "fagerholm site:tkk.fi".

I strongly suggest avoiding Springer's current paperback print. It is a print-on-demand text with the print quality set to "might as well be a jpeg". I'll be returning this and looking for a hardback copy. The content of this book appears very good. I have skimmed it and here is my impression: It takes a biased approach to linear algebra, which is not a bad thing - the author's motivation to present linear algebra the way he would like to have seen it leads to a very cohesive work. The exercises force true understanding, unlike the rote understanding stressed by most introductory courses on the subject, and that is why it is regarded as a "second course" text in linear algebra. The content is quite approachable to any student versed in proofs. Likely, even, to those not: the motivation in the text and basic reasoning of the arguments are very clear and logical, and creates an understanding of the "bigger picture". As other reviewers have mentioned, the print quality in the paperback is terrible. To stress just how bad this is: letters in ordinary roman text are wrinkled around the edges, and sometimes the ink will cut off partway through a letter. The italic text reminds me of a compressed bitmap image, and actually looks *pixelated*. I do not want to even get into the math text. Subscripts and superscripts, where they appear, are extremely light and cut off regularly, and parenthesis and other brackets are wobbly and cut off as well. This harms legibility in reading the formulas. The text hurts my eyes, and I cannot imagine actually reading the book. The sad part is that this book is beautifully typeset, and a lot of work went into making it readable. It has margin notes providing context, clear and skimmable visual indicators of "proof", "theorem", "definition", running heads for sections. The benefits of all of this are seriously outweighed by the horrible print.

I had my first exposure to Linear Algebra last semester, and Axler's text was the one we used. In my opinion, it was a terrible read for a first time lin-al student. Axler seems arrogant, and half proves many things, imploring the reader with "...as you should prove", just as the clouds are STARTING to clear. There are a lot of build ups and "I ALMOST 'get this' " involved with this text. I'm speaking strictly as a beginner/novice in linear algebra. I went into the class hungry for the knowledge, feeling linear algebra is one topic that has many practical applications; I left feeling as though the rumors of a kiss on the first date weren't true- Axler left me high and dry. Great lead off, crummy follow through. I see merit in this text for a 2nd course in Lin Al, or graduate work. (yes, the text certainly claims TO BE for a "second course in Lin Al", i know - but my prof evidently felt otherwise) With a strong background understanding of lin-al, this text could potentially reward the reader with new ideas and inspiration; but if independent study and mere curiosity of the world of linear algebra is your motivation- you'd be better served to spend your money on that new video game you've been eying. In a nutshell, unless you already know a fair bit about linear algebra, you'll be back on amazon with a scowl and bad attitude refining your search for a linear algebra intro text.

This text is easy to understand, formulated quite generally (for example the author uses vector spaces defined over arbitrary fields, not just R or C), and contains lots of examples (possibly too many examples). I don't think I can overstate how good this textbook is. In conjunction with a good text on the computational/algorithmic aspects of matrices you would have a complete undergraduate-level education in linear algebra. I know its a stupidly horrid title, but it is a really good book. The only con would be its shortness; I think there is a lot more room for other topics.

A masterpiece with great mathematical beauty. It should be the required reading for every student in math, science, and engineering. It is not only useful as linear algebra is central to applied math and engineering, but also cultivates mathematical thinking ability of the readers. I truly appreciate the effort the author put into this wonderful textbook.

My favourite linear algebra text. I read this book to brush up on my linear algebra, and it gave a fresh new perspective to a lot of concepts that I never fully grasped when reading other texts.

Bought as a present for my husband. He loved it.

gift