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.