Edit: The order I presented the books is not a ranking simply the chronological order I went through them and this happened to be the order I wrote them in the review. I would recommend H&K.
I'll try to answer two questions in my reviews below: first, the quality of exposition and shortcomings, second the exercises. (full disclosure: This is my first time reviewing a maths book so apologies for any mediocrity and I also did not purchase them or borrow them from the library - I'll leave you to make your deductions).
1 LADR by Axler (recently brought out a 3rd edition but I haven't read this).
i) The pedagogical approach and concept of this book sounds good but in practice it was handled terribly. First, he treats polynomials as functions, rather than formally, and though this can be altered if you know what you're doing, it is hardly appropriate for a beginner. Secondly, he delays matrices far too long - I think after chapter 3, when linear maps were done, the concept of a matrix and determinant could have been beautifully integrated and streamlined into the exposition - needless to say this opportunity was wasted. Everything falls apart, material wise, after chapter 6, though I still got some benefit out of it.
ii) The problems are exceptionally easy. I mean this in reference to knowing the material of the book - as long as you can remember the material in the chapters, you will have no trouble doing them - I didn't spend more than a few seconds on each exercise before seeing what to do (except when coming up with counterexamples).
2 Hoffman & Kunze:
i) Matrices are introduces straight away and it covers more material than Axler. I like the practicality of the book, though its terseness is not for everyone. Addresses many shortfalls of Axler.
ii) Good problem set, some challenging problems, many good 'routine' questions to drill concepts.
3 Halmos: Linear Algebra Problem Book (with reference to FDVS)
i) You actually answer questions to make your own exposition, though Halmos is as perfect as usual. Needs a secondary text for a first time student (FDVS would be good) and the material follows his classic examples format.
ii) Excellent, perfect set of questions. He goes over many examples, you are asked to prove many theorems and search for the kind of things beginning students trip up over. For instance, in the first chapter - an introduction to groups and fields- you are 'hand held' through showing how things like associativity etc. are non-trivial, which group axioms are independent, why it is important for multiplication and addition to interact 'sensibly', and then you can move all of this theory into the second chapter on vector spaces, again beautifully motivated with examples and uses the preceding material. This format goes on, and the questions are both illuminating and yet also I would say completely approachable for a student. There are even hints with solutions the following section, so you can peak at a hint, then the solution if really stuck. However, the questions are mainly easy, though about ten in the whole book did stump me.
Companion to LAPB: FDVS by Halmos.