This is a linear algebra book written by a functional analyst, and the crux of the book is a treatment of the spectral theorem for self-adjoint operators in the finite-dimensional case. It's a beautiful, wonderful book, but not a very good reference for traditional linear algebra topics or applications. You also have to read a fair distance before you even see a linear map, and the exercises are mostly too easy, with a few too hard. But this book was where I first learned about tensor products, and why the matrix elements go the way they do and not the other way (Halmos is very careful on this point).
[Pete Clark] I own this book and read through it often, but it's never taught me linear algebra per se. Let's agree that it's too abstract for a reasonable first introduction to linear algebra; it's really meant for students who already know (some) linear algebra to read through and appreciate one particular, and particularly elegant, presentation of the material. If you want to know about the linear algebra which surrounds functional analysis, then by all means read this book, but much of the material is nonstandard and a bit curious from the perspective of mainstream linear algebra; projections seem to be the most important linear map, and there are many sections lovingly devoted to commuting projections, decomposing projections, etc. I still am not sure why Halmos deifies the [,] as much as he does, and quite honestly, I would learn multilinear algebra anywhere but here.
If you can stand terrible typesetting and an unexciting prose style, this tiny little book is a good rigorous reference for traditional linear algebra (i.e., it doesn't assume you're a tree). A nice bonus at the end is the Wedderburn theorem for division algebras over R, although the lack of sophistication makes for some unmotivated technical carpentry. I look in here whenever I can't remember what a positive-definite matrix is.
You may never need The Book on linear algebra. But one day, you may just have to know fifteen different ways to decompose a linear map into parts with different nice properties. On that day, your choices are Greub and Bourbaki. Greub is easier to carry. End of story.