I've just been looking on Amazon to see how some of my favorite old math texts are doing. I used this one about twenty years ago as a supplementary reference in a graduate course, and I still have my copy.
Everybody with some mathematical background knows the name of Paul Richard Halmos. I saw him speak at Kent State University while I was an undergraduate there (some twenty-odd years ago); to this day I remember the sheer elegance of his presentation and even recall some of the specific points on which, like a magician, he drew gasps and applause from his audience of mathematicians and math students.
This book displays the same elegance. If you're looking for a book that provides an exposition of linear algebra the way mathematicians think of it, this is it.
This very fact will probably be a stumbling block for some readers. The difficulty is that, in order to appreciate what Halmos is up to here, you have to have _enough_ practice in mathematical thinking to grasp that linear algebra isn't the same thing as matrix algebra.
In your introductory linear algebra course, linear transformations were probably simply identified with matrices. But really (i.e., mathematically), a linear transformation is a special sort of mathematical object, one that can be _represented_ by a matrix (actually by a lot of different matrices) once a coordinate system has been introduced, but one that 'lives' in the spaces with which abstract algebra deals, independently of any choice of coordinates.
In short, don't expect numbers and calculations here. This book is about abstract algebraic structure, not about matrix computations.
If that's not what you're looking for, you'll probably be disappointed in this book. If that _is_ what you want, you may still find this book hard going, but the rewards will be worth the effort.
