When i started i found two books to be very useful; i used them simultaneously:
Linear Algebra: Introduction to Abstract Mathematics
and
http://mathematicsbooks.org/0130198579.html
The first is a bit abstract but does a good job of opening you to abstract arguments and generalizations.
The second was the book from which i learned matrix operatios, the eigenvalue problem and other things that could be computed.
Start with it and after a month or so the open the Valenza book.
Once you know the basic theorems (of which there are few) the Valenza book can expose you to learning about the spectral thm, triangulation and inner products and the dual. These topics rarely get touched in the first (U.S.) linear algebra course and the they often get omitted from the second course wherein the student does proofs of the stuff in the first course. (ofte syllabi claim to plan to get to these topics but time invariably runs out )But all these topics are pretty important.
When you are ready to move on and need to learn cayley-hamilton thm and minimal polyomials and invariant subspaces and about quad forms and
similarity and Jordan form, nilpotet operators...etc..
then get:
http://vig.prenhall.com/catalog/academic/product/0,1144,0135367972.html,00.html
Hoffman, Kenneth and Kunze, Ray. Linear Algebra
this book is excellent in exposition and its pretty comprehesive.
ow everybody always says you can't learn math without doing lots of problems. I ever found this more true than with linear algebra.
You can learn a thm but until you see and perform a calculation that illustrates it or uses it you never really learn its meaning. Linear algebra is a great subject to use for learing more serious mathematics and it has endless applications (A very well known mathematician once told me "my biggest problem is that i do'nt know enough linear algebra...you can ever know enough" this is a guy who many people would know...at first i laughed and then i realized how serious he was. now, he knows way more than is covered in these books but the point is to keep on with this subject if you are planing a career in science.)
But if you want to do that you will have to do much more than is expected of you in the standard courses at most universities.