Personally, I prefer: Linear Algebra (2nd Edition): Kenneth M Hoffman, Ray Kunze
Although I must warn you, it is an older text, and the typography is horrid (it was written before Latex made math type-setting more sophisticated).
Another more recent text that has many followers is:
Linear Algebra Done Right (Undergraduate Texts in Mathematics): Sheldon Axler
which is somewhat unusual in that it delays mention of determinants until near the very end, and thus considerably alters the way it handles the Cayley-Hamilton theorem and eigenvalues, for example.
Both of these texts are, in my opinion, unsuitable for a FIRST text in Linear Algebra, but rather form a more cohesive and abstract view of Linear Algebra from a more unified standpoint. They are much easier to digest if one has a fair amount of experience working with vectors and matrices in low-dimensional spaces.