A textbook at a somewhat lower level than the one by Strang that we are using is
David Lay, Linear Algebra and its ApplicationsThe computational approach to the subject is much the same, and there are lots of applications.
There are many, many traditional theorem-proof type books. Two examples are
Noble and Daniel, Applied Linear Algebra
Otto Bretscher, Linear Algebra with Applications
Two other interesting references are
Sheldon Axler, Linear Algebra done right
Peter Lax Linear Algebra and its Applications
A classic of numerical linear algebra is
Golub and Van Loan,Matrix Computations,
In addition, there is an interesting review of several linear algebra texts in the March 2005 issue of The American Mathematical Monthly