[jperras]

Axler's book, "Linear Algebra Done Right", is widely considered to be one of the best texts on linear algebra that focuses less on the mechanics and more on the structure of linear operators on vector spaces.

http://linear.axler.net/

[CalChris]

Yes, and the abridged version (no proofs, examples, and exercises) is downloadable for free. He avoids determinants altogether (well, until the 10th chapter).

http://linear.axler.net/LinearAbridged.html