Strang's hand-wavy teaching style in my opinion kinda falls apart towards the later part of the course. I feel like the whole second half is kinda half baked. I didn't come away really having a good intuitive understanding of what the SVD was or why I should care about eigenvalues. I definitely think it's a good place to start though, but if you want something a little more organized I'd really recommend.

"Matrix Analysis and Applied Linear Algebra"

http://www.amazon.com/Matrix-Analysis-Applied-Algebra-Soluti...

It's very clearly written and all the proofs are not too long nor too short. It pretty quickly goes through all the stuff Strang covers and moves on to more difficult things.

Thank you for that recommendation; I had not heard about the book before.

An extra nice thing: the book's chapters are available for free --- for downloading and viewing, and not for other uses --- from the book's website: http://matrixanalysis.com

Strang is not adequate for someone looking for a rigorous presentation of linear algebra. Axler's Linear Algebra Done Right, and Halmos's (older) Finite-Dimensional Vector Spaces are superior in rigor.