The course is taught by Gilbert Strang, who has written an excellent introductory textbook on the subject:

http://www.amazon.com/Introduction-Linear-Algebra-Fourth-Gil...

Strang has another great book, Linear Algebra and its Applications: http://www.amazon.com/Linear-Algebra-Applications-Gilbert-St... Everything I know about linear algebra I learned from this book.

A slightly more advance books which I love is: Matrix Analysis and Applied Linear Algebra by Carl Meyer. http://www.amazon.com/gp/product/0898714540

You can find a free PDF version online.

I like it more then Strang because it's a lot more concise, covers some more advanced topics and unlike Strang everything is said very accurately. I think Strang's rather hand-wavy way of explaining things starts faltering when he talks about more advance topics.

I would read Strang and listen to his lectures to get a good feel for Linear Algebra (to build up the intuition), and if you feel like you want more then pick up Meyer's book

Seconded. I'm an applied math PhD student (optimization) and I use linear algebra constantly. This book taught me everything.

Thirded. We used an earlier edition at my school for Linear Algebra, and in spite of having a Professor who was retiring at the end of the class (complete with "I'm getting too old for this shit" demeanor), the book was approachable enough for us to get by.