Amazon.com: Used and New: Linear Algebra and Its Applications (3rd Edition)
This is the one I learned Linear Algebra from in undergrad. It was one of the first proof classes we had, so I don't think it will be too bad if you are not totally familiar with rigorous mathematics.
If you are looking for more advanced properties of linear algebra, Dummit and Foote's text on abstract algebra has a lot of interesting more advanced stuff assuming knowledge of the basics. Modules over PIDs, Jordan Canonical Form, RCF, tensors, Dual space, all that sort of stuff.
I think the first one would be more helpful for you as a chemist, but just my opinion.