I'm looking to teach myself linear algebra this summer. I prefer a textbook that is more proof-oriented and theoretical in its approach as opposed to an applications text. What are your recommendations?
I like the text by Larry Smith just for the reasons you cited.
Amazon.com: Linear Algebra (Undergraduate Texts in Mathematics): Books: Larry Smith
An excellent book is what I think of it.
However, if you already know linear algebra and want to learn it on a graduate level (infinite Dimensional space , Hilbert spaces , ...) then that is not the book for you.
An Excellent Offer. MIT University released a course (full or almost full) on Linear Algebra and Differencial Equations FOR FREE. I can tell you how to view them.
I believe the textbooks
1) Linear Algebra with Applications, 7th Edition - Steven J. Leon
2) Linear Algebra and Its Applications, 3rd Edition - David C. Lay
may be useful for you. I used the textbooks to study Linear algebra before.
The bookstore there may be also good choice for buying textbooks. Wish some help.