What mathematical knowledge do I need to begin studying linear algebra? In particular, how much calculus do I need to know?
Also, do you have a favorite linear algebra book you can recommend?
I have great news! You do not really need any calculus to begin studying linear algebra. You do need to understand functions and high-school level algebra to start learning linear algebra. As you progress higher through linear algebra, you could hit a level where dot products get replaced by generalized inner products, and you will deeply wish for the ease of only relying on real and complex spaces - but that's relatively advanced, and there is plenty of material that relies only on skills obtained in high school.
For references, check Where to start learning Linear Algebra? (math.stackexchange.com/questions/4335/where-to-start-learning-linear-algebra).
Calculus is not a prerequisite for Linear Algebra.
A very good book on the subject is "Finite Dimensional Vector Spaces" by Paul R. Halmos. However, it might be difficult to read depending on how much mathematics you have encountered so far.
If you are familiar with Algebra of Matrices, and a bit of set theory, then you can start learning Linear Algebra.
A complete set of reference books for Linear Algebra can found at this link. My favorite book on Linear Algebra is : "Linear Algebra by Friendberg, Insel and Spence."
I'm surprised nobody has metioned Axler's Linear Algebra Done Right. I think this is a perfect book for a beginning student of mathematics who is interested in learning to do proofs. Axler is very clear and careful in his explanations and although there are some sections toward the end where calculus would be useful, these can be skipped on a first reading and revisited once you have calculus under your belt.
Now, while Axler is very good and gives you a firm grounding in the basics, it is not comprehensive. If you really like linear algebra and want to dive deeper, you might want to explore Roman's Advanced Linear Algebra. It is quite a bit more advanced than Axler and presupposes much more mathematical maturity, although technically it is self-contained. I think once you digested everything that's in Axler though you would be in a position to start looking at it.
If you still haven't had enough linear algebra after all that, you could survey Greubs' Linear Algebra and Multilinear Algebra. There is a considerable overlap between Greub's Linear Algebra text and Roman's but in my opinion Greub takes a little more effort to read. In his multilinear algebra text, however, you will find topics that are rarely covered elsewhere. Unfortunately, it is out of print but Google here is your friend.
High school algebra, geometry, and a course on calculus should suffice. But as always, the more mathematics you know, the easier it will be to learn a new math subject. You can check out the free online book at:
there are others, but I used this one(it provides solutions to the exercises which is always helpful)
I quite liked Shilov's Linear Algebra, but it may be a bit dense. As others have said, calculus is not a prerequisite for linear algebra, but be prepared to refer to a math dictionary (or Wikipedia) on occasion if you go for Shilov. It can get a little dry.