Unfortunately, I had to give up on the course last year because my math background is even more limited than yours (you'll need to know how to construct proofs). So time for me to learn calc, I guess. :)
I don't believe Calculus is a prerequisite to Linear Algebra. Constructing proofs is something that's used in all branches of math as far as I know. If proofs are the only thing stopping you then I'd recommend just trying your hand at a few to get the hang of them. I can't imagine they'd be taught in a Calculus course either.
Oh, I realize Calculus isn't a necessary prereq to linear algebra. But my impression is that people normally learn proofs alongside Calc than learn proofs alongside Linear Algebra. To be honest, I was writing more out of a desire to be humorous than accurate; sorry!
Anyways, I've picked up "The Haskell Road to Logic, Maths, and Programming," so hopefully I can learn proofs alongside programming and logic, which is probably the best route for me at this time.
Unfortunately, I had to give up on the course last year because my math background is even more limited than yours (you'll need to know how to construct proofs). So time for me to learn calc, I guess. :)