Linear Algebra is quite self-contained. You'd need Calculus in there to really make sense of the probability and stats, though. Not sure about the vector calculus part and diffeq's, maybe that can be cut and the whole thing might fit in 4 years.
Edit: Reviewing Bretscher (Linear Algebra with Applications, 5th), it appears you are correct; it's not until you study the applications that you need Calculus. Most Unis I'm familiar with, however, will require multivariable (Calc 3) at least as a pre-req for Linear Algebra.