Linear algebra for quantum physics

Quantum mechanics "lives" in a Hilbert space, and Hilbert space is "just" an infinite-dimensional vector space, so that the vectors are actually functions. Then the mathematics of quantum mechanics is pretty much "just" linear operators in the Hilbert space.

Quantum mechanics    Linear algebra
-----------------    --------------
wave function        vector
linear operator      matrix
eigenstates          eigenvectors
physical system      Hilbert space
physical observable  Hermitian matrix

Well, learn linear algebra. An advanced text (on linear algebra over "field" number systems) is these lecture notes [html] from UC Davis.

Once you get that done, you should study differential equations. Or if you want to skip ahead, perhaps Fourier analysis. A free reference would be my notes [pdf]. It's mildly physics-oriented, but connects the ideas back to linear algebra.

Quantum mechanics, when you boil it down, is Fourier analysis. (Instead of the "frequency domain" you have "momentum space", etc.)

Well, if you want to gain any quantitative insights into QM, you'd have to pick up some calculus as well - mainly differential equations, and if you really insist, Fourier analysis too. I was taught decent basic calculus in high school, so you may already know some of the basics.