This is basically the syllabus for a math course I might take next year, I've already completed the required course in linear algebra for the physics program at my school. Would the following knowledge be helpful in any way? Or should I just take a random elective (e.g. "Greek and Roman Epic"). Abstract vector spaces: subspaces, dimension theory. Linear mappings: kernel, image, dimension theorem, isomorphisms, matrix of linear transformation. Changes of basis, invariant spaces, direct sums, cyclic subspaces, Cayley-Hamilton theorem. Inner product spaces, orthogonal transformations, orthogonal diagonalization, quadratic forms, positive definite matrices. Complex operators: Hermitian, unitary and normal. Spectral theorem. Isometries of R2 and R3.