Includes definition of a vector space, subspaces, spans, linear independence, and basis/dimension.

Includes maps, nullspace/kernel, injective/surjective/bijective spaces, isomorphisms, matrices, rank nullity, and product/quotient spaces.

Includes eigenvalues, eigenvectors, characteristic polynomials, invariant subspaces, upper triangular matrices, eigenspaces, and diagonalization.

Includes generalized Cauchy-Schwarz inequalities, orthonormal bases, Gram-Schmidt process, and orthonormal complements.

Includes adjoint operators, normal operators, the Spectral Theorem, positive operators, isometries, and singular value decomposition.

Relates trace and determinant to the previous, more abstract ideas, to tie together loose ends.