This is a course on advanced topics in linear algebra, with emphasis on the study of vector spaces from an abstract, coordinate-free point of view, and on the proving of generalizations of results covered in an elementary linear algebra course. Topics include: abstract vector spaces, linear transformations, matrix representation, nullity and rank, properties of determinants, diagonalization, Cayley-Hamilton theorem, inner product spaces, normal and self-adjoint operators, unitary and orthogonal operators, bilinear forms, dual spaces, canonical forms, spectral theorem.