Preparation

Please review the following material in preparation for the course. There is an exercise sheet included as the first part of the lecture notes that will not be discussed in the seminars, but that is meant to aid you with some of your preparations.

Linear algebra

Vector spaces, linear subspaces, quotient vector spaces, bases, linear maps and matrices. Examples of vector spaces: Euclidean space, vector spaces of polynomials, vector spaces of matrices, etc.

Group Theory

Groups, subgroups, normal subgroups, quotient groups, group homomorphisms, Cayley's theorem, Lagrange's theorem. Examples of groups: symmetric groups, dihedral groups, groups of matrices, the integers modulo n, etc.

Ring Theory

Rings, subrings, ideals, quotient rings, ring homomorphisms, fields, division rings. Examples of rings: polynomial rings, matrix rings, the integers modulo n, etc.