This is the first half of a year long course in algebra, with emphasis on the ways in which algebra and geometry interact and complement each other. We will cover groups, subgroups, homomorphisms, equivalence relations and cosets, quotient groups, products of groups, generators and relations, calculations in the symmetric group, the isomorphism theorems, fields and vector spaces, basis and dimension, linear transformations and matrices, change of basis, direct sums, eigenvectors, characteristic and minimal polynomials, diagonalization, the Cayley-Hamilton theorem, Jordan canonical form, the fundamental theorem of algebra, matrix exponentials, solving linear ODE, representations of finite groups: invariant subspaces, complete reducibility.

A basic goal of the course is to use the abstract theory to develop an intuition in concrete examples and learn to understand and produce sound mathematical arguments.