Representation theory is about using linear algebra to understand and exploit symmetry to the fullest.
As such it plays a major role in many subjects in mathematics and physics. For example it provides a framework for understanding special functions and generalizes Fourier analysis to a non-commutative setting. Also modular forms in number theory are intimately related to representations of the Galois group. In physics one describes particles scattering into smaller elementary particles in terms of the corresponding representation decomposing into irreducible representations.
Concretely representation theory is about how a given subgroup of GL(V) decomposes vector space V into invariant subspaces. Even more concretely this often amounts to block-diagonalizing matrices.
Working with representations of algebras we are able to introduce the theory for both Lie algebras and finite groups. Along the way we will also meet tensor products and categories.
Written exam. Bi-weekly homework counts as bonus.
Linear algebra 1,2, Algebra 1.
Etingof, Introduction to Representation theory
Freely available on the author's website