Representing algebraic objects in a concrete way (say as matrices or permutations) is one of the ways in which abstract algebra attacks concrete problems. Here we are concerned with representations as matrices (equivalently, linear operators on finite dimensional vector spaces). Most members in the algebra group have studied such represenations in some cases (rings, Lie algebras, etc.).
A particular form of representation theory is the study of finite-dimensional algebras (sometimes generalized to artinian rings). For a brief account of this theory, select one of the following: