Publisher Summary This chapter discusses commutator operators and focuses on the algebra of linear operators acting on a linear vector space. The particular form of the operators, or of the elements of the linear vector space, is irrelevant. The situation observed in this chapter leads to considering operators as the vectors of the space a set of matrices. The linear operators are the somewhat peculiar looking ones. Because these are linear operators acting on a linear vector space, the results can be applied to them as appropriate. Groups that may be either finite or infinite are described; however, this chapter is concerned only with infinite groups (that is, groups that contain an infinite number of distinct elements) with finite dimensionality. Lie groups and their infinitesimal transformations are discussed in detail.