Physicists have always found it more convenient to deal with the matrix representations of a group rather than the abstract group itself. If the matrix representation of a group is reducible, then one or more invariant subspaces of it must exist with respect to the group. These invariant subspaces may be reducible or irreducible. The importance of group representations stems from the fact that various irreducible invariant subspaces of a group can be used to accommodate elementary particles with specific characteristics. Before we begin with the group representation theory, we summarize some of the material needed for it. In this context, linear vector spaces and linear operators have special significance.