Remark 1

Suppose that M is finite-dimensional over K. Let $(e_1, . . . , e_n)$ be a basis of V, and let $(e^*_1, . . . , e^*_n)$ be its dual basis. Denote by $(\pi_{ij}(a))$ the matrix of $\pi (a)$ with respect to the basis $(e_1, . . . , e_n)$ of V; we have $\pi_{ij}=$ $c_{\pi}(e_j, e^*_i)$. The mapping $a\mapsto (\pi_{ij}(a))$ is a homomorphism from A to $\mathbf{M}_n(K)$; such a homomorphism is sometimes called a matrix representation of the algebra A.