Chapter 4.2, Problem 17E

Let L be a linear operator on a vector space V. Let A be the matrix representing L with respect to an ordered basis $\{v_1,\mathrm{...},v_n\}$ of V [i.e., $L(v_j)={\displaystyle \underset{i=1}{\overset{n}{\sum }}{a}_{ij}{v}_i,j=1,\mathrm{...},n}$ ]. Show that $A^m$ is the matrix representing $L^m$ with respect to $\{v_1,\mathrm{...},v_n\}$ .