Let $A$ be the matrix representing a linear map $\Phi: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ with respect to the bases $\left\{\mathbf{b}_{1}, \ldots, \mathbf{b}_{n}\right\}$ of $\mathbb{R}^{n}$ and $\left\{\mathbf{c}_{1}, \ldots, \mathbf{c}_{m}\right\}$ of $\mathbb{R}^{m}$, so that $\Phi\left(\mathbf{b}_{i}\right)=A_{j i} \mathbf{c}_{j}$. Let $\left\{\mathbf{b}_{1}^{\prime}, \ldots, \mathbf{b}_{n}^{\prime}\right\}$ be another basis of $\mathbb{R}^{n}$ and let $\left\{\mathbf{c}_{1}^{\prime}, \ldots, \mathbf{c}_{m}^{\prime}\right\}$ be another basis of $\mathbb{R}^{m}$. Show that the matrix $A^{\prime}$ representing $\Phi$ with respect to these new bases satisfies $A^{\prime}=C^{-1} A B$ with matrices $B$ and $C$ which should be defined.