Exercise 10.4.9

Suppose that $V$ and $W$ are finite-dimensional vector spaces with bases $B$ and $C$, respectively. Let $T:V\to W$ be a linear transformation such that

for $i=1,\ldots,n$. Let $M$ be the matrix whose columns are $\left[\mathbf{v}_{1}\right]_{B},\ldots,\left[\mathbf{v}_{n}\right]_{B}$, and let $N$ be the matrix whose columns are $\left[\mathbf{w}_{1}\right]_{C},\ldots,\left[\mathbf{w}_{n}\right]_{C}$. Suppose that $M$ is invertible. Show that $\left[T\right]_{C,B}=NM^{-1}$.