Exercise 6.2.2

Assume that $T:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$ is a linear transformation, and that $\mathbf{u}_{1},\ldots,\mathbf{u}_{n}$ is a basis of $\mathbb{R}^{n}$. For all $i=1,\ldots,n$, let $\mathbf{v}_{i}=T(\mathbf{u}_{i})$. Let $A$ be the matrix that has $\mathbf{u}_{1},\ldots,\mathbf{u}_{n}$ as its columns, and let $B$ be the matrix that has $\mathbf{v}_{1},\ldots,\mathbf{v}_{n}$ as its columns. Show that $A$ is invertible and the matrix of $T$ is $BA^{-1}$.