Exercise 2.4.21

First we prove the independence of $\{T_{\mathit{ij}}\}$. Suppose that ${\sum <!--\; FUNCTION\; APPLICATION\; -->}_{i,j}{a}_{\mathit{ij}}{T}_{\mathit{ij}}=0$. We have that

This means $a_{\mathit{ik}}=0$ for all proper $i$ since $\{w_i\}$ is a basis. And since $k$ is arbitrary we have that $a_{\mathit{ik}}=0$ for all $i$ and $k$.

Second we prove that ${[T_{\mathit{ij}}]}_{\beta }^{\gamma }=M^{\mathit{ij}}$. But this is the instant result of

and

for $k\ne j$. Finally we can observe that $\Phi (\beta )=\gamma$ is a basis for $M_{m\times n}(𝔽)$ and so $\Phi$ is a isomorphism by Exercise 2.4.15.