Exercise 3.C.5

Proof. I will use facts from section 3F, although this was not what Axler intended.

Consider the dual basis ${\psi }_1,\dots ,{\psi }_n$ of $w_1,\dots ,w_n$ and the dual map $T^{\prime }$ of $T$ (note that $T^{\prime }$ is map from $W^{\prime }$ to $V^{\prime }$). By the previous exercise, there exists a basis of ${\phi }_1,\dots ,{\phi }_m$ of $V^{\prime }$ such that all entries in the first column of $M(T^{\prime })$ (with respect to the bases we have here of $W^{\prime }$ and $V^{\prime }$) are $0$ except for possibly a $1$ in the first row, first column. By Exercise 31 in section 3F, there exists a basis $v_1,\dots ,v_m$ of $V$ such that its dual basis is ${\phi }_1,\dots ,{\phi }_m$. Moreover, by 3.114, we have $M(T)$ (with respect to the bases shown here of $V$ and $W$) is equal to ${(M(T^{\prime }))}^t$, which shows that $M(T)$ satisfies the desired property. □