Suppose $V$ and $W$ are finite-dimensional and $T \in \mathcal L(V,W)$. Prove that $\dim \text{range }T = 1$ if and only if there exist a basis of $V$ and a basis of $W$ such that with respect to these bases, all entries of $\mathcal M(T)$ equal $1$.

Suppose $\dim \text{range }T = 1$. We need to construct a basis $v_1,\dots,v_n$ of $V$ and $w_1,\dots,w_m$ of $W$ such that $Tv_k =$

TODO

Now suppose all entries of $\mathcal M(T)$ equal $1$ with respect to bases $v_1,\dots,v_n$ of $V$ and $w_1,\dots,w_m$ of $W$. Then

Thus $(w_1+\dots+w_m)$ is a basis for $\text{range }T$, so $\dim \text{range }T = 1$.