Show that matrix of $T$ has at least dim range $T$ nonzero entries. Suppose $V$ and $W$ are finite-dimensional and $T \in \mathcal{L}(V, W)$. Show that with respect to each choice of bases of $V$ and $W$, the matrix of $T$ has at least dim range $T$ nonzero entries.
I have no idea on how to solve this.

Suppose we have a basis {$v_{1},...,v_{n}$} of $V$ and {$w_{1},...,w_{m}$} of $W$, and so $M(T)$ is an $m$ by $n$ matrix. Let $p$ denote the number of zero-columns in the matrix. Thus, we know the dimension of the null space is at least $p$. And so by the fundamental theorem of linear maps, $n = \dim \operatorname{ran}(T)+\dim \operatorname{null}(T) \geq \dim \operatorname{ran}(T)+p$. Therefore, $n-p \geq \dim \operatorname{ran}(T)$, and we are done.
