Exercise 3.C.1

Proof. Let $v_1,\dots ,v_m$ denote a basis of $V$. The dimension of $\mathrm{range}<!--\; FUNCTION\; APPLICATION\; -->T$ is the same as the dimension of $\mathrm{span}<!--\; FUNCTION\; APPLICATION\; -->(T{v}_1,\dots <!--\; FUNCTION\; APPLICATION\; -->,T{v}_m)$. So if the matrix of $T$ has less than $\dim <!--\; FUNCTION\; APPLICATION\; -->\mathrm{range}<!--\; FUNCTION\; APPLICATION\; -->T$ nonzero entries, it follows from the defintion of $M(T)$ that the list $T{v}_1,\dots ,T{v}_m$ has less than $\dim <!--\; FUNCTION\; APPLICATION\; -->\mathrm{range}<!--\; FUNCTION\; APPLICATION\; -->T$ non-zero vectors, which is not possible, because its span is $\mathrm{range}<!--\; FUNCTION\; APPLICATION\; -->T$. □