Exercise 3.B.16

Proof. We will prove the contrapositive.

Suppose $V$ is infinite-dimensional and $T$ is a linear map on $V$. Assume that $\mathrm{null}<!--\; FUNCTION\; APPLICATION\; -->T$ is finite-dimensional. We need to show that $\mathrm{range}<!--\; FUNCTION\; APPLICATION\; -->T$ is infinite-dimensional.

Let $v_1,\dots ,v_n$ be a basis of $\mathrm{null}<!--\; FUNCTION\; APPLICATION\; -->T$. Because $V$ is infinite-dimensional, we can extend this list to a linearly independent list $v_1,\dots ,v_n,\dots ,v_m$ of any length $m$. By the same reasoning used in 3.22, it follows that $T{v}_{n+1},\dots ,T{v}_m$ is a linearly independent list in $\mathrm{range}<!--\; FUNCTION\; APPLICATION\; -->T$. Since $m$ was arbitrary, by Exercise 14 in 2.A, we have that $\mathrm{range}<!--\; FUNCTION\; APPLICATION\; -->T$ infinite-dimensional. □