The question states "Find the range and the kernel (those are new words for the column space and nullspace) of $T$.
Part a is:
$T(v_1, v_2) = (v_2, v_1)$
Is the kernel
$\begin{bmatrix} 0 \\ 0 \end{bmatrix}$
because the only way to get $(v_2, v_1) = 0$ is for both $v_1$ and $v_2$ to be $0$, and the range
$\begin{bmatrix} 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \end{bmatrix}$
due to the fact that the columns space becomes
$v_2 \begin{bmatrix} 1 \\ 0 \end{bmatrix}+ v_1 \begin{bmatrix} 0 \\ 1 \end{bmatrix}$
and in a related note, if the original question becomes
$T(v_1,v_2,v_3) = (v_1,v_2)$
does $v_3$ in both the kernel and range become irrelevant due to the fact $v_3$ seems to become a free variable? In othe words, the kernel is $(0, 0, 1)$ and the range is $(1, 0, 0)$ and $(0, 1, 0)$.