= Solution
{c}
Here $\dim V=2$. By the <rank-nullity theorem>, $r(T)+n(T)=2$, and a <dimension> is a nonnegative integer. All three possibilities occur:
$$
\boxed{\begin{array}{c|c}
(r(T),n(T))&T(x,y)\\\hline
(0,2)&(0,0,0)\\
(1,1)&(x,0,0)\\
(2,0)&(x,y,0)
\end{array}}
$$
The <images of a linear map> in these examples have <dimensions> $0,1,2$, respectively, while the corresponding <kernel of a linear map> is $\mathbb R^2$, $\{(0,y):y\in\mathbb R\}$ and $\{0\}$. The identity above excludes all other pairs.
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