Past exam of the mathematics course of the University of Oxford / 2015 / Prelims / Mathematics I / 1 / c / i

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: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.