Question

Prove that there does not exist a linear map T : R5 ! R5

such that range T = null T.

Answer 1

By the rank-nullity theorem we have $\dim\operatorname{Im}T+\dim\ker T=\dim\mathbb R^5=5$

If $\operatorname{Im}T=\ker T$, then $\dim\operatorname{Im}T=\dim\ker T=\frac{5}{2}$. It is impossible, so $\operatorname{Im}T\neq\ker T$