Kernel and Image (of endomorphism) inclusions Are these propositions always true (for any $u \in \mathscr{L}(E) \neq 0$) ? 


*

*$\mathrm{Ker}(u) \subsetneq \mathrm{Ker}(u^2)$

*$\mathrm{Im}(u^2) \subsetneq \mathrm{Im}(u)$


If it is, how can I prove it ? The inclusions are obvious, but I can't figure out how to prove the non-equality ..
Thanks

Both statements are not true. Just consider $u=\text{id}$, the identity map.
