Example of endomorphism $\phi$ such that $V \neq Im(\phi) \oplus Ker(\phi)$ where $V$ is a finite dimensional vector space.
Is it possible to get one as I was thinking the basis to basis map, I calculated $V=\Bbb R^2$ so $\phi(x,y)=(x+y,0)$ then I realized that $ Im(\phi)=Sp\{(1,0)\}$ and $ Ker(\phi)=Sp\{(0,1)\}$. So it will not work.
Then I became confused remembering the proof of Rank-Nullity theorem. So is it possible to get one in finite dimensional space. Might not be!
I know for projection $\pi:V \to V$ $V=Im(\pi) \oplus Ker(\pi)$. What about others?