Let $V$ be finite dimensional complex vector space, $d:V\to V$ be linear map such that $d^2=0$ and $\Delta=dd^{\star}+d^{\star}d$, where $d^{\star}$ is the adjoint of $d$. Prove that
(a) $dd^{\star}x=0$ imply $d^{\star}x=0$ and $d^{\star}dx=0$ imply $dx=0$
(b) $Ker \ \Delta=Ker \ d\ \cap \ Ker \ d^{\star}$
(c) There is the orthogonal decompsition $V=Ker \ \Delta \oplus Im \ d \oplus Im \ d^{\star}$
I've proved (a) and (b) but I'm stuck at (c). I'm not sure how to prove the existence of a orthogonal decompsition with three subspaces. I've proved that $ Im \ d^{\perp}\ \cap Im \ d^{\star^{\perp}}= Ker\ \Delta$. Is this sufficient to say the orthogonal decomposition exist?