Let $A \in M_{n,n}(\mathbb F)$. Is $\mathbb F^n$ the direct sum of $R(A)$ and $N(A)$ ?
If I could show that the orthogonal complement of $R(A)$ is $N(A)$ the result would follow, but the orthonal complement of $R(A)$ is $N(A^T)$ which aren't neccesarily equal to $N(A)$ ?
It's not necessary that $$\Bbb F^n=R(A)\oplus N(A)$$ since it's possible that $$R(A)\cap N(A)\ne\{0\}$$ and even we might find that $$R(A)\subset N(A)$$ in the case when $A$ is nilpotent with index $2$.