Concerning this question I asked (specifically my last comment), is it true that the formula $L\oplus \ker S =H$ holds in a finite dimensional vector space, where $S:H\rightarrow H$ is linear operator, $(b_1,\ldots,b_k)$ is a basis of the image of $S$ and $L$ consists of the preimages of the $b_i$ under $S$, i.e. $l_j\in L$ is such that $l_j\in S^{-1} (b_j)$ ?
If we equip $H$ with an inner product, such that $S$ becomes selfadjoint, would then $\text{im} S\oplus \ker S =H$ hold ? (Or can you give a counterexample?)