Well, I have been thinking of this for quite a bit now. If $V$ and $W$ are vector spaces over a field $F$ and $f$ is a linear transformation $V \to W$, the Rank-Nullity Theorem states that the dimension of finite dimensional vector space $V$ is given by $$\textbf{dim} V = \textbf{rank} f + \textbf{nullity} f$$.
My question is suppose that $f$ is just a transformation of $V$ to $V$ itself, and that the intersection of the kernel and the image of $f$ is just the trivial space (the space that contains only the zero vector), then wouldn't that imply that:
$V=\textbf{ker} f \oplus \textbf{Im} f$?