Let $T \in \mathcal{L}(V, F)$, and $u \in V$ such that $T(u) \neq 0$. Prove that $V = \text{null}(T) + \text{span}(u)$
$F$ is just $\mathbb{R}$. My first thought was about the fundamental theorem of linear maps, that is dim$V = $dim null $T + $ dim range $T$, since it's of a similar format, but it doesn't seem to apply. I think the key is that it's a linear map to $\mathbb{R}$, since $T$ sends everything to $0$ or not $0$, but then I think what I don't understand is what exactly is the span of a vector in just $\mathbb{R}$?