Proposition 1 (Rank-Nullity Theorem). Let $V$, $W$ be vector spaces over a field $F$. For any linear function $f : V \to W$,
\[\rank f + \null f = \dim V.\]As a corollary:
Proposition 2. Let $V$, $W$ be vector spaces over a field $F$, $f : V \to W$ be a linear function and $w \in W$. By LF > Proposition 4, $f^{-1}[\{w\}]$ is a subspace. We have