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