Proposition 9

Let $E, F$ be two vector spaces over a field $K$ and $u : E \to F$ a linear mapping. (i) $\mathrm{rg}(u) \leq \inf(\dim E, \dim F)$. (ii) Suppose that $E$ is finite-dimensional; in order that $\mathrm{rg}(u) = \dim E$, it is necessary and sufficient that $u$ be injective. (iii) Suppose that $F$ is finite-dimensional; in order that $\mathrm{rg}(u) = \dim F$, it is necessary and sufficient that $u$ be surjective.

This follows immediately from relations (12) and (13).