Proposition 10

Let E, F be two vector spaces over the same field K and u : E \to F a linear mapping.

(i) For u to be injective (resp. surjective), it is necessary and sufficient that $^t u$ be surjective (resp. injective).

(ii) $\operatorname{rg}(u) \leq \operatorname{rg}(^t u)$ and $\operatorname{rg}(u) = \operatorname{rg}(^t u)$ if $\operatorname{rg}(u)$ is finite.

The second assertion follows from the above and Theorem 4.