Corollary

Let $E$ be a vector space of finite dimension $n$ and $u$ an endomorphism of $E$. The following properties are equivalent: (a) $u$ is bijective; (b) $u$ is injective; (c) $u$ is surjective; (d) $u$ is right invertible; (e) $u$ is left invertible; (f) $u$ is of rank $n$.

If E is an infinite-dimensional vector space, there are injective (resp. surjective) endomorphisms of E which are not bijective (Exercise 9).

Let K, K' be two fields, $\sigma : K \to K'$ an isomorphism of K onto K', E a vector K-space, E' a vector K'-space and $u : K \to K'$ a semi-linear mapping relative to $\sigma$ (§ 1, no. 13); the dimension of the subspace $u(E)$ of E' is also called the rank of $u$. It is also the rank of $u$ considered as a linear mapping of E into $\sigma_*(E')$, for every basis of $u(E)$ is also a basis of $\sigma_*(u(E))$.