Injective K-Homomorphisms are K-Automorphisms

Question

I just read in Chambert-Loirs A field guide to algebra:

Let $K \subset L$ be a finite extension and let $\sigma: L \rightarrow L$ be any $K$-Homomorphism. Like any morphism of fields, $\sigma$ is injective. It follows that $\sigma(L)$ is a K-Vector space of dimension $[L:K]$, hence $[\sigma(L):K]$ = $[L:K]$.

I don't quite understand the part "It follows that $\sigma(L)$ is a K-Vector space of dimension $[L:K]$". Where does this come from? I'm sure that I'm missing something obvious here..

Answer 1

For a field extension $L/K$, by definition $L$ is a vector space over $K$ of degree $[L:K]$. The $K$-vector space structure of $L$ now is carried over to the field $\sigma(L)$, since $\sigma$ is an injective $K$-homomorphism. And, for finite-dimensional vector spaces, injective implies surjective (rank-nullity theorem).

Answer 2

The homomorphic image of a finite dimensional vector space by an injective homomorphism has the same dimension as the vector space.