Proposition 21

Let $K$ be a field and $L$ an extension field of $K$.

(i) For every vector space $E$ over $K$, $\dim_L(E_{(L)}) = \dim_K E$.

(ii) For every $K$-linear mapping $u : E \to F$, where $E$ and $F$ are vector spaces over $K$, $\operatorname{rg}(u_{(L)}) = \operatorname{rg}(u)$.

If $(e_i)_{i \in I}$ is a basis of $E$ over $K$, $(1 \otimes e_i)_{i \in I}$ is a basis of $E_{(L)}$ over $L$ (§ 5, no. 1, Proposition 4), whence the first assertion; the second follows from the first and the fact that $u_{(L)}(E_{(L)})$ is canonically identified with $(u(E))_{(L)}$ by the Corollary to Proposition 18.