Proposition 3

*Let $V_1, V_2$ be two right vector spaces over $\mathbf{K}$ and $V'_1, V'_2$ $\mathbf{K}'$-structures on $V_1, V_2$ respectively. $V_1$ (resp. $V_2$) is canonically identified with $V'_1 \otimes_{\mathbf{K}'} \mathbf{K}$ (resp. $V'_2 \otimes_{\mathbf{K}'} \mathbf{K}$) (no. 1, Proposition 1).

(i) *The mapping $f' \mapsto f' \otimes 1_{\mathbf{K}} = f'_{(\mathbf{K})}$ is a bijection of $\mathrm{Hom}_{\mathbf{K}'}(V'_1, V'_2)$ onto the set of $\mathbf{K}$-linear mappings of $V_1$ into $V_2$ which are rational over $\mathbf{K}'$; the inverse bijection associates with every $\mathbf{K}$-linear mapping $f : V_1 \to V_2$ rational over $\mathbf{K}'$ the $\mathbf{K}'$-linear mapping $f' : V'_1 \to V'_2$ with the same graph as the restriction of $f$ to $V'_1$.

(ii) *For every $\mathbf{K}$-linear mapping $f : V_1 \to V_2$ rational over $\mathbf{K}'$,

(i) Clearly with the above identifications, if $f' : V'_1 \to V'_2$ is a $\mathbf{K}'$-linear mapping, $f'_{(\mathbf{K})} = f' \otimes 1_{\mathbf{K}}$ is rational over $\mathbf{K}'$ and $f'$ is the mapping with the same graph as the restriction of $f'_{(\mathbf{K})}$ to $V'_1$. Conversely, if $f : V_1 \to V_2$ is a $\mathbf{K}$-linear mapping rational over $\mathbf{K}'$ and $f' : V'_1 \to V'_2$ has the same graph as the restriction of $f$ to $V'_1$, $f$ and $f'_{(\mathbf{K})}$ coincide on $V'_1$, which is a generating system of $V_1$ over $\mathbf{K}$, hence $f = f'_{(\mathbf{K})}$.

(ii) If $f = f' \otimes 1_{\mathbf{K}}$, then $f(V_1) = f(V'_1 \otimes_{\mathbf{K}'} \mathbf{K}) = f'(V'_1) \otimes_{\mathbf{K}'} \mathbf{K}$ and, as $f'(V'_1) \subset V'_2, f(V'_1) = f'(V'_1) \cap V'_2$ (§ 7, no. 9, Proposition 19); the formula $f^{-1}(V'_2) = V'_1 + \mathrm{Ker}(f)$ follows immediately.