Proposition 4

For a $\mathbf{K}$-linear mapping $f : V_1 \to V_2$ to be rational over $\mathbf{K}'$, it is necessary and sufficient that its graph $\Gamma$ be rational over $\mathbf{K}'$ for the product $\mathbf{K}'$-structure on $V_1 \times V_2$.

Let $g$ be the mapping $x_1 \mapsto (x_1, f(x_1))$ from $V_1$ to $V_1 \times V_2$; this is a $\mathbf{K}$-linear mapping such that $\Gamma = g(V_1)$; if $f$ is rational over $\mathbf{K}'$, so is $\Gamma$ by virtue of Corollary 2 to Proposition 3. Conversely, suppose that $\Gamma$ is rational over $\mathbf{K}'$ and let it have the $\mathbf{K}'$-structure induced by that on $V_1 \times V_2$; it follows immediately from the definitions that the restrictions $p_1, p_2$ to $\Gamma$ of the projections $\mathrm{pr}_1, \mathrm{pr}_2$, are $\mathbf{K}$-linear mappings, rational over $\mathbf{K}'$ of $\Gamma$ into $V_1$ and $V_2$ respectively. As $p_1$ is bijective, its inverse mapping $q_1$ is rational over $\mathbf{K}'$ (Corollary 1 to Proposition 3) and hence so is $f = p_2 \circ q_1$.