Corollary 1

Every sum and every intersection of vector sub-K-spaces of V which are rational over K' is a rational subspace over K'.

The assertion concerning sums is obvious. On the other hand, if $(W'_i)_{i \in I}$ is a family of vector sub-$\mathbf{K}'$-spaces of $V'$, then $\left( \bigcap_{i \in I} W'_i \right) \otimes_{\mathbf{K}'} \mathbf{K} = \bigcap_{i \in I} (W_i \otimes_{\mathbf{K}'} \mathbf{K})$ (§ 7, no. 7, Corollary to Proposition 14), which proves the corollary.

A basis $B$ of $V$ over $\mathbf{K}$ is said to be rational over $\mathbf{K}'$ if it consists of vectors which are rational over $\mathbf{K}'$.