Definition 2

Let V be a right vector space over K, with a K'-structure V'. A vector of V is said to be rational over K' if it belongs to V'. A vector sub-K-space W of V is said to be rational over K' if it is generated (over K) by vectors rational over K'.

Let (v'_i)_{i \in I} be a basis of V' over K', which is therefore also a basis of V over K (no. 1, Proposition 1). For a vector x = $\sum_i v'_i \xi_i$ of V to be rational over K', it is necessary and sufficient that $\xi_i \in K'$ for all $i \in I$.

If W is a vector sub-K-space of V which is rational over K', it follows from Definition 2 that $W' = W \cap V'$ is a vector sub-K'-space of W, which generates W over K; on the other hand every subset of W' which is free over K' is also free over K since it is contained in V' (no. 1, Proposition 1). It follows (no. 1, Proposition 1) that W' is a K'-structure on W said to be induced by the K'-structure V' on V.

For every vector sub-K'-space W' of V', we shall denote by W'.K the vector sub-K-space of V consisting of the linear combinations of elements of W' with coefficients in K.