Corollary 2

Let K be a commutative field, E, F two vector spaces over K and L a commutative extension field of K. Let u be an element of $E \otimes_K F$, M and N its associated subspaces and u' the canonical image of u in $(E \otimes_K F)_{(L)}$ (canonically identified with $E_{(L)} \otimes_L F_{(L)}$, cf. § 5, no. 1, Proposition 3); then $\mathrm{rg}(u') = \mathrm{rg}(u)$ and the subspaces associated with u' are canonically identified with $M_{(L)}$ and $N_{(L)}$.

If $u = \sum_{i=1}^r x_i \otimes y_i$, where the families $(x_i)$ and $(y_i)$ are free, then $u' = \sum_{i=1}^r (1 \otimes x_i) \otimes (1 \otimes y_i)$ and the families $(1 \otimes x_i)$ and $(1 \otimes y_i)$ are free in $E_{(L)}$ and $F_{(L)}$ respectively (§ 5, no. 1, Proposition 4).