basis of vector spaces

Question

Let $F\subseteq E$ be a extension of fields (not required finite extension), $W$ is an $F$-module and $V$ is an $E$-module such that $W\subseteq V$. We assume that the dimension of $V$ over $E$ is $n<\infty$.

Assume there exist $e_1,...,e_n\in W$ which form a $E$-basis of $V$ , then I want to know if we can deduce that $e_1,...,e_n$ form a basis of $W$ over $F$?

This is important in the proof of theorem 2 in this paper.

Thanks!

Answer 1

Let $E=F(x)$ (field of rational functions). Take $W=V=E$. Then $W$ is infinite dimensional over $F$, but obviously finite dimensional over $E$.