# Intersection of vector subspaces and extension of the ground field.

Let $V$ be a vector space over a field $F$ and $\{W_i\}_{i\in I}$ a collection of subspaces of $V$. Let $K$ be an extension field of $F$. If $I$ is finite then it is easy to see that $$\big(\bigcap_{i\in I} W_i\big)\otimes_F K=\big(\bigcap_{i\in I} W_i\otimes_F K \big).$$ Is this true also when $I$ is infinite?

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But, nevertheless, the intersection has a basis (Axiom of Choice?), and extends to a basis of each $W_i$. It is not hard to prove that an "extension of scalars" of a free module produces a free module with "the same" generators, in the sense that they are images of the originals.
Thus, the intersection of the $W_i\otimes_F K$'s is no larger than the tensor product with the intersection.