**Problem**. Let be a vector space over some field and Let be a finite collection of the subspaces of and suppose that for any with we have Prove that

**Solution**. The proof is by induction over . There is nothing to prove if Suppose and put By the induction hypothesis for all So for any there exists some Since the set is linearly dependent. Thus there exists such that for some But if then and so We also have that for all Therefore

Advertisements

It seems that this problem is also true for every finitely generated torsion-free module over an integral domain. We should replace “dimension” with “rank” of the module, and use localization.

Hey Rasool

You’re right. Thanks for the comment.