(adsbygoogle = window.adsbygoogle || []).push({}); The problem statement, all variables and given/known data

Let V be a vector space over an infinite field. Prove that V is not the union of finitely many proper subspaces of V.

The attempt at a solution

Suppose V is the union of the proper subspaces U_{1}, ..., U_{n}. Let u_{i}be a vector not in U_{i}. If u_{1}+ ... + u_{n}is in the union, then there must be some subspace that contains it. But then that subspace contains a sum where one of the terms doesn't belong to it. I'm hoping this isn't possible but I can't think of anything contradictory.

In any case, I think my approach is wrong because I haven't really used the fact that V is defined over an infinite field and that the union is a finite one.

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# Homework Help: Union of Proper Subspaces Problem

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