Proof Verification: If a vector space V is a union of k subspaces of its subspaces, prove that V is equal to one of its subspace.

Question

Suppose the subspaces are S1,S2,S3.....Sk

Take a from S1, b from S2....k from Sk

then a+b+....k belongs to V and union of all subspaces.

By the definition of union, a+b+....k must belong to atleast one of the subspace which will make it equal to V.

Is this proof correct? Or am I missing something?

Answer 1

Actually, if a vector space over a field $F$ is the union of $k$ proper subspaces, then $|F|<k$. So if the field is infinite, one of the subspaces is equal to $V$. This result is known as the avoidance lemma for subspaces.

For a proof you can take a look at my answer to this closely related question