Hey,

Can you check to see if this problem is correct? Thank you!

Prove that if, If and are both subspaces of the vector space , then is not a vector space unless one of the subspaces or are contained in the other.

Let and be vector spaces. Clearly if either or is

contained in the other, then is a vector space because

their union would simply be the space that the other one in

contained in. Now, suppose for the sake of contradiction that is a vector space and neither nor are contained

within the other. Then obviously, since neither is a subset of the

other, there exist elements and such that

and . Consider the vector . Since we are assuming that is a vector space, so that or .

If , then by closure

since . This obviously contradicts that

.A similar argument can be made if , so

is not a vector space. Therefore, for to be a

vector space, one of the spaces must be contained within the other.