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.