a) Suppose we have two spaces subspaces U and W of V
To prove U W is a subspace of V, it is sufficient to prove that any linear combination also belongs to U W.
Let us choose two vectors u and w belong to U W.
i.e. u,w U W
i.e. u,w U and u,w W
Since U and W are vector spaces and u,w U and u,w W,so their linear combination is also belong to that spaces.
i.e. U and W where are any scaler come from underlying scaler field.
i.e. U W
It shows that any linear combination of the vectors u and w also belongs to U W.
This prove that U W is the subspace of V. (proved)
b) To give an example of two vector spaces whose union is not a vector space we have know the following which give the proper scenario when the union of two vector spaces is again a subspace of the vector spaces.
Suppose U and W are to subspaces. Then U W is a subspace of V if and only if either U W or W U.
Now using this we can construct an example.
We know that is a vector space over .
Let us assume the subspaces of :
But here neither U W nor W U.
So U W is not a subspace of here.