12 Prove that the union of two subspaces of V is a subspace

12. Prove that the union of two subspaces of V is a subspace of V if and only if one of the subspaces is contained in the other.

(Recall that the union of two sets A and B is denoted

AUB and is defined by AU B = {112 that r could also be in both sets.) A or x E B). A mathem atical \"or\" means

Solution

The reason why this can happen is that all vector spaces, and hence subspaces too, must be closed under addition (and scalar multiplication). The union of two subspaces takes all the elements already in those spaces, and nothing more. In the union of subspaces W1 and W2, there are new combinations of vectors we can add together that we couldn\'t before, like v1+w2 where v1W1and w2W2.

For example, take W1W1 to be the xx-axis and W2 the yy-axis, both subspaces of R2.
Their union includes both (3,0) and (0,5)( whose sum, ((3,5), is not in the union. Hence, the union is not a vector space.