The text does not address the question of whether the union

The text does not address the question of whether the union of subspaces is a subspace. Conjecture: If W and T are subspaces of the vector space V then W union T is a subspace. Prove or disprove the conjecture.

Solution

W and T are subspaces of the vector space V then W U T is a subspace if and only if W is a subset of T or T is a subset of W

Proof of the above statement:

If W is subset of T, W U T = T

assume p1,p2 belongs to T U W, then p1 and p2 will also belong to T since W U T = T and (p1+p2) will also belong to T

Hence the argument is true when either W U T = T or T U W = W

Now taking the case when one is not the subset of the other set

p1 belongs to T and p1 doesn\'t belongs to W, or p2 belongs to W and p2 doesn\'t belong to T

Now p1+p2 will belong to T U W i.e. it implies that either (p1+p2) belongs to U or (p1+p2) belongs to W but both the statements can\'t be true

Hence the given subspaces W U T is not the vector space of V