a Suppose that U and W are subspaces of a vector space V Sho

a) Suppose that U and W are subspaces of a vector space V. Show that the intersection of U and W is a subspace of V.

b) Suppose that U and W are subspaces of a vector space V. Show that the union of U and W is NOT a subspace of V. Take V to be R2 and U and W be subspaces in V.

Solution

U (int) W = {v:v belongs to U and v belons to W}

Since both U and W are subspaces of V

=> 0 belongs to U and 0 belongs to W

=> 0 belongs to U (int) W

so if u.w belongs to U (int) W

then it must belongs to both the subspaces

i.e. u.w belongs to U and u,w belongs to W

since U and W are subspaces, hence it must be closed under addition

(u + w) belongs to U and (u+w) belongs to V

Hence the addition, scalar and identity all elements are present in the intersection, hence U (int) W is a subspace of V

2) Proving by counter example

u = [1 0] belongs to U and U U W

w = [0 1] belongs to W and U U W

u + w = [1 1] doesn\'t belongs to U or W, hence U or W is not the subspace of V