Q4.

(a) Let U and W be subspaces of a vector space V. Prove that UnW is a subspace of V.

(b) Let V be a finite-dimensional vector space, and U and W subspaces of V. Prove that

codim(UnW) ≤ codim (U) + codim(W).

[For a subspace W of a finite-dimensional vector space V, the codimension of W in

V is defined to be codim(W) = dim(V) - dim(W).]

(c) Using part (b), prove that any two distinct planes in R3 that pass through the origin

intersect in a line.