Let W be a finitely generate vector space, and U, V ⊆ W. Let B = {z1, . . . , zk} be a basis of U∩V ,

with the convention that if U∩V = {0}, then k = 0 and B = ∅. Extend B to a basis of U by adding

C = {u1, . . . , um} (i.e. B ∪ C is a basis of U). Again we use the convention that if U ∩ V = U,

then m = 0, and C = ∅. Similarly, extend B to a basis of V by adding D = {v1, . . . , vn}. Show

that B ∪ C ∪ D is a basis of U + V . Conclude dim(U + V ) = dim(U) + dim(V ) − dim(U ∩ V ).