Confusing linear algebra problem

[AkilMAI]

Homework Statement

Let U1 and U2 be two subspaces of a finite dimensional vector space V , let {u_1,u_2...,u_m} be a basis of U1 /\U2(where /\ means intersection) and let {u_1,u_2...,u_m, v_1,v_2...,v_k} be an extension to a basis of U1 . Let W = span{v_1,v_2...,v_k}. I need to prove that that W /\U2 = {0}
and that, if U1 + U2 = V , then W # U2 = V,where W#U2 is the direct sum of W and U2.

Homework Equations

The Attempt at a Solution

I proved that W /\U2 = {0} with some help but I need help with the second part...but I'm more interested in an explanation as I find this problem...confusing

 

[AkilMAI]

anyone?

 

[AkilMAI]

By the dimension theorem dim(W # U2)=dim W+dim U2+ dim(W /\U2)...since W /\U2=0 dim(W /\U2)=0 and since W+U2=V =>dimW+dimU2=V=>W # U2 = V,is this correct?