Let V = R^2 and W be the subspace generated by w=(2,3). Let U be the subspace generated by u=(1,1). Show that V is the direct sum of W and U. Can you generalize this to any two vectors u and w?
My attempt to generalize:
V=W+U
=r(w1 w2) + s(u1 u2)
=(rw1 rw2) + (su1 su2)
=(rw1+su1, rw2+su2) E R^2
W=U
r(w1 w2)=s(u1 u2)
(rw1 rw2) = (su1 su2)
rw1=su1
rw2=su2
Since this can be true for more than just r and s=0, we cannot conclude that U intersect W = {0}. Therefore, the direct sum cannot be generalized for any two vectors.
Is this correct logic?