Let u_1, u_2 be finite dimensional subspaces of a vector space V. Show that the sum u_1 + u_2 is direct if dim(u_1 +u_2) = dim(u_1) + dim (u_2) Solution we know that dim(U1+U2)= dim(U1)+dim(U2)-dim(U1 intersection U2) given that dim(U1+U2)= dim(U1)+dim(U2) i.e dim(U1 intersection U2) =0 U1 intersection U2 = {0} hence sum U1+U2 is direct.