Suppose V is finite dimensional, with dimV=n≥1. Prove that there exist 1-dimensional subspaces U1,…,Un of V such that
V=U1⊕⋯⊕Un
Let v1,…,vn be a basis for V, and set Uj=span(vj).
Apply 1.44 and note that
0=u1+⋯=un=a1v1+⋯+anvn
implies every aj=0 by independence, thus each uj=0 as desired and we have a direct sum.