Suppose $V$ is finite dimensional, with $\dim V = n \ge 1$. Prove that there exist 1-dimensional subspaces $U_1,\dots,U_n$ of $V$ such that

Let $v_1,\dots,v_n$ be a basis for $V$, and set $U_j = \text{span}(v_j)$.

Apply 1.44 and note that

implies every $a_j = 0$ by independence, thus each $u_j = 0$ as desired and we have a direct sum.