Let $U_1,U_2,...,U_k$ be subspaces of vector space $V$. Prove that $V=U_{1}\oplus U_{2}\oplus...\oplus U_{k}$ iff every vector $v\in V$ can be written uniquely as $v=u_{1}+u_{2}+...+u_{k}$ while $\forall i\in\{1,...,k\}\quad u_{i}\in U_{I}$.
Thoughts I know how to prove $\Leftarrow$ but I'm confused in the other direction. I think I need to use the setting of direct sum that $0=u_{1}+u_{2}+...+u_{k}$ iff $\forall i\in\{1,...,k\}\quad u_{I}=0$, or maybe display the $U_i$ bases.