# Direct sum definition

We say that $V$ is the $\textit{direct sum}$ of subspaces $U_1,\ldots,U_m$ if every element $v \in V$ can be written uniquely as the a sum $u_1 + \cdots + u_m$ where each $u_j \in U_j.$

I need help understanding the definition. What do we mean by $v$ can be $\textit{"written uniquely"}$ as a sum $u_1+ \cdots + u_m?$

That means there does not exist another set of $\{w_1, \dots,w_m\}$, with $w_j\in U_j$ for each $j$, and where $u_j \neq w_j$ for at least one $j$, but $w_1 + \dots + w_m = v \in V$.
• do you mean "but $w_1+ \cdots+w_m=v \in V$?" – user167857 Oct 18 '15 at 2:05