> [!NOTE] Theorem > Let $n \geq 1.$ Let $\mathbb{R}^{n}$ denote the [[Real n-Space|set of all n-tuples of real numbers]]. Let $+$ denote [[Addition in Real n-Space|real vector addition]]. Then $+$ is [[Commutativity|commutative]]: that is, for all $\underline{v}, \underline{w}\in \mathbb{R}^{n}$ $\underline{v}+\underline{w} =\underline{w}+\underline{v}.$ **Proof**: Let $\underline{v}=(a_{1},a_{2},\dots,a_{n})$ and $\underline{w}=(b_{1},\dots,b_{n}).$ Then by definition, $\underline{v}+\underline{w}=\begin{pmatrix} a_{1} + b_{1} \\ a_{2}+b_{2} \\ \vdots \\ a_{n} +b_{n} \end{pmatrix} = \begin{pmatrix} b_{1} + a_{1} \\ b_{2}+a_{2} \\ \vdots \\ b_{n} +a_{n} \end{pmatrix}=\underline{w}+\underline{v}.$