> [!NOTE] Lemma (Zero Vector) > Let $n\geq 1.$ Let $\mathbb{R}^{n}$ denote the [[Real n-Space|real n-space]]. The [[Real Zero Vector|zero vector]] (or the origin of $\mathbb{R}^{n}$) is the [[Identity element of a binary operation|identity element]] of [[Addition in Real n-Space|real vector addition]]: that is, for all $\underline{v}\in \mathbb{R}^{n},$ $\underline{v}+\underline{0}=\underline{v}=\underline{0}+\underline{v}.$ Proof: Let $\underline{v}=(a_{1},a_{2},\dots,a_{n})\in \mathbb{R}^{n}$ by definition $\underline{v}+\underline{0}= \begin{pmatrix} a_{1}+0 \\ a_{2}+0 \\ \vdots \\ a_{n}+0 \end{pmatrix} = \begin{pmatrix} a_{1} \\ a_{2} \\ \vdots \\ a_{n} \end{pmatrix}= \begin{pmatrix} 0+a_{1} \\ 0+a_{2} \\ \vdots \\ 0+a_{n} \end{pmatrix} = \underline{0}+\underline{v}. $