> [!NOTE] Lemma > Let $\underline{v}=(a_{1},\dots,a_{n})\in \mathbb{R}^{n}.$ The [[Inverse under a binary operation|inverse]] of $\underline{v}$ under [[Addition in Real n-Space|real vector addition]] is given by the [[Scalar Multiplication in Real n-Space|scalar multiple]] $-\begin{pmatrix} a_{1} \\ \vdots \\ a_{n} \end{pmatrix} =\begin{pmatrix}-a_{1} \\ \vdots \\ -a_{n} \end{pmatrix}$ **Proof**: Let $\underline{0}$ denote the zero vector. Then by [[Zero Vector is Identity of Addition in Real n-Space]], $\underline{0}$ is indeed the identity of real vector addition. Clearly $-\underline{v}+\underline{v}=\underline{0}=\underline{v}+(-\underline{v}).$