> [!NOTE] Theorem > Let $n\geq 1.$ Then $\mathbb{R}^{n},$ the [[Real n-Space|real n-space]] forms an [[Groups|Abelian group]] with respect to [[Addition in Real n-Space|real vector addition]]. **Proof**: It follows from [[Zero Vector is Identity of Addition in Real n-Space]], the identity of vector addition, the zero vector, is in $\mathbb{R}^{n}.$ It follows from [[Additive Inverse of Real Vector]] that every element of $\mathbb{R}^{n}$ has an inverse under $+$ in $\mathbb{R}^{n}.$ It follows form [[Associativity of Addition in Real n-Space]] that for all $u,v,w\in \mathbb{R}^{n},$ $u+(v+w)=(u+v)+w.$ By definition, $\mathbb{R}^{n}$ is closed under vector addition. It follows from [[Commutativity of Addition in Real n-Space]] that vector addition is commutative.