> [!NOTE] Lemma > 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]]. Let $\lambda\in \mathbb{R}.$ Let $\underline{v},\underline{w}\in \mathbb{R}^{n}.$ Then $\lambda(\underline{v}+\underline{w})=\lambda \underline{v}+\lambda \underline{\lambda}$where $\lambda \underline{w}$ denotes a [[Matrix Scalar Multiplication|scalar multiple]] of $\lambda.$ Proof: Let $\underline{v}=(a_{1},a_{2},\dots ,a_{n}),\underline{w}=(b_{1},b_{2},\dots,b_{n}).$ By definition and [[Distributivity of Multiplication over Addition of Real Numbers]], $\lambda(\underline{v}+\underline{w})=\begin{pmatrix} \lambda(a_{1}+b_{1})\\ \lambda(a_{2}+b_{2}) \\ \vdots \\ \lambda(a_{n}+b_{n}) \end{pmatrix}= \begin{pmatrix} \lambda a_{1} +\lambda b_{1} \\ \lambda a_{2} + \lambda b_{2} \\ \vdots \\ \lambda a_{n}+ \lambda b_{n} \end{pmatrix} = \lambda \underline{v} + \lambda \underline{w}.$