> [!NOTE] Lemma > Let $n\geq 1.$ Let $\underline{v}\in \mathbb{R}^{n}$ be a [[Real n-Space|n-tuple of reals]]. Let $||\underline{v}||$ denote the [[Euclidean Norm|length]] of $\underline{v}.$ Let $\lambda\in \mathbb{R}.$ Let $\lambda \underline{v}$ denote a [[Scalar Multiplication in Real n-Space|scalar multiple]] of $\underline{v}.$ Then $||\lambda \underline{v}||=|v|||\underline{v}||$where $|\lambda|$ denotes the [[Absolute value function|absolute value]] of $\lambda.$ **Proof**: Let $\underline{v}=(a_{1},\dots,a_{n}).$ Then $||\lambda\underline{v}||=(\lambda a_{1})^{2}+\dots+ (\lambda a_{n})^{2} =\lambda^{2}(a_{1}^{2}+\dots+a_{2}^{2} )= \lambda^{2} ||\underline{v}||^{2}$and taking positive square roots (hence the modulus sign for $|\lambda|$) gives the result.