> [!NOTE] Lemma > Let $\underline{v},\underline{w}\in \mathbb{R}^{n}$ be elements of the [[Real n-Space|real n-space]]. Then $\underline{v}$ and $\underline{w}$ are [[Linearly Independent Subset of Real n-Space|linearly dependent]] iff $\underline{v}$ and $\underline{w}$ are [[Collinearity in Real n-Space|collinear]]. **Proof**: Suppose $\underline{v}$ and $\underline{w}$ are linearly dependent then WLOG there exists $\lambda_{1}\neq 0$ and $\lambda_{2}\in \mathbb{R}$ so that $\lambda_{1}\underline{v}+\lambda_{2}\underline{w}=\underline{0}.$ Rearranging gives $\underline{v}=-(\lambda_{2}/\lambda_{1})\underline{w}.$ Thus the vectors are collinear. Conversely, suppose $\underline{v}$ and $\underline{w}$ are collinear. WLOG there exists $\lambda\in \mathbb{R}$ so $\underline{v}=\lambda \underline{w}.$ Rearranging gives $\underline{v}-\lambda \underline{w}=\underline{0}$ which is a linear dependent relation since the coefficient of $\underline{v}$ is $1\neq 0.$