> [!NOTE] Definition (Linearly Independent Real Vectors) > Let $\underline{v}_{1},\underline{v}_{2},\dots,\underline{v}_{s}\in \mathbb{R}^{n}$ be a [[Finite Set|finite set]] of [[Real n-Space|n-tuples of reals]]. They are linearly independent iff for all $\lambda_{1},\lambda_{2},\dots,\lambda_{s}\in \mathbb{R},$ $\lambda_{1}\underline{v}_{1}+\lambda_{2}\underline{v}_{2}+\dots+\lambda_{s}\underline{v}_{s}=\underline{0}\implies \lambda_{1}=\lambda_{2}=\dots=\lambda_{s}=0$where $\underline{0}$ denotes the [[Real Zero Vector|zero vector]]. **Note**: Negating gives that the vectors are linearly dependent iff there exists $\lambda_{1},\lambda_{2},\dots,\lambda_{s}\in \mathbb{R}$ so that $\lambda_{1}\underline{v}_{1}+\dots+\lambda_{s}\underline{v}_{s}=\underline{0}$ and for some $i=1,2,\dots,s,$ $\lambda_{i}\neq 0.$ # Properties By [[Linearly Independent Subset of Real n-Space contains at Most n Elements]], if $S\subset \mathbb{R}^{n}$ is linearly independent then $|S|\leq n.$