> [!NOTE] Lemma > Let $S\subset \mathbb{R}^{n}$ be a [[Basis of Real n-Space|basis]]. Then $S$ is [[Finite Set|finite]] with [[Cardinality|cardinality]] $|S|=n.$ **Proof**: By definition $S$ is a spanning set of $\mathbb{R}^{n}$ and linearly independent. Thus the result follows directly from [[Linearly Independent Subset of Real n-Space contains at Most n Elements]] and [[Spanning Set of Real n-Space contains at Least n Elements]].