> [!NOTE] Lemma > Let $V$ be a [[Finite Dimensional Real Vector Space|finite dimension real vector space]]. Then there exists $n\in \mathbb{N}^{+}$ so that there exists a [[Linear Isomorphism|linear isomorphism]] between the [[Real n-Space|real n-space]] and $V$: that is, $V\cong \mathbb{R}^{n}.$ **Proof**: By [[Existence of Basis of Finite Dimensional Real Vector Space contains Basis (Sifting Lemma)|sifting lemma]], there exist a basis $B=\{ v_{1},v_{2},\dots,v_{n} \}$ of $V.$ By [[Coordinate Map with respect to Basis for Finite Dimensional Real Vector Space is Linear Isomorphism]], the [[Coordinate Map with respect to Basis for Finite Dimensional Real Vector Space|coordinate map]] of $V$ with respect to $B$ gives a linear isomorphism from $V$ to $\mathbb{R}^{n}.$ By [[Linear Isomorphism from Finite Dimensional Vector Space Preserves Dimension]], $n$ is unique.