# Statement(s) > [!NOTE] Statement 1 (General Form of Isometry on Real n-Space) > Suppose $T:\mathbb{R}^n\to \mathbb{R}^n$ is an [[Isometry|isometry]]. Then there exists a unique [[Orthogonal endomorphisms of Euclidean spaces|orthogonal matrix]] $A\in \mathbb{R}^{n\times n}$ and some vector $\mathbf{b}\in \mathbb{R}^n$ such that for all $\mathbf{x}\in \mathbb{R}^n,$ $T(\mathbf{x})=A\mathbf{x}+\mathbf{b}.$ # Proof(s) **Proof of statement 1:** Notice that $T$ is an isometry iff $L(\mathbf{x}):=T(\mathbf{x})-T(\mathbf{0})$ is a [[Linear maps|linear]] isometry, since [[Euclidean Metric is Translation Invariant]]. It follows from [[Isometry group of real n-space with standard metric]] that $L(\mathbf{x})=A\mathbf{x}$ for some orthogonal $A.$ Hence $T(\mathbf{x})=A\mathbf{x}+\mathbf{b}$ where $\mathbf{b}=T(0).$ $\blacksquare$ # Application(s) **Consequences**: **Examples**: # Bibliography