> [!NOTE] Definition (Orthogonal group) > Let $n\in\mathbb{N}^{+}.$ Let $\mathbb{F}$ be a [[Field (Algebra)|field]]. The $n$th [[Orthogonal Group|orthogonal group]] on $\mathbb{F}$ (*or the orthogonal group on* $\mathbb{F}^{n}$), denoted $O_{n}(\mathbb{F})$ or $O(n,\mathbb{F}),$ is the following [[Subsets|subset]] of the [[General Linear Group|general linear group]] $\{ M\in \text{GL}_{n}(\mathbb{F}) \mid M^{T} = M^{-1} \},$ where $M^T$ denotes the [[Dual of linear map|transpose]] of $M,$ under [[Function Composition|composition]]. > [!Example] > The [[Second Orthogonal Group Over The Reals|orthogonal group on]] $\mathbb{R}^{2}$ is the set of isometries on $\mathbb{R}^{2},$ denoted $O_{2}(\mathbb{R}).$ # Properties > [!NOTE] Proposition ($O(n,\mathbb{F})$ is a group) > Contents # Applications