The Orthogonal Group#
Theorem: The Orthogonal Group
Let \((V, \langle \cdot, \cdot \rangle)\) be a finite-dimensional real inner product space.
The set of all orthogonal transformations \(f: V \to V\) is a subgroup of the general linear group.
Definition: The Orthogonal Group
We call this subgroup the orthogonal group.
Notation
We denote the orthogonal group from \(V\) as \(O(V)\). When \(V\) is the real vector space \(\mathbb{R}^n\), we also write \(O(n, \mathbb{R})\).
Proof
TODO