The Special Orthogonal Group#
Theorem: The Special Orthogonal Group
Let \((V, \langle \cdot, \cdot \rangle)\) be a finite-dimensional real inner product space.
The set of all orthogonal transformations \(f: V \to V\) whose determinant is \(+1\) form a subgroup of the orthogonal group
Definition: The Special Orthogonal Group
We call this subgroup the special orthogonal group.
Notation
We denote the special orthogonal group from \(V\) as \(\operatorname{SO}(V)\). When \(V\) is the real vector space \(\mathbb{R}^n\), we also write \(\operatorname{SO}(n, \mathbb{R})\).
Proof
TODO