I know how to show that multiplying by an orthogonal matrix preserves the angle and distance between two vectors. I have seen everywhere that Orthogonal matrices are kind of related to rotations and reflections.
What is a good definitive proof that reflection/rotation $\iff$ orthogonal matrix. Both for the $2$x$2$ case and the $3$ x $3$ case. (both directions)
What is a proof that Given an isometry $T$, there exists an $n$ × $n$ orthogonal matrix $A$ and a vector $u$ such that the associated map $T$ on position vectors is given by $T (p) = Ap + u$. Is this bit when there is a combination of rotation and reflection?