Length Preservation of Real Orthogonal Matrices

Theorem: Length Preservation of Real Orthogonal Matrices

Multiplying a column vector $\vec{v}\in {\mathbb{R}}^n$ by a real orthogonal matrix $A\in {\mathbb{R}}^{n\times n}$ preserves its length.

$$||A\vec{v}||=||\vec{v}||$$

PROOF

TODO