Orthonormal Bases from Orthogonal Matrices

Theorem: Orthonormal Bases from an Orthogonal Matrix

If $A\in {\mathbb{R}}^{n\times n}$ is a real orthogonal matrix, then:

• the columns of $A$ form an orthonormal basis of the vector space ${\mathbb{R}}^n$;

• the columns of its transpose $A^{\mathsf{T}}$ also form an orthonormal basis of the vector space ${\mathbb{R}}^n$.

PROOF

TODO