Orthonormal Bases from Unitary Matrices

Theorem: Orthonormal Bases from a Unitary Matrix

If $U\in {\mathbb{C}}^{n\times n}$ is a unitary matrix, then:

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

• the columns of its Hermitian transpose $U^{†}$ also form an orthonormal basis of the vector space ${\mathbb{C}}^n$.

PROOF

TODO