How do unitary preserve inner products? Why is $\langle U x, U y \rangle = \langle x, U^\dagger U y \rangle$?

Question

On many links online [1], [2], [3], etc. they mention the following:

$U$ is a unitary matrix, i.e. $UU^\dagger = U^\dagger U = I$, where $U^\dagger$ is the Hermitian conjugate of $U$.

Then, $$\langle U x, U y \rangle = \langle x, U^\dagger U y \rangle$$

Why is this so?

Answer 1

Since $\langle u,v\rangle=u^\dagger v$, we generally have $$\langle Tu,\,v\rangle=(Tu)^\dagger v=u^\dagger T^\dagger v=\langle u,\,T^\dagger v\rangle$$ for any vectors $u,v$ and any linear map $T$.