I managed to prove that if $A$ is unitary than for any base orthonormal base $\{u_i\}$: $\{A^*u_i\}$ is also orthonormal base.
I need some help proving that it's an if and only if, and not only if. I only managed to say that $A$ is invertible since $\{A^*u_i\}$ is also a base.
Edit: I'm using the definition "$A$ is unitary if $AA^*=A^*A=I$.