Unitary matrix matrix problem I need to proof this question Show that for any
Recall that the unitary group $U(n)$ consists of all $A \in M_n(C)$ with $A^*A = I$. Show that a matrix $A \in M_n(C)$ is in $U(n)$ if and only if $\langle Ax,Ayi\rangle = \langle x,y\rangle$ for all $x, y \in C$.
So I just need the $\Leftarrow$ implication; I made it but I am not sure if my justification is good enough. 
$\langle Ax,Ay\rangle = \langle x,A^*Ay\rangle = \langle x,y\rangle = x^*A^*Ay = x^*y$ so {problem here} $A^*A = I$
therefore $A \in M_n(C)$ is in $U(n)$.
So do you guys think that this is good enough or do is their some problems in this argument?

It seems that what you need to show is that
$$
x^*(A^*A)y = x^*y \text{ for all } x,y \in \Bbb C^n \implies
A^*A = I
$$
In general: suppose that $x^*Ay = x^*By$ for all $x,y \in \Bbb C^n$.  How can we use this to conclude that $A = B$?
