Proving an Identity Involving Matrices and Inner Products

[devious_]

Is there a non-ugly proof of the following identity:
$$\langle Ax,y \rangle = \langle x,A^*y \rangle$$
where A is an nxn matrix over, say, $\mathbb{C}$, A* is its conjugate transpose, and $\langle \cdot , \cdot \rangle$ is the standard inner product on $\mathbb{C} ^n$.

 

[Galileo]

Just start writing out the left side using $$<a,b>=a^\dagger b$$.

 

[devious_]

Thanks.

Although using my definition for the standard inner product I had to use $\langle a , b \rangle = a^t \bar{y}$, but it all worked out in the end.