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.