Consider the vector space $\displaystyle M_{2} (C)$ of all 2x2 matrices with complex entries. If A= $\displaystyle \left(\begin{array}{cc}a&b\\c&d\end{array}\right)$ then A* denotes the conjugate transpose of A.

A.) For A,B belong to $\displaystyle M_{2} (C)$, show that <A,B>=tr(AB*) defines an inner product.

B.) Find an orthonormal basis for $\displaystyle M_{2} (C)$ with respect to this inner product.