Consider the vector space of all 2x2 matrices with complex entries. If A= then A* denotes the conjugate transpose of A.
A.) For A,B belong to , show that <A,B>=tr(AB*) defines an inner product.
B.) Find an orthonormal basis for with respect to this inner product.
I think your basis is already orthonormal.
The dimension of is three so either , either the dimension of is 1. To find a basis of , you can take a random matrix which trace does not equal 0 (say for example) and see how you can express it has a sum of matrices of and of one matrix which is not in . Then, check if this last matrix is in and... I let you take it from here.how would i describe U perp?