I meant to reply days ago, sorry for the delay.

It seems to me that your first statement, that

is in fact what you're supposed to prove, that the change of basis transformation preserves the inner product. In other words that the basis transformation is an orthogonal linear transformation. So assuming i've interpreted your notation correctly then for

denoting a basis for D then

represents the basis for D in terms of the basis for

, ie is a basis transformation. Hence for

and

you need to prove that you can use the matrix A, as you defined, to directly compute the inner product in D.

If it were me i would start with the fact that the matrix representation of the inner product in

is the identity matrix I, so

.

What do you notice about the matrix

?