Let $T:\mathbb{C}^2 \to \mathbb{C}^2$ be a inner product space, $T(x,y)=(2ix+y, -x-iy)$ and the inner product $<(x_1, x_2), (y_1, y_2)>=4x_1\overline{y_1}+9x_2\overline{y_2}$.
Find $T^*(x,y)$
I found out that the canonical basis $C=\left \{(1,0), (0,1) \right \}$ isn't orthonormal (but orthogonal) for the given inner product, so I orthonormalized it and got $C'= \left \{(1/2,0), (0,1/3) \right \}$.
Then I used the definition for adjoint transformations:
$<T^*(1/2, 0), (x,y)>=<(1/2,0), T(x,y)>$
$<(1/2, 0), (2ix+y, -x-iy)>=-4ix+2y$
$<T^*(0, 1/3), (x,y)>=<(0, 1/3), T(x,y)>$
$<(0, 1/3), (2ix+y, -x-iy)>=-3x-3iy$
So $T^*(x,y)=(-4ix+2y, -3x-3iy)$, but the correct answer is $T^*(x,y)=(-2ix-9/4y, 4/9x+iy)$.
What am I doing wrong?