...Let $f : \mathbb C^3 \to \mathbb C$ given by $f(x, y, z) = 5x - iz$, see that $f \in (\mathbb C^3)^*$. Calculate $T^t(f)$.

Question

Consider the linear transformation $T : \mathbb C_2 \to \mathbb C_3$ given by $T(x, y) = (2x + y, y -x, iy)$. Let $f : \mathbb C^3 \to \mathbb C$ given by $f(x, y, z) = 5x - iz$, see that $f \in (\mathbb C^3)^*$. Calculate $T^t(f)$.

The transpose of $T, T^t : (\mathbb C^3) \to (\mathbb R^2)$, is given by $T^t(f)(x, y) = f(T(x; y)) = f(2x + y, x – y, iy)$.

And

Considering the functional $f1 : \mathbb C^3 \to \mathbb C$, given by $f(x, y, z) = 5x – iz$. then $T^t(f)(x, y) = f(2x + y, x – y, iy) = 5(2x+y) + 0(x-y) -i(iy) = 10x + 6y$.

Did I understand correct? Is it just that? Did I do it right?