# Linear transformation over complex numbers

I just want to verify that I'm on the right track for this:

If $T$ is the linear transformation $T: \mathbb{C} \to \mathbb{C}$ (where $\mathbb{C}$ is a vector space over $\mathbb{R}$, and $T(x) = (a + bi) \cdot x$, find the matrix that represents $T$ for the basis $\{1, i\}$.

So $T(1) = (a + bi) \cdot 1 = a + bi$ and $T(i) = ai + bi^2 = ai - b$, and therefore the matrix that represents $T$ is $\left( \begin{array}{c} a + bi \\ ai - b \end{array} \right)$?

Or is it a linear matrix? Or is it totally wrong?

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You should express the result in the basis $\{1,i\}$ as well, that is your matrix should be $\begin{bmatrix} a & b \\ -b & a\end{bmatrix}$.