I was asked the following question:
Let $A: \mathbb R^2 \to \mathbb R^2$ be a linear transformation, represented by the matrix $\begin{bmatrix}a & b \\ c & d\end{bmatrix}$ in the standard basis.
For $a,b,c,d \in \mathbb R$, show that $A$ can be written $A(z)=\alpha z+\beta \overline z$ where $\alpha, \beta \in \mathbb C$ are some combination of $a,b,c,d$.
I don't completely understand the question, because I'm not sure I understand how linear transformations are defined on complex numbers.
For example, if $z=a+ib$, then $A(z) = \begin{bmatrix}a & b \\ c & d\end{bmatrix} \begin{pmatrix}a \\b \end{pmatrix}$? or is it $\begin{pmatrix}a \\ib \end{pmatrix}$
Would appreciate help understanding the question, please do not solve it for me.
An explanation of the question:
The idea here is that we think of $\Bbb C$ as a vector space over $\Bbb R$ with basis $1$ and $i$, so that $A$ denotes the transformation matrix with respect to the basis $\{1,i\}$
That is, for any $x,y \in \Bbb R$, we define $$ T(x + yi) = A \pmatrix{x\\y} = \pmatrix{ax + by\\cx + dy} = (ax + by) + (cx + dy)i $$