Affine transformations that correspond to entire complex functions

I assume the following is a standard consideration and question, but I don't know how to prove it:

There is a trivial one-to-one correspondence between affine transformations $$f = (f_x,f_y)$$ of the plane $$\mathbb{R}^2$$:

$$\begin{bmatrix} f_x \\ f_y \\ \end{bmatrix} = \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix}\begin{bmatrix} x \\ y \\ \end{bmatrix} + \begin{bmatrix} e \\ f \\ \end{bmatrix}$$

with $$ad - bc \neq 0$$ and complex functions $$g(u + iv) = g_u(u,v) + ig_v(u,v)$$ with

$$\begin{bmatrix} g_u \\ g_v \\ \end{bmatrix} = \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix}\begin{bmatrix} u \\ v \\ \end{bmatrix} + \begin{bmatrix} e \\ f \\ \end{bmatrix}$$

Some such functions are entire functions, e.g. the identity function

$$\begin{bmatrix} g_u \\ g_v \\ \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix}\begin{bmatrix} u \\ v \\ \end{bmatrix}$$

others are not, e.g. the complex conjugate

$$\begin{bmatrix} g_u \\ g_v \\ \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & -1 \\ \end{bmatrix}\begin{bmatrix} u \\ v \\ \end{bmatrix}$$

which corresponds to reflection at the real axis.

I guess that the only affine transformations that correspond to entire complex functions are compositions of (uniform) scalings, rotations, and translations. Especially reflections don't qualify. Of non-uniform scalings and shearings (= non-uniform translations) I doubt it.

The qualifiying transformations would simply correspond to functions

$$g(z) = z_1\cdot z + z_0$$

with $$z_0 = e + if$$ and

$$\begin{bmatrix} a & b \\ c & d \\ \end{bmatrix} = r\begin{bmatrix} \cos(\varphi) & \sin(\varphi) \\ -\sin(\varphi) & \cos(\varphi) \\ \end{bmatrix}$$

for $$z_1 = re^{i\varphi}$$.

My questions are:

1. Is this correct?

2. How to prove it?

Yes, it is correct. I suppose that you know how to prove that functions of the type $$z\mapsto z_1\times z+z_0$$ are entire. Now, suppose that $$a,b,c,d,e,f\in\mathbb R$$ and that you consider the map$$\begin{array}{rccc}F\colon&\mathbb C&\longrightarrow&\mathbb C\\&x+yi&\mapsto&ax+by+e+(cx+dy+f)i.\end{array}$$Is it entire? It is clear that it is entire if and only if the map$$\begin{array}{ccc}\mathbb C&\longrightarrow&\mathbb C\\x+yi&\mapsto&ax+by+(cx+dy)i\end{array}$$is entire. But if it is entire, then it is holomorphic and the Cauchy-Riemann equations tell you then that $$a=d$$ and that $$c=-b$$. So, take $$z_1=a+bi$$ and $$z_0=e+fi$$. Then, $$F(z)=z_1\times z+z_0$$.