Let $a$ and $b$ be complex numbers.
Let $T_{a}(z)=za$ and $S_{b}(z)=bz^{*}$ be $R^{2}\rightarrow R^{2}$ transformations.
How can we show than every linear $R^{2}\rightarrow R^{2}$ transformation can be written as a unique sum $T_{a}+S_{b}$?
We know that $T_{a}$ is a scaling plus rotation transformation, and $S_{b}$ is a reflextion, rotation and scaling transformation (in that order).
Take an arbitrary linear transformation $f$:$$f(x,y)=(\alpha x+\beta y,\gamma x+\delta y).\tag1$$Then you want to write as $T_a+S_b$. If $a_1=\operatorname{Re}a$, $a_2=\operatorname{Im}a$, $b_1=\operatorname{Re}b$, and $b_2=\operatorname{Im}b$, you want to express $(1)$ as\begin{multline}(a_1x-a_2y,a_2x+a_1y)+(b_1x+b_2y,b_2x-b_1y)=\\=\bigl((a_1+b_1)x+(-a_2+b_2)y,(a_2+b_2)x+(a_1-b_1)y\bigr),\end{multline}So, solve the system$$\left\{\begin{array}{l}a_1+b_1=\alpha\\-a_2+b_2=\beta\\a_2+b_2=\gamma\\a_1-b_1=\delta.\end{array}\right.$$