An affin transformation of the complex plane called $f$ can be represented by its analytic expression in the canonical base $\left(1,i\right)$, the general form of that expression is:

$$\left\{\begin{array}{ccc}\hfill {x}^{\prime}& \hfill =\hfill & {t}_{1}+ax+by\hfill \\ \hfill {y}^{\prime}& \hfill =\hfill & {t}_{2}+cx+dy\hfill \end{array}\right.$$

That analytic expression will be represented by the list [t1+i*t2, a+i*c, b+i*d], ie: [ f(0), f(1)-f(0), f(i)-f(0)], that list will be briefly called (improperly) matrix of the transformation $f$. The two last elements of that list: [ a+i*c, b+i*d] represent the matrix of the linear part of $f$:Lf$=f-f\left(0\right)$.