Linearität der affinen Abbildung

Die affine Abbildung $\mathit{P}\text{'}=\mathit{T}(\mathit{P})$ ist durch folgendes Gleichungssystem dargestellt:

$$\begin{array}{rcl}\mathit{x}\text{'}& =& {\mathit{a}}_{11}\mathit{x}+{\mathit{a}}_{12}\mathit{y}\\ \mathit{y}\text{'}& =& {\mathit{a}}_{21}\mathit{x}+{\mathit{a}}_{22}\mathit{y}.\end{array}$$

Wir betrachten jetzt die Punkte ${\mathit{P}}_1=({\mathit{x}}_1,{\mathit{y}}_1)$ und ${\mathit{P}}_2=({\mathit{x}}_2,{\mathit{y}}_2)$ und erzeugen daraus einen dritten Punkt $\mathit{P}=\alpha {\mathit{P}}_1+\beta {\mathit{P}}_2=(\alpha {\mathit{x}}_1+\beta {\mathit{x}}_2,\alpha {\mathit{y}}_1+\beta {\mathit{y}}_2)$. Transformieren wir den Punkt $\mathit{P}$ in $\mathit{P}\text{'}$ mittels $\mathit{T}$, so erhalten wir:

$$\begin{array}{rcl}\mathit{x}\text{'}& =& {\mathit{a}}_{11}(\alpha {\mathit{x}}_1+\beta {\mathit{x}}_2)+{\mathit{a}}_{12}(\alpha {\mathit{y}}_1+\beta {\mathit{y}}_2)\\ \mathit{y}\text{'}& =& {\mathit{a}}_{21}(\alpha {\mathit{x}}_1+\beta {\mathit{x}}_2)+{\mathit{a}}_{22}(\alpha {\mathit{y}}_1+\beta {\mathit{y}}_2)\end{array}$$

oder

$$\begin{array}{rcl}\mathit{x}\text{'}& =& \alpha ({\mathit{a}}_{11}{\mathit{x}}_1+{\mathit{a}}_{12}{\mathit{y}}_1)+\beta ({\mathit{a}}_{11}{\mathit{x}}_2+{\mathit{a}}_{12}{\mathit{y}}_2)\\ \mathit{y}\text{'}& =& \alpha ({\mathit{a}}_{21}{\mathit{x}}_1+{\mathit{a}}_{22}{\mathit{y}}_1)+\beta ({\mathit{a}}_{21}{\mathit{x}}_2+{\mathit{a}}_{22}{\mathit{y}}_2).\end{array}$$

Dies ist

$$\mathit{P}\text{'}=\alpha \mathit{T}({\mathit{P}}_1)+\beta \mathit{T}({\mathit{P}}_2),$$

was die Eigenschaften einer linearen Transformation definiert:

Lineare Transformation

$$\mathit{T}(\alpha {\mathit{P}}_1+\beta {\mathit{P}}_2)=\alpha \mathit{T}({\mathit{P}}_1)+\beta \mathit{T}({\mathit{P}}_2).$$