Rotation \( \mathrm{R}= \left( \begin{array}{rr} \cos(\theta) & \sin(\theta) \\ -\sin(\theta) & \cos(\theta) \end{array} \right), \;\;\; \) Scaling \(\mathrm{S}= \left( \begin{array}{rr} a & 0 \\ 0 & b \end{array} \right), \;\;\; \) Translation \((x+t_x, y+t_y)\) (not linear)

Cannot express conveniently composition b/w several rotation, scaling, translation.

Then translation can be expressed linearly \(p'=\mathrm{T}\,p\), with \( p'= \underbrace{\left(\begin{array}{rrr}1&0&t_x \\ 0&1&t_y \\ 0&0&1\end{array}\right)}_{\mathrm{T}} \left(\begin{array}{r}x\\y\\1\end{array}\right)=\left(\begin{array}{c}x+t_x \\ y+t_y \\ 1\end{array}\right) \)

Similarily with rotation \( \mathrm{R}= \left( \begin{array}{rrr} \cos(\theta) & \sin(\theta) & 0\\ -\sin(\theta) & \cos(\theta) & 0 \\ 0&0&1 \end{array} \right), \;\;\; \) and scaling \(\mathrm{S}= \left( \begin{array}{rrr} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & 1 \end{array} \right) \).