Derive the transformation matrix for the reflection of points in the line $y = 2 - x$

Question

I have to use the basic transformations (translation,scaling and rotation) to derive the reflection. How can I proceed to begin ?

Answer 1

I'll assume you know how to find the transformation matrix of a reflection in a line through the origin. In your case the line $l$ it's not through the origin, so you must first fix an arbitrary point $\textbf p$ on the line $l$ and apply the $T_{-\textbf p}$ translation, in this way you obtain a line $l'$ through the origin and parallel to $l$. Now apply the reflection $S_\phi$ with respect to $l'$, where $\phi$ is the angle between $l$ and the $x$ axis. Finally, you apply the inverse translation $T_{\textbf p}$. The reflection is given by $$S = T_{\textbf p} \circ S_\phi \circ T_{-\textbf p}$$ and it does not depend on the choice of the point $\textbf p$ on the line $l$.