Past exam of the year 1 of the computer science course of the University of Oxford / 2017 / Digital systems, linear algebra, introduction to formal proof / 5 / d

With column-vector homogeneous coordinates, the translation matrix isIt maps $(x,y,1{)}^T$ to $(x+a,y+b,1{)}^T$. The final coordinate incorporates the constant displacement into matrix multiplication.

Reflection in $x=0$ sends $(x,y)$ to $(-x,y)$. Therefore its homogeneous-coordinate matrix isIts square is the identity, as required for applying the same reflection twice.

Translate the mirror line to zero, reflect there, then translate back. Under the column-vector convention the rightmost operation acts first:This composition of affine transformations maps $(x,y)$ to $(4-x,y)$, fixing exactly the points with $x=2$.