Past exam of the year 1 of the computer science course of the University of Oxford / 2018 / Digital systems, linear algebra, introduction to formal proof / 6 / b

A unit vector along the line is $v=(3,4{)}^T/5$. Reflection fixes its component parallel to $v$ and negates its perpendicular component, so the reflection matrix isIndeed $H(3,4{)}^T=(3,4{)}^T$, $H(-4,3{)}^T=-(-4,3{)}^T$ and $H^2=I$. These characterize the orthogonal reflection in that line.

The line passes through $p=(0,1{)}^T$ and has the same direction as in part (i). Translate by $-p$, reflect by $H$, then translate back. In homogeneous coordinates the affine reflection isAct on column $(x,y,1{)}^T$, reading the product from right to left. Every point with $y=4x/3+1$ is fixed. The translation column is $p-Hp$, as for any reflection in an affine hyperplane.