Skill: Model transformations using matrices

A $2\times2$ matrix transforms a point’s position vector. Its first column is where $(1,0)$ goes, and its second column is where $(0,1)$ goes. To apply a sequence of transformations, multiply the matrices in reverse order: the transformation performed first goes on the right.

Example: Reflect the point $P=(2,1)$ across the $x$-axis, then rotate it $90^\circ$ anticlockwise about the origin.

The reflection matrix is

$$F=\begin{pmatrix}1&0\\0&-1\end{pmatrix},$$

since it leaves the $x$-coordinate unchanged and changes the sign of the $y$-coordinate.

The $90^\circ$ anticlockwise rotation matrix is

$$R=\begin{pmatrix}0&-1\\1&0\end{pmatrix}.$$

Because the reflection happens first, the combined transformation is represented by $RF$:

$$RF = \begin{pmatrix}0&-1\\1&0\end{pmatrix} \begin{pmatrix}1&0\\0&-1\end{pmatrix} = \begin{pmatrix}0&1\\1&0\end{pmatrix}.$$

Now apply it to the position vector of $P$:

$$\begin{pmatrix}0&1\\1&0\end{pmatrix} \begin{pmatrix}2\\1\end{pmatrix} = \begin{pmatrix}1\\2\end{pmatrix}.$$

So the point ends at $(1,2)$. You can check the steps: reflection sends $(2,1)$ to $(2,-1)$, and the rotation then sends $(2,-1)$ to $(1,2)$.