IB Maths AI 3.8 Notes

Geometric transformations

In this section, we study how matrices can be used to transform points in two dimensions. A transformation changes the position, orientation, or size of an object. Using matrices allows these changes to be described clearly and efficiently.

A point $(x,y)$ can be written as the column vector $\begin{pmatrix}x\\y\end{pmatrix}$. A matrix transformation acts on this vector to produce a new image point.

In general, a transformation can be written as:

$\begin{pmatrix}x'\\y'\end{pmatrix}=A\begin{pmatrix}x\\y\end{pmatrix}+\begin{pmatrix}e\\f\end{pmatrix}$

where $A$ is a $2\times2$ matrix and $\begin{pmatrix}e\\f\end{pmatrix}$ is a translation vector.