# Iterated Linear Maps in the Plane

The next lab that has caught my attention and would like to explore further is the second to last chapter in our book: Iterated Linear Maps in the Plane. I like this one because, although very similar to our first lab, is very graphical and includes matrix operations.

Before, we were iterating the function $f(x) = ax+b$. This sort of iteration is called an affine map. In this lab we will be doing linear maps or maps where the constant term $b = 0$. Further, because we are in the plane, our function will have vector valued inputs and will have vector-valued output. So our function will look something like:

$$f(x, y) = (a_{11}x + a_{12}y, a_{21}x + a_{22}y)$$

Or similarly, we can write our equation:
$$\left({} \begin{array}{c} x_{n+1} \\{} y_{n+1} \end{array} \right){} = f(x_n, y_n) = A \left({} \begin{array}{c} x_n \\{} y_n \end{array} \right){}$$

where $A = \left({} \begin{array}{c} a_{11} & a_{12} \\{} a_{21} & a_{22} \end{array} \right){}$.

Some questions this lab seeks to answer are similar to our first lab: it will ask us to try some different variations of our matrix $A$ and/ or our initial values and see if we can notice a pattern.