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.