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.