**Purpose**: This program illustrates the geometric
relationship between a $2 \times 2$ real matrix $A$ and the plane
linear transformation $T$ with standard matrix $A$ (namely
multiplication by $A$). The effect of $T$ is shown on the unit
square marked with the letter F.
If$\newcommand{\Reals}{\mathbf{R}}\newcommand{\Vec}[1]{\mathbf{#1}}$
the eigenspaces of $T$ are real, they are shown in pink, either as
the entire plane, or as two lines, or as one line.

**Usage**: There are two ways to change the matrix
$A$. First, you can use the mouse to modify the F: click once on
the *lower right corner* of the green box containing the F
(the “base of the F”) to “activate”
$T(\Vec{e}_{1})$; move the mouse, and click again to
deactivate. Similarly, click on the *upper left corner* of
the green box (the “spine of the F”) to
activate/deactivate and move $T(\Vec{e}_{2})$.

You can also type directly into the form fields to set the
entries of $A$. Press `Tab` or `Return` to
update the canvas.

**Explanation**: A linear transformation
$T:\Reals^{2} \to \Reals^{2}$ has a *standard matrix*
$A = [T]_{\operatorname{std}}^{\operatorname{std}}$. The base of
the F is $T(\Vec{e}_{1})$, the first column of $A$. The spine of
the F is $T(\Vec{e}_{2})$, the second column of $A$.