Geometry of Linear Transformations

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 $A$.