$\newcommand{\Reals}{\mathbf{R}}\newcommand{\Vec}[1]{\mathbf{#1}}$ A linear transformation $T:\Reals^{2} \to \Reals^{2}$ has a standard matrix $A = [T]_{\operatorname{std}}^{\operatorname{std}}$. This program illustrates the geometric relationship between $T$ and $A$ by mapping the unit square marked with the letter F. 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 $A$. Type coefficients into the form fields to change $A$ directly, or click the corner at the base or spine of the F and drag the mouse.