# Math Insight

### Applet: Linear transformation in two dimensions

The linear transformation $\vc{T}=A\vc{x}$, with $A$ specified in the upper left hand corner of the applet, is illustrated by its mapping of a quadrilateral. The original quadrilateral is shown in the $xy$-plane of the left panel and the mapped quadrilateral is shown in the $x'y'$-plane of the right panel. You can change the linear transformation by typing in different numbers and change either quadrilateral by moving the points at its corners. The determinant of $A$ (shown in upper left hand corner) determines how much $\vc{T}$ stretches or compresses area and whether or not it reverses the orientation of the region. The orientation of each quadrilateral can be determined by examining the order of the colors while moving in a counterclockwise direction around its perimeter. Use the + and - buttons of each panel to zoom in and out. (When $\det A =0$, you cannot drag points in the right panel.)

Applet file: linear_transformation_2d.ggb