See how a 2×2 matrix A acts on the plane. The left panel is the domain; the right shows its image under T(x,y) = (ax+by, cx+dy). Draw with the mouse in the left panel and watch the image appear on the right. Green tracks the image of (1,0), blue the image of (0,1).
A modern JavaScript + Canvas port of a 2002 Java applet by Kim Chi Tran
(originally linearMap.java / linearGrid.java).