Here are some things to try with the applet. Try each thing for several matrices A.

Choose A by changing a,b,c,d, and experiment with the applet. In each case, try to relate what you see with not only the computations you are learning about, but also with the concepts and language of linear algebra that you are studying.

• When you click a vector, try clicking the same point using 'circles' and 'boxes'. Think about the result for several points and several matrices.
• Pick any 2 by 2 matrix that occured in the text or homework and use it for experiments. Try to illustrate or visualize whatever computation was done in the example.
• When you are learning about a new term like "column space", try looking at the applet and talking (if nobody is around) about the new word and to what extent it describes part of the picture you are looking at.
• For several matrices A, try to find an approximate solution to Ax=b by trial and error when b is, say, (1,1), or (0,1). You can click several points to do this, or think about the results of the 'boxes' or 'circles' options to get a clue where x might be. (note: if you can find an x for which Ax is something like (56.35,57.9) then you have nearly solved Ax=(1,1). Why?)
• Using 'vectors', click again on Ax, repeating this several times to iterate the mapping. Do you see any pattern? Does the pattern depend on A?
• Compare the column space of A = [.6 1; 1.2 2+u] for various u, including 0. What is special about 0? What looks different in the 'boxes' option, for u > 0 versus u < 0?
• Try to find vectors x for which x and Ax are aligned. What is this called? Do all (2 by 2 real) matrices have such things?
• Once you've seen what your matrix A does to the boxes and circles that the applet draws, try to figure out what A does to things the applet does not know how to draw, like pentagons, or faces.
• For each topic in the class, find an example A and use the applet to make a simple picture which illustrates something about the topic.
• Write your own applet that is better than this one. Maybe you can do 3 or 4 dimensions, or solve Ax=b numerically, or plot what A does to a curve that you scribble with the mouse, or whatever.

Don't miss the map applet, which is not restricted to linear transformations.