This Mathwright Web Activity is for exploring the geometry of linear maps of the plane. There are two screens. On the first, shown below, there are separate graphs to display vectors in the domain and their images in the range. The user has a choice of several interaction styles:

• Select individual points with the mouse
• Create polygons by selecting vertices
• Create circles by selecting their centers and one incident point
• Drag the domain point and see the image move in real time

In this last mode, vectors may be displayed as points, or segments drawn from the origin, and the domain values may be restricted to a specified length.

The second page is very similar, but superimposes both domain and range on a single graph (see the screen shot below.)

This page has the added feature of permitting the image vector to be drawn as a segment starting from the position of the domain vector, when in the animated drag-point mode. That is mainly of value for visualizing flows subject to the differential equation x' = Ax.

Sample activities:

1. Explore the geometry of a variety of structured matrices: symmetric, diagonal, skew symmetric, etc. For this activity, an instructor can provide a work sheet with the different matrix types to explore, and can include reflections, rotations, dilation/contractions, etc.

2. Eigenvector visualization, with a symmetric matrix. Using the second page, with the superimposed domain and range, check off the box for show position vectors, but leave the others unchecked. Drag a point around in the graph, and look for places where the two displayed vectors line up. What is the algebraic significance of lining up in this way. Experiment with both circular and radial movements. Are there any points where this lining up occurs? If so, how many?

Next, clear the screen, and use the circle button to draw a circle centered at the origin, of radius 4. Then, click in the last check-off box, and restrict domain points to have length 4. Finally, without clearing the display, use the drag point feature again. Now as you move the mouse, the domain point will traverse the circle while the range point traces an ellipse. Again look for points where the two lines coincide. What do you notice about the locations of these points?