Loci: Resources

Linear Transformations of Points

by Marc Renault (Shippensburg Univ.)

Applet Description

This interactive Geogebra applet allows exploration of a linear transformation in terms of images of a set of points.  The Geogebra interface allows dragging of points and vectors to make for versatile explorations of basic linear algebra ideas. Suggested activities and exercises using the tool are included on the applet page and separately for easy printing.

Geogebra is an open source exploration/development tool that allows the creation of applets using a Java-based "player" understood by most browsers.  If you have difficulty loading the applet, see www.geogebra.org for more information on the system/browser/java requirements for viewing Geogebra applications.

Open the applet in a new browser window

Instructions and Activities

These instructions and suggested activities also appear on the html page with the applet. They are repeated here for easy viewing and printing.

Instructions

Click and drag the red and orange arrows to change the matrix A of the linear transformation.
(Alternatively, you can set the matrix by using the text boxes below the applet.)
You can show or hide points on the screen by using the checkboxes.
Drag the points u, v, w... around to see how their images Au, Av, Aw... change.

Exploration

1. Move u and v around and see how Au and Av change.  Confirm that the point Au is really the result of applying the matrix A to the point u.
2. Put several points on the screen and place them in a straight line. What do you see?
3. Which transformation makes Au = u, Av = v, Aw = w, etc.?
4. Find a transformation that reflects points across the x-axis. Same for y-axis.
5. Find a transformation that multiplies vectors by 2. So, Au = 2u, Av = 2v, etc.
6. Find a transformation that projects points "orthogonally" onto the x-axis. So, for example A(3, 5) = (3, 0) and A(-13, -7) = (-13, 0).  More generally, A(x, y) = (x, 0).
7. In the previous example, what is the range of the transformation?  The codomain?  What are their dimensions?
8. Find a matrix A so that Au always lies somewhere on the line y = x, no matter where u is.
9. Place points u, v, w on the screen so that u + v = w.  What do you notice about Au, Av, Aw?  Does this relationship hold for all linear transformations?
10. Place u at the point (1, 0), place v at the point (0, 1), and then try a few different transformations.  How are Au and Av related to the columns of A?  Can you prove this relationship?

Note for Windows users:
Ctrl + [click and drag] on the background to move the coordinate axes around.
Ctrl + [scroll wheel] to zoom in and out.