Visualizing Linear Transformations of Points

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

Click and drag the red and orange arrows to change the matrix of the linear transformation.
Drag the points around to see how their images change.
You can add up to six points by using the checkboxes

  1. Move u and v around and see how Au and Av change.
  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).
  7. In the previous example, what is the dimension of the range?
  8. Find a transformation that projects points onto the line y = x.
Note:
Ctrl + [click and drag] on the background to move the coordinate axes around.
Ctrl + [scroll wheel] to zoom in and out.

Marc Renault, Created with GeoGebra