Loci: Resources

Images of F

by Steve Phelps (Madeira High School)

Applet Description

This interactive Geogebra applet allows exploration of a linear transformation in terms of images of a closed figure that happens to be in the shape of the letter F initially.  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 page 2 of this posting and as a separate pdf file 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.

Click here or on the screen shot above to open the applet in a separate window.

Investigations

In the Images of F applet on page 1, the columns of the matrix are the elementary vectors e₁ and e₂. The blue figure is a pre-image initially in the shape of an F. The green figure is the image of the blue F under the transformation given by the matrix.

To answer the questions below, you can drag the tips of the elementary vector to set up the appropriate matrices. You may also need to drag the vertices of the blue F as well.

Warm Up:  Set up the following matrices one at a time. Pay particular attention to the lattice points of F and to the lattice points of the image of F.

Investigation 1: Drag the tips of the elementary vectors to set up the following matrices. Discuss the transformations and the resulting image of F under these matrix transformations.

1. transformations with matrices of the form

   \left[ \begin{array}{cc} k & 0 \\ 0 & 1 \end{array} \right]
2. transformations with matrices of the form

   \left[ \begin{array}{cc} 1 & 0 \\ 0 & k \end{array} \right]
3. transformations with matrices of the form

   \left[ \begin{array}{cc} k & 0 \\ 0 & k \end{array} \right]
4. transformations with matrices of the form

   \left[ \begin{array}{cc} 0 & k \\ k & 0 \end{array} \right]
5. transformations with matrices of the form

   \left[ \begin{array}{cc} 1 & 0 \\ k & 1 \end{array} \right]
6. transformations with matrices of the form

   \left[ \begin{array}{cc} 1 & k \\ 0 & 1 \end{array} \right]

Investigation 2: Drag the tips of the elementary vectors to set up matrices that will perform the following transformations. Pay attention to the orientation of the vectors.

1. Reflection over the  x – axis
2. Reflection over the  y – axis
3. 90-degree clockwise rotation around the origin
4. Half-turn around the origin
5. 90-degree counterclockwise rotation around the origin
6. Reflection over the line  y = x 
7. Reflection over the line  y = -x