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Transformations of the Plane

Author:Mr. Ellis

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Below is a linear transformation matrix composed of two column vectors. The two column vectors are the unit vectors i and j, and will change as you drag points B and C around the plane. As you are dragging B and C around, notice how the Linear Transformation Matrix and its determinant change with respect to the area of the parallelogram. Click the refresh symbol in the top right of the Geogebra window to reset your applet at anytime. Answer the following questions: 1. After you've finished playing, click the refresh symbol in the top right of the Geogebra window. Now, let's try out some transformations.
  • Record the matrix with your unit vectors in standard position
  • Record the matrix after a 90 degree rotation around the origin
  • Record the matrix after a 180 degree rotation around the origin (you may need to refresh first)
  • Record the matrix after a 270 degree rotation
  • What can you conclude about how the matrix changes after a rotation?
2. Great job! Let's continue finding transformations. This time, we'll reflect over certain lines. Make sure to go ahead and hit that refresh symbol so your unit vectors are back in standard position.
  • Record the matrix with your unit vectors in standard position
  • Record the matrix after a reflection over the y-axis
  • Record the matrix after a reflection over the x-axis
  • Record the matrix after a reflection over the line y=x
  • What can you conclude about how the matrix changes after a reflection?
3. Set the magnitude of vector i to 3 and the direction to 0 degrees. Set the magnitude of vector j to be 3 and the direction to 90 degrees. How would you describe how the transformation that just took place? What is the new matrix? 4. What is the determinant of the matrix when the area of the parallelogram is 1? 5. What happens to the determinant of the matrix when you make the area 0?

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