This worksheet explores the effect of a linear transformation (in ), and its relationship with the eigenvectors, eigenvalues and determinant. Drag the point around the unit circle, and see how its image changes. Can you identify the eigenvectors and eigenvalues?

The large blue point is a point on the unit circle. Its image under the transformation T is shown as the smaller point. Drag around the unit circle and see how the image changes. Where are the eigenvectors? What (approximately) are the eigenvalues? Click 'Show eigenvectors' at top-right to check your answer. Click 'Show basis vectors' to see the effect of the transformation on the standard basis vectors , (also called ). You can enter a new linear transformation by entering values in the matrix at top-left. You can also drag the images of the basis vectors to change . Some interesting transformations to try:

• - enter this as
• - enter this as

Questions to consider:

• What do the eigenvalues represent, geometrically?
• What does the determinant represent, geometrically?
• What is the relationship between the determinant () and the eigenvalues?
• What does it mean geometrically if the determinant is negative? positive? zero?
• Look at where (if anywhere) the image of the unit circle intersects with the unit circle. What is the significance of these intersection points? Under what conditions on the eigenvalues do the curves intersect?
• What is the relationship between the images and the matrix ?

Clock image from Wikipedia by David Ilff - https://en.wikipedia.org/wiki/File:Clock_Tower_-_Palace_of_Westminster,_London_-_May_2007.jpg