This worksheet illustrates the effect of a linear transformation [math]T[/math] (in [math]R^2[/math]) on a unit vector. Drag the point [math]P[/math] around the unit circle, and see how its image [math]TP[/math] changes. Can you identify the eigenvectors and eigenvalues of [math]T[/math]? Click 'Show eigenvectors' at top-right to check your answer. You can enter a new linear transformation by entering values in the matrix [math]T[/math] at top-left.

Questions to consider: [list] [*]What happens if you try a rotation (like [math]T = \begin{bmatrix}0 & 1\\-1 & 0\end{bmatrix}[/math])? A reflection (like [math]T = \begin{bmatrix}0 & 1\\1 & 0\end{bmatrix}[/math])? [*]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? [/list] Clock image from Wikipedia by David Ilff - [url]https://en.wikipedia.org/wiki/File:Clock_Tower_-_Palace_of_Westminster,_London_-_May_2007.jpg[/url]