Matrices as linear transformations (and eigenvectors)

Visualizing Linear Transformations

Consider the 2x2 matrix

. We can multiply

on the right by a column vector

to get another column vector in

. This operation is an example of a linear transformation (

transforms the vector

into

), and it can be visualized using the applet above. This applet can also be used to visualize changes of basis representations and eigenvectors.

Sliding the slider to the right "applies" the linear transformation to the basis vectors

and

, displayed above. Clicking "Show/Hide Lattice" reveals a lattice of

generated by the vectors

and

, and thus a visualization of how

transforms the entire plane. Clicking "Show/Hide V" reveals a third vector,

, that will also transform. By changing the relevant values in each box, you can change

, and