rotation matrix

Why is rotation a linear transformation? Let's check it out geometrically. In the applet above, the slider labeled T is the amount of counterclockwise rotation in radians. Let's call the rotation by T . The vectors and are arbitrary; you can change them by dragging the blue dots. Geogebra has helpfully computed , which is labeled as . Geogebra also gives and . One of the linearity properties is: . That is, we need to verify that if we

add, then rotate

we get the same result as if we

rotate, then add

Add, then rotate means: we compute , then rotate the result by T radians. Rotate, then add means: we rotate and (to get and , respectively), then add those two together. You can see visually (if you trust Geogebra) that either way, we end up with the vector labeled . So distributes over addition of vectors, which is one half of being linear. (Can you think of what it would look like to verify that respects scaling?)