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 [math]R_T[/math].[br][br]The vectors [math]v_1[/math] and [math]v_2[/math] are arbitrary; you can change them by dragging the blue dots.[br][br]Geogebra has helpfully computed [math]v_1+v_2[/math], which is labeled as [math]v_{sum}[/math]. Geogebra also gives [math]v_3=R_T(v_1)[/math] and [math]v_4=R_T(v_2)[/math].[br][br]One of the linearity properties is: [math]R_T(v_1+v_2)=R_T(v_1)+R_T(v_2)[/math]. That is, we need to verify that if we[br][center]add, then rotate[/center]we get the same result as if we[br][center]rotate, then add[/center][br]Add, then rotate means: we compute [math]v_{sum}[/math], then rotate the result by T radians.[br]Rotate, then add means: we rotate [math]v_{1}[/math] and [math]v_{2}[/math] (to get [math]v_{3}[/math] and [math]v_{4}[/math], respectively), then add those two together.[br][br]You can see visually (if you trust Geogebra) that either way, we end up with the vector labeled [math]v_{sum2}[/math]. So [math]R_T[/math] distributes over addition of vectors, which is one half of being linear. (Can you think of what it would look like to verify that [math]R_T[/math] respects scaling?)