Let e_{1} = (1, 0) and e_{2} = (0, 1) be the standard basis of the Cartesian plane. Find the matrix relative to this basis of the rotation about the origin through the angle theta.
Find the matrix with respect to the basis e in the previous exercise of the reflection in the line through the origin that has angle of elevation theta/2 (counterclockwise from the positive direction along the first coordinate axis).
When g is the basis of the Cartesian plane with g_{1} = (2, 2) and g_{2} = (-2, 2), what is the matrix of the rotation about the origin through the angle pi/2 relative to g ?
When h is the basis of the Cartesian plane with h_{1} = (a, b) and h_{2} = (c, d), what is the matrix of the rotation about the origin through the angle pi/2 relative to h ? (Assume that a d - b c <> 0. )