(b) Given a 2 × 2 rotation matrix R represented as
$\begin{aligned} R = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}, \end{aligned}$
show that it preserves the standard inner product, i.e., for all x, y ? R², we have x?y = (Rx)?(Ry).
(c) Now, let us consider an inner product in R² defined by the 2 × 2 matrix
$\begin{aligned} D = \begin{bmatrix} 2 & 1 \\ 1 & 3 \end{bmatrix}. \end{aligned}$
Find the matrix D' (in terms of R and D) such that the inner product defined by D is preserved
under the rotation by R, i.e., for all x, y ? R², we have x?Dy = (Rx)?D'(Ry).
(d) For ? = ?/4, compute D' explicitly.
(e) Consider u = [1, 1]? ? R² and v = [2, -1]? ? R². Compute the angle between u and v under the
inner product defined by D, and the angle between Ru and Rv under the inner product defined by
D'.
