Chapter 4.2, Problem 5E

Find the standard matrix representation for each of the following linear operators:

(a) L is the linear operator that rotates each x in ${ℝ}^2$ by $45°$ in the clockwise direction.

(b) L is the linear operator that reflects each vector x in ${ℝ}^2$ about the $x_1$ axis and then rotates it $90°$ in the counterclockwise direction.

(c) L doubles the length of x and then rotates it $30°$ in the counterclockwise direction.

(d) L reflects each vector x about the line $x_2=x_1$ and then projects it onto the $x_1$ −axis.