For a given real number theta find a 2 \times 2 matrix R_{theta} for which the linear function rho defined by rho(x) = R_{theta} x is the counterclockwise rotation of the plane through the angle of (radian) measure theta. Hint: First work out the four special cases where theta takes the values 0, pi/2, pi, and 3pi/2.
Find a 3 \times 3 matrix S for which the linear function sigma given by sigma(x) = S x is the reflection of R^{3} in the xz plane (where the 2^{nd} coordinate y = 0).
Possibly very difficult at this stage: Find a 3 \times 3 matrix T for which the linear function sigma given by sigma(x) = T x is the reflection of R^{3} in the plane 2 x - 2 y + z = 0 .