(a) Prove that an orthogonal 2x2 matrix must have the form

[a -b

b a]

OR

[a b

b -a]

where

[a

b] is a unit vector.

(b) Using part (a), show that every orthogonal 2x2 matrix is of the form

[cosx -sinx

sinx cosx]

OR

[ cosx sinx

sinx -cosx]

where 0<x < 2 pai.

(c) Show that every orthogonal 2x2 matrix corresponds to either a rotation or a reflection in R^2.

(d) Show that an orthogonal 2x2 matrix Q corresponds to a rotation in R^2 if detQ=1 and a reflection in R^2 if det Q=-1.