Prove that linear transformation associated with the improper orthogonal matrix
( Cos[theta] Sin[theta] )
( Sin[theta] -Cos[theta] )
is a reflection through the line that makes an angle 1/2 theta with the x-axis.
Show that the composition of two such reflections, with angles theta1 and theta2, is a rotation. What is the angle of the rotation? Does the composition depend upon the order of the two reflections?