Chapter 2.3, Problem 30E

Consider the lines P and Q in ${ℝ}^2$ in the accompanying figure. Consider the linear transformation $T\left(\stackrel{\to }{x}\right)={\text{ref}}_Q\left({\text{ref}}_p\left(\stackrel{\to }{x}\right)\right)$ ; that is, we first reflect $\stackrel{\to }{x}$ about P andthen we reflect the result about Q.

a. For the vector $\stackrel{\to }{x}$ given in the figure, sketch $T\left(\stackrel{\to }{x}\right)$ .What angle do the vectors $\stackrel{\to }{x}$ and $T\left(\stackrel{\to }{x}\right)$ enclose?What is the relationship between the lengths of $\stackrel{\to }{x}$ and $T\left(\stackrel{\to }{x}\right)$ ?
b. Use your answer in part (a) to describe the transformation T geometrically, as a reflection, rotation,shear, or projection.
c. Find the matrix of T.
d. Give a geometrical interpretation of the linear transformation $L\left(\stackrel{\to }{x}\right)={\text{ref}}_P\left({\text{ref}}_Q\left(\stackrel{\to }{x}\right)\right)$ , and find thematrix of L.