Chapter 5.3, Problem 73E

Let n be an even positive integer. In both parts of this Problem let V be the subspace of all vectors $\stackrel{\to }{x}$ in ${ℝ}^n$ such that $x_{j+2}=x_j+x_{j+1}$ , for all $j=1,\mathrm{...},n-2$ .
(In Exercise 70 we consider the special case $n=4$ .) Consider the basis $\stackrel{\to }{v},\stackrel{\to }{w}$ of V with
   $\stackrel{\to }{a}=\left[\begin{array}{l}1\\ a\\ a^2\\ ⋮\\ a^{n-1}\end{array}\right],\stackrel{\to }{b}=\left[\begin{array}{l}1\\ b\\ b^2\\ ⋮\\ b^{n-1}\end{array}\right]$ ,
where $\stackrel{\to }{a}=\frac{1+\sqrt{5}}{2}$ and $b=\frac{1-\sqrt{5}}{2}$ .
(In Exercise 4.3.72 we consider the case $n=4$ .)
a. Show that $\stackrel{\to }{a}$ is orthogonal to $\stackrel{\to }{b}$ .
b. Explain why the matrix P of the orthogonal projection onto V is a Hankel matrix. Sec Exercises 71 and 72.