Chapter 5.3, Problem 28E

Consider an $n\times n$ matrix A. Show that A is an orthogonal matrix if (and only if) A preserves the do product, meaningthat $\left(A\stackrel{\to }{x}\right)\cdot \left(A\stackrel{\to }{y}\right)=\stackrel{\to }{x}\cdot \stackrel{\to }{y}$ forall $\stackrel{\to }{x}$ and $\stackrel{\to }{y}$ in ${ℝ}^n$ . Hint: In Summary 5.3.8. show that statement(iv)implies (vi), and (vi) implies (ii).