Past exam of the Probability, statistics and options II course of the University of Oxford / 2011 / P / P4 / b

Let $r=\Vert u{\Vert }_2>0$ and choose $s=1$ when $u_1\ge 0$, otherwise $s=-1$. Set $w=u+sr{e}_1$, which is nonzero because $r+s{u}_1>0$. ComputeThus the reflecting term in $H(w)u$ is exactly $w$, givingIts first coordinate is nonzero and all remaining coordinates vanish. The choice of sign also avoids subtracting nearly equal vectors when the first component is large.