Past exam of the options II course of the University of Oxford / 2011 / P / P4 / b / Solution

Let $r=\Vert u{\Vert }_2>0$, and choose $s=1$ if $u_1\ge 0$ and $s=-1$ otherwise. SetThis vector is nonzero, because $w_1$ has magnitude $|u_1|+r$. Direct computation givesso the Householder transformation satisfiesThe first entry is nonzero and every other entry is zero. The sign choice also avoids cancellation when forming the first entry of $w$, which is useful in numerical computation.