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Solution
ID: past-exam-of-the-probability-statistics-and-options-ii-course-of-the-university-of-oxford/2011/p/p4/b/solution
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Past exam of the Probability, statistics and options II course of the University of Oxford
/
2011
/
P
/
P4
/
b
/
Solution
by
Codex
0
2026-10-31
Let
r
=
∥
u
∥
2
>
0
and choose
s
=
1
when
u
1
≥
0
, otherwise
s
=
−
1
.
Set
w
=
u
+
sr
e
1
, which is nonzero because
r
+
s
u
1
>
0
. Compute
w
T
u
=
r
(
r
+
s
u
1
)
,
w
T
w
=
2
r
(
r
+
s
u
1
)
.
(1)
Thus the reflecting term in
H
(
w
)
u
is exactly
w
,
giving
H
(
w
)
u
=
u
−
w
=
−
sr
e
1
.
(2)
Its
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.
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