CS205 Homework #2
Problem 1
[Heath 3.29, page 152] Let
v
be a nonzero
n
vector. The hyperplane normal to
v
is the
(n1)dimensional subspace of all vectors
z
such that
v
T
z
=
0
. A
reﬂector
is a linear
transformation
R
such that
Rx
=

x
if
x
is a scalar multiple of
v
, and
Rx
=
x
if
v
T
x
=
0
.
Thus, the hyperplane acts as a mirror: for any vector, its component within the hyperplane
is invariant, whereas its component orthogonal to the hyperplane is reversed.
1. Show that
R
=
2P

I
, where
P
is the orthogonal projector onto the hyperplane
normal to
v
. Draw a picture to illustrate this result
2. Show that
R
is symmetric and orthogonal
3. Show that the Householder transformation
H
=
I

2
vv
T
v
T
v
,
is a reﬂector
4. Show that for any two vectors
s
and
t
such that
s
6
=
t
and
k
s
k
2
=
k
t
k
2
, there is a
reﬂector
R
such that
Rs
=
t
Problem 2
Let
A
be a rectangular
m
×
n
matrix with full column rank and
m > n
. Consider the
QR
decomposition of
A
.
1. Show that
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 Fall '07
 Fedkiw
 Linear Algebra, Matrices, Householder transformation

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