Math 425, Spring 2011
Jerry L. Kazdan
Problem Set 5
DUE: Thurs. Feb. 24
Late papers will be accepted until 1:00 PM Friday
.
1. In
R
4
the vectors
U
1
:
= (
1
,
1
,
1
,
1
)
,
U
2
:
= (
1
,
1
,

1
,

1
)
,
U
3
:
= (
2
,

2
,
2
,

2
)
,
U
4
:
= (
1
,

1
,

1
,
1
)
are orthogonal, as you can easily verify.
a) Use these to ﬁnd an orthonormal basis
e
k
:
=
α
k
U
k
,
k
=
1
,...,
4.
b) Write the vector
v
:
= (
0
,

2
,
2
,
5
)
using this basis:
v
=
a
1
e
1
+
a
2
e
2
+
a
3
e
3
+
a
4
e
4
.
c) Find the projection,
Pv
, of
v
into the plane spanned by
U
2
and
U
3
.
d) Compute
k
Pv
k
.
2. Let
X
be a linear space with an inner product (not necessarily
R
n
) and let
P
:
X
→
X
be an
orthogonal projection
, so
P
2
=
P
and
P
=
P
*
. Write
V
for the image of
P
; it is the space into
which vectors are projected. Given
x
∈
X
, write
x
=
v
+
w
, where
v
=
Px
is the projection of
x
into
V
. Show that
w
is orthogonal to
V
.
3. Let
f
(
x
)
be a 2
π
periodic function. Use Fourier series to investigate ﬁnding 2
π
periodic
solutions of

u
00
(
x
)+
u
=
f
(
x
)
,
so we want
u
and all of its derivatives to be 2
π
periodic.
This is routine – and short. Expand
f
in a Fourier series, so
f
(
x
) =
∑
∞
k
=

∞
a
k
e
ikx
and seek the
solution as a Fourier series
u