MATH 223, Linear Algebra
Winter, 2008
Assignment 9, due
in class
Wednesday, April 2, 2008
1. Let
V
be the vector space of realvalued functions defined and continuous
on [

π
2
,
π
2
]
.
(a) Show that, for any
f
∈
V
,
(
Z
π
2

π
2
f
(
x
) cos
xdx
)
2
≤
2
Z
π
2

π
2
f
(
x
)
2
cos
xdx.
Identify those function
f
for which equality holds.
(b) Show that, for any
f
∈
V
,
(
Z
π
2

π
2
f
(
x
) cos
2
xdx
)
2
≤
4
3
Z
π
2

π
2
f
(
x
)
2
cos
xdx.
Identify those function
f
for which equality holds.
2.
(a) Suppose that
V
is any inner product space, and that
~v
and
~w
are
orthogonal vectors in
V
. Show that

~v
+
~w

2
=

~v

2
+

~w

2
.
(b) Now suppose that
V
is finitedimensional and
W
is a subspace of
V
. For any
~u
∈
V
, show that
Proj
W
~u
is the "closest" vector to
~u
that's in
W
; that is, if
~x
=
Proj
W
~u
and
~
y
6
=
~x
but
~
y
∈
W
, then

~u

~x

<

~u

~
y

.
3. Let
V
=
C
5
with the standard inner product. Let
W
=
Span
1
1
0
1

i
1 +
i
,
2

4
i
4
1
1

6
i
4 +
i
,

2 + 4
i
6

2
i
6

2
i
2
2 + 4
i
.
(a) Find an orthonormal basis for each of
W
and
W
⊥
.
(b) Find
Proj
W
1
0
i
0
1
.
4. For each of the following Hermitian matrices
A
, find a unitary matrix
U
such that
U
T
AU
is diagonal, and find the diagonal matrix.
A
=
3
2
2
2
3
2
2
2
3
,
A
=
0
1 +
i
2 +
i
1

i
1
3

i
2

i
3 +
i
4
.
1