MATH 235
Inner Product Spaces
Assignment 7
Hand in questions 4,5,6,7,8,9 by 9:30 am on Wednesday November 15, 2006.
1. If
V
=
P
3
with the inner product
< f,g >
=
R
1

1
f
(
x
)
g
(
x
)
dx
, apply the GramSchmidt
algorithm to obtain an orthogonal basis from
B
=
{
1
,x,x
2
,x
3
}
.
2. Consider the vector space
C
[0
,
1] of all continuously diﬀerentiable functions deﬁned
on the closed interval [0
,
1]. The inner product in
C
[0
,
1] is deﬁned by
< f,g >
=
R
1
0
f
(
x
)
g
(
x
)
dx.
Find the inner products of the following pairs of functions and state
whether they are orthogonal
(a)
f
(
x
) = cos(2
πx
) and
g
(
x
) = sin(2
πx
)
(b)
f
(
x
) =
x
and
g
(
x
) = e
x
(c)
f
(
x
) =
x
and
g
(
x
) = 3
x
3. Show that
<
u
,
v
>
=
u
1
v
1
+ 8
u
2
v
2
deﬁnes an inner product in
R
2
. Under this inner
product, ﬁnd a vector
y
which is orthogonal to
x
= (1
,
1)
T
.
4. Deﬁne an inner product on
R
2
by
<
(
x,y
)
,
(
x
0
,y
0
)
>
= 2
xx
0

(
xy
0
+
yx
0
) + 2
yy
0
and let
e
1
= (1
,
0) and
e
2
= (0
,
1).
(a) Prove directly that
<
v
,
v
>
≥
0 for all
v
∈
R
2
.
(b) Find

e
1

,

e
2

and
<
e
1
,
e
2
>
with respect to the inner product deﬁned above.
(c) Apply the GramSchmidt process to
e
1
,
e
2
to obtain a basis for
R
2
which is
orthonormal with respect to the inner product deﬁned above.
5. (a) Let
A
=
±
a
1
a
2
a
3
a
4
²
and
B
=
±
b
1
b
2
b
3
b
4
²
be real matrices. Show that
< A,B >
=
a
1
b
1
+
a
2
b
3
+
a
3
b
2
+
a
4
b
4
is not an inner product on
M
22
.
(b) Let
p
(
x
) and
q
(
x
) be polynomials in
P
2
. Show that
< p,q >
=
p
(0)
q
(0) +
p
(
1
2
)
q
(
1
2
) +
p
(1)
q
(1) is an inner product on
P
2
.
6. Let
W
= Span
{
1
,
cos(
x
)
,
cos(2
x
)
} ∈
C
[

π,π
] with inner product
< f,g >
=
R
π

π
f
(
x
)
g
(
x
)
dx
. It is known that
{
1
√
2
π
,
1
√
π
cos(
x
)
,
1
√
π
cos(2
x
)
}
is an or
thonormal basis for
W
. A function
f
∈
W
satisﬁes
R
π

π
f
(
x
)
dx
=
a
,
R
π

π
f
(
x
) cos(
x
)
dx
=
b
and
R
π

π
f
(
x
) cos(2
x
)
dx
=
c
. Find an expression for

f

in terms
of
a,b
and
c
.
1