ECEN 601: Assignment 7
Problems:
1. Let
V
be the subspace of
R
[
x
] of polynomials of degree at most 3. Equip
V
with the inner
product
(
f
|
g
)
=
integraldisplay
1
0
f
(
t
)
g
(
t
)
dt.
(a) Find the orthogonal complement of the subspace of constant polynomials.
(b) Apply the Gram-Schmidt process to the basis
{
1
,x,x
2
,x
3
}
.
2. Let
W
be a finite-dimensional subspace of an inner product space
V
, and let
E
be the
orthogonal projection of
V
on
W
. Prove that
(
Eα
|
β
)
=
(
α
|
Eβ
)
for all
α
,
β
in
V
.
3. Let
S
be a subset of an inner product space
V
. Show that (
S
⊥
)
⊥
contains the subspace
spanned by
S
. When
V
is finite-dimensional, show that (
S
⊥
)
⊥
is the subspace spanned by
S
.
4. Let
X
and
Y
be vector spaces over the same set of scalars. Let
LT
[
X,Y
] denote the set of
all linear transformations from
X
to
Y
. Let
L
and
M
be operators from
LT
[
X,Y
]. Define
an addition operator between
L
and
M
as
(
L
+
M
)(
x
) =
L
(
x
) +
M
(
x
)
for all
x
∈
X
. Also define scalar multiplication by
(
aL
)(
x
) =
a
(
L
(
x
))
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- Fall '08
- Staff
- Linear Algebra, Inner product space, orthogonal complement
-
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