Math 235 Assignment 4
Due 9:15am, Wednesday Feb 14, 2007.
1. From the Text
§
5.3 #4, #18, and #26.
§
5.4 #16 and #20.
2.
Rubric:
1
In the following questions, the parts should be done in the
order in which they are listed.
(a) Let
P
:
V
→
V
be a linear operator di
ff
erent from
I
,
the identity
operator in
V
,
and from
O
,
the zero operator on
V
,
and such that
P
2
=
P
(
P
is said to be an
idempotent
operator).
i. Prove that
I

P
is an idempotent operator on
V
.
ii. Recall that the
range space
of
P
is
P
V
:=
{
P
u
:
u
∈
V}
.
Prove
that
A. ker(
P
) = (
I

P
)
V
,
B. ker(
I

P
) =
P
V
.
(
Hint:
Note that
P
(
I

P
) =
O
and that
I
= (
I

P
) +
P
.
The
latter is called a
partition of unity
.)
iii. Prove that ker
P
∩
ker(
I

P
) =
{
0
}
.
iv. Find the set of distinct eigenvalues of
P
.
(This is called the
spectrum
of
P
,
and is denoted by
spec
(
P
)
.
)
v. Prove that
P
is not invertible.
vi. Prove that if
v
∈
V
then there exists a unique pair (
v
1
,
v
2
)
of vectors with
v
1
∈
ker
P
and
v
2
∈
ker(
I

P
) such that
v
=
v
1
+
v
2
.
(
Comment:
In this case we say that
V
is a
direct sum
of the
subspaces ker
P
and ker(
I

P
)
,
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 Fall '08
 ANDREWCHILDS
 Math, Linear Algebra, Algebra, Vector Space, Ker, b. Ker, A. ker

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