Problem set 4
Math 110 Lec 2
Due:
Wed. 7/20
1,2
can be solved after 7/11.
3,4
can be solved after 7/12.
5,6
can be solved after 7/13.
7,8
can be solved after 7/14.
1.
(Axler 3.F.14) Define
T
:
P
(
R
)
→ P
(
R
)
,
(
Tp
)(
x
) :=
x
2
p
(
x
) +
p
00
(
x
)
.
(i) Define
ϕ
∈ P
(
R
)
0
by
ϕ
(
p
) :=
p
0
(4)
.
Describe
T
0
(
ϕ
).
(ii) Define
ϕ
∈ P
(
R
)
0
by
ϕ
(
p
) :=
R
1
0
p
(
x
) d
x.
Evaluate (
T
0
ϕ
)(
x
3
)
.
2.
Let
V
be a vector space.
(i) Prove that
T
:
V
0
×
V
0
→
(
V
×
V
)
0
,
T
(
ϕ, ψ
)(
v, w
) :=
ϕ
(
v
) +
ψ
(
w
)
is an isomorphism.
(ii) Consider the
diagonal map
Δ :
V
→
V
×
V, v
7→
(
v, v
)
.
After identifying (
V
×
V
)
0
with
V
0
×
V
0
as in (i), the dual map becomes a map
Δ
0
:
V
0
×
V
0
→
V
0
.
Describe this map.
3.
(Axler 5.A.26, 5.A.27) Suppose
V
is finitedimensional and
T
∈ L
(
V
).
(i) Assume that every nonzero vector in
V
is an eigenvector of
T
. Prove that
T
is a scalar
multiple of the identity operator.
(ii) Assume that every hyperplane (subspace
U
⊆
V
of dimension dim(
U
) = dim(
V
)

1) is
invariant under
T
. Prove that
T
is a scalar multiple of the identity operator.
Continued on the back
4.
(Axler 5.A.35, 5.A.36) Let
T
∈ L
(
V
) be an operator and
U
⊆
V
be an invariant subspace.
Let
T
denote the quotient operator
T
∈ L
(
V/U
)
,
T
(
v
+
U
) :=
T
(
v
) +
U.
(i) Assume that
V
is finitedimensional. Prove that every eigenvalue of
T
is also an eigen
value of
T
.
(ii) Give a counterexample to show that when
V
is infinitedimensional,
T
can have an eigen
value that is not an eigenvalue of
T
.
5.
(Axler 5.B.20) Suppose
V
is a finitedimensional complex vector space and
T
∈ L
(
V
).
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 Fall '08
 GUREVITCH
 Math, Linear Algebra, Algebra