Problem set 8
Math 110 Lec 2
Due:
Never. This problem set consists of 8 short problems and is just for “fun”.
1,2,3,4
can be solved already.
5,6
can be solved after 8/8.
7,8
can be solved after 8/9.
1.
Let
P
2
(
R
) denote the space of real polynomials of degree at most 2, together with the
inner product
h
f, g
i
:=
Z
1
0
f
(
x
)
g
(
x
) d
x.
Consider the map
S
:
P
2
(
R
)
→ P
2
(
R
)
,
p
(
x
)
7→
p
(1

x
)
.
Is
S
an isometry?
2.
(Axler 7.C.2) Suppose
T
is a positive operator on
V
. Suppose
v, w
∈
V
are such that
Tv
=
w
and
Tw
=
v.
Prove that
v
=
w.
3.
We have seen several times that, if
S
and
T
are diagonalizable operators that commute
(
ST
=
TS
) then
S
and
T
are simultaneously diagonalizable. No such result holds for the
Jordan normal form.
Let
A
=
1
1

1

1
∈
C
2
,
2
. It is clear that
A
and

A
commute. Prove that there is no
basis with respect to which
A
and

A
are simultaneously in Jordan normal form.
4.
Let
T
∈ L
(
V
) be a normal operator on a finitedimensional complex vector space
V
.
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 Fall '08
 GUREVITCH
 Linear Algebra, Algebra, Polynomials