7
Let
T
be the linear operator on
R
4
which is represented in the standard ordered basis by the matrix
M
=
0
0
0
0
a
0
0
0
0
b
0
0
0
0
c
0
Under what conditions on
a, b, c
is
T
diagonalizable?
7
Let
T
be a linear operator on the
n
dimensional vector space
V
, and suppose that
T
has
n
distinct characteristic
values. Prove that
T
is diagonalizable.
7
Let
A
and
B
∈
F
n
×
n
. Prove that if (
I

AB
) is invertible then (
I

BA
) is invertible with inverse
I
+
B
(
I

AB
)

1
A
.
7
Prove for
A, B
∈
F
n
×
n
that
AB
and
BA
have precisely the same characteristic values in
F
.
7
Suppose that
A
∈
R
2
×
2
is symmetric. Prove that
A
is similar over
R
to a diagonal matrix.
7
Let
N
∈
C
2
×
2
such that
N
2
= 0. Prove that either
N
= 0 or
N
is similar over
C
to
0
0
1
0
.
7
If
A
∈
C
2
×
2
show that
A
is similar over
C
to either a diagonal matrix or a matrix of the form
s
0
1
s
.
7
Let
V
=
F
n
×
n
. Let
A
be a fixed element of
V
. Let
T
be the linear operator on
V
given by left multiplication
by the matrix
A
. Is it true that
A
and
T
have the same characteristic values?
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 Spring '09
 Linear Algebra, Vector Space, linear operator

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