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Kunze Linear Algebra
· Problem 7.2.1
Rational and Jordan Forms
Mathematics — Methods · cyclic-decomposition
Problem
Let
T
T
T
be the linear operator on
F
2
F^2
F
2
which is represented in the standard ordered basis by the matrix
[
0
0
1
0
]
.
\begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix}.
[
0
1
0
0
]
.
Let
α
1
=
(
0
,
1
)
\alpha_1 = (0, 1)
α
1
=
(
0
,
1
)
. Show that
F
2
≠
Z
(
α
1
;
T
)
F^2 \neq Z(\alpha_1; T)
F
2
=
Z
(
α
1
;
T
)
and that there is no non-zero vector
α
2
\alpha_2
α
2
in
F
2
F^2
F
2
with
Z
(
α
2
;
T
)
Z(\alpha_2; T)
Z
(
α
2
;
T
)
disjoint from
Z
(
α
1
;
T
)
Z(\alpha_1; T)
Z
(
α
1
;
T
)
.
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