Section 5.4: 15:
To prove the CayleyHamilton theorem for matrices, we can use the
isomorphism between
n
×
n
matrices
M
n
×
n
(
F
) and linear transformations
L
(
F
n
) given by
A
7→
L
A
. The key additional fact that we need about this isomorphism, is that it also
preserves multiplication, and thus a simple induction shows that for any polynomial
g
(
t
)
∈
P
(
F
),
g
(
L
A
) =
L
g
(
A
)
. We also know, by deﬁnition, that the characteristic polynomial
f
(
t
)
of
A
and of
L
A
are the same. Thus by CayleyHamilton for linear operators, we know
0 =
f
(
L
A
) =
L
f
(
A
)
, and since the map
A
7→
L
A
is onetoone, we get
f
(
A
) = 0.
Section 5.4: 20:
If
V
is
T
cyclic with generator
v
0
, and if
U
commutes with
T
, then
we can write
U
(
v
0
) =
g
(
T
)(
v
0
) for some polynomial
g
(
t
) (Problem 13). Then since
U
and
T
commute, we know by a simple induction that
U
and
T
j
also commute so we can
write
U
(
T
j
(
v
0
)) =
T
j
(
U
(
v
0
)) =
T
j
(
g
(
T
)
v
0
), but
g
(
T
) also commutes with
T
j
so we get
U
(
T
j
(
v
0
)) =
T
j
(
g
(
T
)
v
0
) =
g
(
T
)(
T
j
v
0
). Since the set of vectors
T
j
v
0
spans
V
, this implies
that
Uv
=
g
(
T
)
v
for any vector
v
∈
V
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 Spring '10
 FUCKHEAD
 Linear Algebra, direct sum, Eigenspaces, direct sum decomposition

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