8.6. RATIONAL CANONICAL FORM
399
with basis
f
e
1
; : : : ; e
n
g
. Let
A
D
.a
i;j
/
be the matrix of
T
with respect to
this basis, so
Te
j
D
P
i
a
i;j
e
i
.
Let
F
be the free
KOExŁ
–module with basis
f
f
1
; : : : ; f
n
g
and define
˚
W
F
!
V
by
P
j
h
i
.x/f
i
7!
P
j
h
i
.T /e
i
. Then
˚
is a surjective
KOExŁ
–module homomorphism. We need to find the kernel of
˚
.
The transformation
T
can be “lifted” to a
KOExŁ
–module homomor
phism of
F
by using the matrix
A
. Define
T
W
F
!
F
by requiring that
Tf
j
D
P
i
a
i;j
f
i
. Then we have
˚.Tf /
D
T ˚.f /
for all
f
2
F
.
I claim that the kernel of
˚
is the range of
x
T
. This follows from
three observations:
1.
range
.x
T /
ker
.˚/
.
2.
range
.x
T /
C
F
0
D
F
, where
F
0
denotes the set of
K
–linear
combinations of
f
f
1
; : : : ; f
n
g
.
3.
ker
.˚/
\
F
0
D f
0
g
.
The first of these statements is clear since
˚.xf /
D
˚.Tf /
D
T ˚.f /
for
all
f
2
F
. For the second statement, note that for any
h.x/
2
KOExŁ
,
h.x/f
j
D
.h.x/
h.T //f
j
C
h.T /f
j
:
Since multiplication by
x
and application of
T
commute, there is a poly
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 Fall '08
 EVERAGE
 Linear Algebra, Algebra, KŒx

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