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3.5. LINEAR ALGEBRA OVER
Z
193
Proof.
Condition (a) trivially implies condition (b). If (b) holds, then each
standard basis element
O
e
j
is in
Z
f
v
1
;:::;v
n
g
,
O
e
j
D
X
i
a
i;j
v
i
:
(3.5.1)
Let
A
D
.a
i;j
/
. The
n
equations
3.5.1
are equivalent to the single matrix
equation
E
D
PA
, where
E
is the
n
–by–
n
identity matrix. Thus,
P
is
invertible with inverse
A
2
Mat
n
.
Z
/
.
If
P
is invertible with inverse
A
2
Mat
n
.
Z
/
, then
E
D
PA
implies
that each standard basis element
O
e
j
is in
Z
f
v
1
;:::;v
n
g
, so
f
v
1
;:::;v
n
g
generates
Z
n
. Moreover, ker
.P/
D f
0
g
means that
f
v
1
;:::;v
n
g
is linearly
independent.
n
We can now combine Proposition
3.5.9
and Lemma
3.5.12
to obtain
our main result about bases of subgroups of
Z
n
.
Theorem 3.5.13.
If
N
is a subgroup of
Z
n
, then
N
is a free abelian group
of rank
s
±
n
. Moreover, there exists a basis
f
v
1
;:::;v
n
g
of
Z
n
, and there
exist positive integers
d
1
;d
2
;:::;d
s
, such that
d
i
divides
d
j
if
i
±
j
and
f
d
1
v
1
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 Fall '08
 EVERAGE
 Linear Algebra, Algebra, Equations

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