168
3. PRODUCTS OF GROUPS
Proof.
It follows from Propostion
3.3.25
that any basis of a ﬁnite di
mensional vector space is ﬁnite. If a ﬁnite dimensional vector space has
two bases
X
and
Y
, then
j
Y
j ± j
X
j
, since
Y
is linearly independent
and
X
is spanning. But, reversing the roles of
X
and
Y
, we also have
j
Y
j ± j
X
j
.
n
Deﬁnition 3.3.27.
The unique cardinality of a basis of a ﬁnite–
dimensional vector space
V
is called the
dimension
of
V
and denoted
dim
.V /
. If
V
is inﬁnite–dimensional, we write dim
.V /
D 1
.
Corollary 3.3.28.
Let
W
be a subspace of a ﬁnite dimensional vector
space
V
.
(a)
Any linearly independent subset of
W
is contained in a basis of
W
.
(b)
W
is ﬁnite dimensional, and
dim
.W /
±
dim
.V /
.
(c)
Any basis of
W
is contained in a basis of
V
.
Proof.
Let
Y
be a linearly independent subset of
W
. Since no linearly
independent subset of
W
has more than dim
.V /
elements, by Proposition
3.3.25
,
Y
is contained in linearly independent set
B
that is maximal among
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 Fall '08
 EVERAGE
 Linear Algebra, Algebra, Vector Space, Natural number, Axiom of choice

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