Page 113 - Advanced Linear Algebra
P. 113
The Isomorphism Theorems 97
It follows from the previous theorem that if dim²= ³ B , then
i
dim²= ³ ~ dim²= ³
since the dual vectors also form a basis for = i . Our goal now is to show that the
converse of this also holds. But first, let us consider an example.
Example 3.3 Let = be an infinite-dimensional vector space over the field
-
- ~ { ~ ¸ Á ¹, with basis . Since the only coefficients in are and , a
8
finite linear combination over is just a finite sum. Hence, is the set of all
=
-
finite sums of vectors in and so according to Theorem 0.12,
8
((= (F8 ( ³ ~ ( (
²
8
On the other hand, each linear functional = i is uniquely defined by
specifying its values on the basis . Since these values must be either or ,
8
specifying a linear functional is equivalent to specifying the subset of on
8
which takes the value . In other words, there is a one-to-one correspondence
between linear functionals on and all subsets of . Hence,
8
=
( i (= ( ~ ( ² F8 ( ³ ( ( 8 ( =
This shows that = i cannot be isomorphic to , nor to any proper subset of .
=
=
i
Hence, dim²= ³ dim²= . ³
We wish to show that the behavior in the previous example is typical, in
particular, that
i
dim²= ³ dim²= ³
with equality if and only if is finite-dimensional. The proof uses the concept
=
of the prime subfield of a field , which is defined as the smallest subfield of
2
2
the field . Since Á 2 , it follows that contains a copy of the integers
2
Á Á ~ b Á ~ b b Á Ã
2
2
If has prime characteristic , then ~ and so contains the elements
{ ~ ¸ Á Á Á Á Ã Á c ¹
which form a subfield of . Since any subfield of contains and , we see
2
2
-
that { and so { - is the prime subfield of . On the other hand, if has
2
2
characteristic , then 2 contains a “copy” of the integers and therefore also
{
the rational numbers , which is the prime subfield of . Our main interest in
r
2
the prime subfield is that in either case, the prime subfield is countable .
Theorem 3.12 Let be a vector space. Then
=
i
dim²= ³ dim²= ³
with equality if and only if is finite-dimensional.
=
