Page 26 - Advanced Linear Algebra
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10 Advanced Linear Algebra
(~ 7)7 !
where 7 ! is the transpose of . This relation is easily seen to be an equivalence
7
relation and we will devote some effort to finding canonical forms for
(
)
congruence. For some base fields such as , or a finite field , this is
-
sd
(
relatively easy to do, but for other base fields such as r ) , it is extremely
difficult.
Zorn's Lemma
In order to show that any vector space has a basis, we require a result known as
Zorn's lemma. To state this lemma, we need some preliminary definitions.
Definition A partially ordered set is a pair ²7Á ³ where is a nonempty set
7
and is a binary relation called a partial order , read “less than or equal to,”
with the following properties:
1 )(Reflexivity ) For all 7 ,
2 )(Antisymmetry ) For all Á 7 ,
and implies ~
3 )(Transitivity ) For all Á Á 7 ,
and implies
Partially ordered sets are also called posets .
It is customary to use a phrase such as “Let be a partially ordered set” when
7
the partial order is understood. Here are some key terms related to partially
ordered sets.
Definition Let be a partially ordered set.
7
,
1 The maximum largest top ) element of , should it exist, is an element
)
(
7
4 7 with the property that all elements of are less than or equal to
7
4, that is,
7 ¬ 4
Similarly, the mimimum least , smallest , bottom) element of , should it
(
7
7
exist, is an element 5 7 with the property that all elements of are
greater than or equal to , that is,
5
7 ¬ 5
2 A maximal element is an element 7 with the property that there is no
)
larger element in , that is,
7
7Á ¬ ~
