Grassmann Algebra

TheComplement.nb 14
Relating exterior and regressive products
The exterior product of an m-element with the complement of another m-element is an nelement,
and hence must be a scalar multiple of the unit n-element 1
����� . Hence:
����� ����� ����������� ����� ����� �����
Α � Β � a� 1 � Αm� Β � a 1 � a
m m
m
����� �����������
m��n�m� ����� �����
� Αm � a � ��1� �Βm �
�����
Αm � a
� ��1� m��n�m� � m
� Β m
�����
� Α � a
m
����� �����
Α � Β � a 1
m m
� Β m
�
�����
Α � a
m
5.6 The Complement of a Complement
The complement of a cobasis element
In order to determine any conditions on the gij required to satisfy the complement of a
complement axiom in Section 5.2, we only need to compute the complement of a complement
of basis 1-elements and compare the result to the form given by the axiom.
The complement of the complement of this basis element is obtained by taking the complement
of expression 5.13 for the case i = 1.
����� � � �
�����
ei
n
j�1
�����
g1�j�ej �����
In order to determine ei
����� �����
, we need to obtain an expression for the complement of a cobasis
element of a 1-element
�����
ej . Note that whereas the complement of a cobasis element is defined,
�����
the cobasis element of a complement is not. To simplify the development we consider, without
loss of generality, a specific basis element e1 .
First we express the cobasis element as a basis (nÐ1)-element, and then use the complement
axiom to express the right-hand side as a regressive product of complements of 1-elements.
����� ����������������������������
e1 ����� � e2 � e3 � � � en � e2
����� � e3
����� � � � en
�����
Substituting for the ei
����� from the definition [5.13]:
�����
e1 ����� � � �g21�e1 ����� � g22�e2 ����� � � � g2�n�en ����� �
� � �g31�e1 ����� � g32�e2 ����� � � � g3�n�en ����� �
� �
� � �gn1�e1 ����� � gn2�e2 ����� � � � gnn�en ����� �
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