Grassmann Algebra

TheRegressiveProduct.nb 25
����ei
� m
� es
p
����
�
�
����ej
�
k
� es
p
����
�
�
����ei
� m
� ej
k
� es
p
From the definition of cobasis elements we have that:
ei
m
ej
k
es
p
ei
m
� es
p
� es
p
� ��1� mk �ej
k
�����
� ei
m
�����
� ei � ej
m
������������� k
� ej
k
� es
p
� 1�
����
� es,
m � k � p � n
� p
Substituting these four elements into the Common Factor Axiom above gives:
��1� mk �ej
k
�����
� ei � 1� ��ei � ej�
����� m
m
������������� k
Or, more symmetrically, by interchanging the first two factors:
ei
m
�����
� ej � 1� ��ei � ej�
�
����� k
m
������������� k
1
����� �ei � ej
� m
������������� k
It can be seen that in this form the Common Factor Axiom does not specifically display the
common factor, and indeed remains valid for all basis elements, independent of their grades.
In sum: Given any two basis elements, the cobasis element of their exterior product is
congruent to the regressive product of their cobasis elements.
The regressive product of cobasis elements
In Chapter 5: The Complement, we will have cause to calculate the regressive product of
cobasis elements. From the formula derived below we will have an instance of the fact that the
regressive product of (nÐ1)-elements is simple, and we will determine that simple element.
First, consider basis elements e1 and e2 of an n-space and their cobasis elements e1
The regressive product of e1 ����� and e2 ����� is given by:
e1 ����� � e2 ����� � �e2 � e3 � � � en���� e1 � e3 � � � en�
Applying the Common Factor Axiom enables us to write:
e1 ����� � e2 ����� � �e1 � e2 � e3 � � � en���e3 � � � en�
We can write this either in the form already derived in the section above:
2001 4 5
�����
3.33
3.34
and e2 ����� .