Grassmann Algebra

TheRegressiveProduct.nb 8
The inverse of an n-element
Axiom �9 says that every (non-zero) n-element has an inverse with respect to the regressive
product. Suppose we have an n-element Α expressed in terms of some basis. Then, according to
n
1.10 we can express it as a scalar multiple (a, say) of the unit n-element 1. n
We may then write the inverse Α�1 of Αn with respect to the regressive product as the scalar
n
multiple 1
����
a of 1 n .
�1 1
Α � a�1 � Α � ���� �1
n n n a n
We can see this by taking the regressive product of a�1 n with 1
����
a �1 n :
Α n � Α n
�1
� �a�1n �� ���
�
1
���� �1
���
� 1 � 1 � 1
a n�
n n n
If Α n is now expressed in terms of a basis we have:
Α n � be1 � e2 � � � en � b
�����
� �1 n
�1 Hence Α can be written as:
n
�1 �
Α � �����
n
b �1 n � �2
Summarizing these results:
Historical Note
�������� �e1 � e2 � � � en
b
�1 1
Α � a�1 � Α � ���� �1
n n n a n
�1 �
Α � be1 � e2 � � � en � Α �
n n
2
�������� �e1 � e2 � � � en
b
In Grassmann's Ausdehnungslehre of 1862 Section 95 (translated by Lloyd C. Kannenberg) he
states:
2001 4 5
3.13
3.14
If q and r are the orders of two magnitudes A and B, and n that of the principal domain,
then the order of the product [A B] is first equal to q+r if q+r is smaller than n, and second
equal to q+rÐn if q+r is greater than or equal to n.