Grassmann Algebra

TheInteriorProduct.nb 5
� ���� 7: The interior product is not associative.
The exterior, regressive and interior products have relations derived directly from the
associativity of the regressive product.
�Α m ���� Β k
�Α m � Β k
� ���� à r
� ���� à r
� Α m ���� �Β k
� Α m ��Β k
� Γ �
r
���� à �
r
By interchanging the order of the factors on the right-hand side of formula 6.5 we can derive an
alternative expression for it.
Α m ���� �Β k
� Γ r
� � ��1� kr �Α m ���� �Γ r
�Α m ���� Β k
� ���� à r
� Β � �
k
� ��1� kr �Α m ���� Γ r
� ����  k
� ���� 8: The unit scalar 1 is the identity for the interior product.
The interior product of an element with the unit scalar 1 does not change it. The interior product
of scalars is equivalent to ordinary scalar multiplication. The complement operation may be
viewed as the interior product with the unit n-element 1
����� .
Α m ���� 1 � Α m
Α m ���� a � Α m � a � a�Α m
1 ���� 1 � 1
����� �����
Α � 1 ���� Αm
m
� ���� 9: The inverse of a scalar with respect to the interior product is its complement.
The interior product of the complement of a scalar and the reciprocal of the scalar is the unit nelement.
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6.5
6.6
6.7
6.8
6.9
6.10
6.11