Grassmann Algebra

TheInteriorProduct.nb 9
Α ����
m ����
Β � Β
�k1
k2
�
� � �  ���
�������������������������
� Α��Β � Β � � � Β � � Αm �
kp�
m k1 k2 kp
����
���� ���� �����
Β � Βk2 � � � Βkp���
�k1
�
���� ���� ����
� ���Α � Β �� Βk2�����
Βkp � ���Αm ���� Β � ���� Β � ���� �� ���� Β
m k1
k1 k2
kp
Α ����
m ����
Β � Β
�k1
k2
�
� � � Β
kp�
Thus, the interior product is left-associative.
��� � ���Α m ���� Β k1
Interior products of elements of the same grade
� ���� Β � ���� �� ���� Β
k2
kp
The regressive product of an m-element and an (nÐm)-element is given by formula 3.21.
Α � Β � �Α � Β ��1 ��
m n�m m n�m
0
The dual of this is given by formula 3.25. By putting 1 equal to 1
n ����� (formula 5.8), we obtain
Α � Β � �Α � Β �� 1
m n�m m n�m
�����
��
n
�����
By putting Β � Β these may be rewritten in terms of the interior product as:
m n�m
Α m ���� Β m
� � Α m � Β m
����� ��1
�����
Α � Β � � Αm ���� Β �� 1
m m
m
�����
Since the grades of the two factors of the interior product Α m ���� Β m
also be called an inner product.
The inner product
The interior product of two elements Α m and Β m
product. We proceed to show that the inner product is symmetric.
Taking the complement of equation 6.23 gives:
����������� �����
Α � Β � � Αm ���� Β
m m
m
�� 1
����� �����
� Αm ���� Β
m
are the same, the product may
of the same grade is (also) called their inner
On the other hand, we can also use the complement axiom [5.3] and the complement of a
complement formula [5.30] to give:
2001 4 5
6.21
6.22
6.23