Grassmann Algebra

TheInteriorProduct.nb 29
This formula may be extended in a straightforward manner to the case where Α and Β are not
m k
simple: since a non-simple element may be expressed as the sum of simple terms, and the
formula is valid for each term, then by addition it can be shown to be valid for the sum.
Extension of the fundamental formula to elements of higher grade
If x is of grade higher than 1, then similar relations hold, but with extra terms on the right-hand
side. For example, if we replace x by x1 � x2 and note that:
�Α � Β� ���� �x1 � x2� � ��Α � Β� ���� x1� ���� x2
m k
m k
then the product rule derived above may be applied successively to obtain the following result.
�Α � Β� ���� �x1 � x2� � �Α ���� �x1 � x2�� � Β �Α��Β���� �x1 � x2��
m k
m k m k
���1�m�1 ���Α ���� x1���Β���� x2� � �Α ���� x2���Β���� x1��
m k
m k
The complementary form of the fundamental formula
Another form of equation 6.54 which will be found useful is obtained by replacing Β by its
k
complement and then taking the complement of the complete equation.
���������������������
�����
���������������������
�����
�Α � Β � ���� x � �Α ���� x�� Β � ��1�
m k
m k
m ���������������������
�����
�Α ��Β ���� x�
m k
By converting interior products to regressive products and complements, applying the
6.66
Complement Axiom [5.3], converting back to interior products, and then interchanging the roles
of Α and Β we obtain:
m k
�Α ���� Β��x � �Α � x� ���� Β � ��1�
m k
m k
m�1�Α ���� �Β ���� x�
m k
This formula is the basis for the derivation of a set of formulae of geometric interest in the next
section: the Triangle Formulae.
The decomposition formula
By putting Β equal to x and rearranging terms, this formula expresses a 'decomposition' of the
k
element Α in terms of the 1-element x:
m
where x � �
2001 4 5
x
������������ �����������
Α m � ��1� m ���Α m � x � � ���� x � � �Α m ���� x � ��x � �
x����x . Note that �Α m � x� � ���� x � and �Α m ���� x � ��x � are orthogonal.
6.67
6.68