Grassmann Algebra

TheComplement.nb 17
The complement of the complement of an m-element
Consider the general m-element Α ��ai�ei . By taking the complement of the complement of
m m
this element and substituting from equation 5.18, we obtain:
����� �����
Α � �
m
�����
� �
� ��1� m��n�m� �Α
m
i
�����
ai ei
m
i
�ai���1� m��n�m� �ei
m
Finally then we have shown that, provided the complement mapping gij is symmetric and the
constant � is such that � 2 �gij� � 1, the complement of a complement axiom is satisfied by
an otherwise arbitrary mapping, with sign ��1� m��n�m� .
Special cases
����� ����� m��n�m� Α � ��1� �Αm
m
The complement of the complement of a scalar is the scalar itself.
����� �����
a � a
The complement of the complement of an n-element is the n-element itself.
����� �����
Α �Αn
n
The complement of the complement of any element in a 3-space is the element itself, since
��1�m��3�m� is positive for m equal to 0, 1, 2, or 3.
Alternatively, we can say that
����� �����
Α �Αm except when Α is of odd degree in an even-dimensional
m
m
space.
5.7 Working with Metrics
� The default metric
In order to calculate complements and interior products, and any products defined in terms of
them (for example, Clifford and hypercomplex products), GrassmannAlgebra needs to know
what metric has been imposed on the underlying linear space � 1 . Grassmann and all those
writing in the tradition of the Ausdehnungslehre tacitly assumed a Euclidean metric; that is, one
in which gij �∆ij . This metric is also the one tacitly assumed in beginning presentations of
the three-dimensional vector calculus, and is most evident in the definition of the cross product
as a vector normal to the factors of the product.
2001 4 5
5.30
5.31
5.32