Grassmann Algebra

TheComplement.nb 7
5.3 Defining the Complement
The complement of an m-element
To define the complement of a general m-element, we need only define the complement of basis
elements, since by the linearity axiom 2
����� , we have that for Α a general m-element expressed as
m
a linear combination of basis m-elements, the complement of Α is the corresponding linear
m
combination of the complements of the basis m-elements.
Α m � � ai�ei
m
The complement of a basis m-element
����� �����
� Α � � ai�ei
m
m
5.11
To define the complement of an m-element we need to define the complements of the basis melements.
The complement of a basis m-element however, cannot be defined independently of
the complements of basis elements in exterior linear spaces of other grades, since they are
related by the complement axiom. For example, the complement of a basis 4-element may also
be expressed as the regressive product of two basis 2-elements, or as the regressive product of a
basis 3-element and a basis 1-element.
���������������������������� ������������� ������������� ��������������������
e1 � e2 � e3 � e4 � e1 � e2 � e3 � e4 � e1 � e2 � e3 � e4
�����
The complement axiom enables us to define the complement of a basis m-element in terms only
of the complements of basis 1-elements.
����������������������������
e1 � e2 � � � em � e1
����� � e2
����� � � � em
�����
5.12
Thus in order to define the complement of any element in a Grassmann algebra, we only need to
define the complements of the basis 1-elements, that is, the correspondence between basis 1elements
and basis (nÐ1)-elements.
Defining the complement of a basis 1-element
The most general definition we could devise for the complement of a basis 1-element is a linear
combination of basis (nÐ1)-elements. For example in a 3-space we could define the complement
of e1 as a linear combination of the three basis 2-elements.
�����
e1 � a12�e1 � e2 � a13�e1 � e3 � a23�e2 � e3
However, it will be much more notationally convenient to define the complements of basis 1elements
as a linear combination of their cobasis elements. Hence in a space of any number of
dimensions we can write:
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