Grassmann Algebra

TheComplement.nb 13
Orthogonality
The specific complement mapping that we impose on a Grassmann algebra will define the
notion of orthogonality for that algebra. A simple element and its complement will be referred
to as being orthogonal to each other. In standard linear space terminology, the space of a simple
element and the space of its complement are said to be orthogonal complements of each other.
This orthogonality is total. That is, every 1-element in a given simple m-element is orthogonal
to every 1-element in the complement of the m-element.
Visualizing the complement axiom
We can use this notion of orthogonality to visualize the complement axiom geometrically.
Consider the bivector x�y. Then
���������
x � y is orthogonal to x�y. But since
���������
x � y �
�����
x �
�����
y this
also means that the intersection of the two (nÐ1)-spaces defined by x
����� and y
����� is orthogonal to
x�y. We can depict it in 3-space as follows:
Graphic showing three pictures
1) A bivector labeled x
����� with vector x orthogonal to it.
2) A bivector labeled y
����� with vector y orthogonal to it.
3) The bivectors shown intersecting in a vector that is orthogonal to the bivector x� y.
Ref JMB p60
The regressive product in terms of complements
All the operations in the Grassmann algebra can be expressed in terms only of the exterior
product and the complement operations. It is this fact that makes the complement so important
for an understanding of the algebra.
In particular the regressive product (discussed in Chapter 3) and the interior product (to be
discussed in the next chapter) have simple representations in terms of the exterior and
complement operations.
We have already introduced the complement axiom as part of the definition of the complement.
Taking the complement of both sides, noting that the degree of Α m � Β k
complement of a complement axiom gives:
Or finally:
2001 4 5
���������
����� �����
� Αm�
Βk
���������
Α � Β
m k
������������ � ��1� �m�k�n���n��m�k�n�� �Αm � Β k
Α � Β � ��1�
m k
�m�k���m�k�n� �Α
m
�����
������������ �����
� Βk
is m+kÐn, and using the
5.26