Grassmann Algebra

Introduction.nb 19
x � ae1 � be2 � ce3
x
� �������������������������������
� ae1 � be2 � ce3 � ae1
����� � be2
����� � ce3
����� � ae2 � e3 � be3 � e1 � ce1 � e2
x
�� ������������������������������������������������������
� ae2 � e3 � be3 � e1 � ce1 � e2 �
������������� ������������� �������������
ae2 � e3 � be3 � e1 � ce1 � e2 � ae1 � be2 � ce3
� x �� � x
More generally, as we shall see in Chapter 5: The Complement, we can show that the
complement of the complement of any element is the element itself, apart from a possible sign.
�� m��n�m�
Α � ��1� �Αm
m
This result is independent of the correspondence that we set up between the m-elements and
(nÐm)-elements of the space, except that the correspondence must be symmetric. This is
equivalent to the commonly accepted requirement that the metric tensor (and inner product) be
symmetric. We call this the Repeated Complement Theorem.
The Complement Axiom
1.18
From the Common Factor Axiom we can derive a powerful relationship between the Euclidean
complements of elements and their exterior and regressive products. The Euclidean complement
of the exterior product of two elements is equal to the regressive product of their complements.
���������
Α � Β ��
� �
Αm � Βk
m k
This result, which we call the Complement Axiom, is the quintessential formula of the
Grassmann algebra. It expresses the duality of its two fundamental operations on elements and
their complements. We note the formal similarity to de Morgan's law in Boolean algebra.
1.19
We will also see that adopting this formula for general complements will enable us to compute
the complement of any element of a space once we have defined the complements of its basis 1elements.
Graphic showing the complement of a bivector as the intersection of two complements
���������
Α � Β �� Α
� �
� Β. With these basic notions, definitions and formulae in hand, we can now proceed to define the
interior product.
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