Grassmann Algebra

TheRegressiveProduct.nb 18
Finally, by putting � e1 � e2 � e3 � 1 n , we have Z expressed as a 1-element:
Z � 1
����� ���x1�y2 � x2�y1��e1 � �x1�y3 � x3�y1��e2 � �x2�y3 � x3�y2��e3�;
�
Thus, in sum, we have the general congruence relation for the intersection of two 2-elements in
a 3-space.
�x1�e1 � e2 � x2�e1 � e3 � x3�e2 � e3�
� �y1�e1 � e2 � y2�e1 � e3 � y3�e2 � e3�
� �x1�y2 � x2�y1��e1 � �x1�y3 � x3�y1��e2 � �x2�y3 � x3�y2��e3
Check by calculating the exterior products
We can check that Z is indeed a common element to X and Y by determining if the exterior
product of Z with each of X and Y is zero. We use GrassmannSimplify to do the check.
DeclareExtraScalars��x1, x2, x3, y1, y2, y3��;
GrassmannSimplify��Z � X, Z � Y��
�0, 0�
3.6 The Common Factor Theorem
Development of the Common Factor Theorem
3.27
The example in the previous section applied the Common Factor Axiom to two general
elements by expanding all the terms in their regressive product, applying the Common Factor
Axiom to each of the terms and then factoring the result. In the case one of the factors in the
regressive product is simple, as is often the case, and expressed as a product of 1-element
factors, there is usually a more effective way to determine a simple common factor. The formula
for doing this we call the Common Factor Theorem.
This theorem is one of the most important in the Grassmann algebra. We will see later that it has
a counterpart expressed in terms of exterior and interior products which forms the principal
expansion theorem for interior products.
Consider a regressive product Α � Β where Α is given as a simple product of 1-element factors
m k m
and m+k = n+p, p > 0. Since the common factor (Γ, say) is a p-element it can be expressed as a
p
linear combination of all the essentially different p-elements formed from the 1-element factors
are defined by:
of Α m : Γ p
2001 4 5
��ai Αi
p
, where the Αi
p
Α �Α1 � Α1 �Α2 � Α2 � � �ΑΝ � ΑΝ
m m�p p m�p p
m�p p
�� m
p �