Grassmann Algebra

TheRegressiveProduct.nb 42
x p �
�Α m � x p �� Β
n�m
����������������������������
Α m � Β
n�m
� � �
Α ��x� Β �
m p n�m
����������������������������
Α � Β
m n�m
If p = 1, there are just two terms, reducing the decomposition formula to:
x �
�Α m � x�� Β
n�m
����������������������������
Α m � Β
n�m
�
Α ��x� Β �
m
n�m
����������������������������
Α � Β
m n�m
3.48
3.49
In Chapter 4: Geometric Interpretations we will explore some of the geometric significance of
these formulae, especially the decomposition formula. The decomposition formula will also find
application in Chapter 6: The Interior Product.
Expressing an element in terms of another basis
The Common Factor Theorem may also be used to express an m-element in terms of another
basis with basis n-element Β by expanding the product Β � Α. n
n m
Let the new basis be �Β1, �, Βn� and let Β �Β1� � � Βn , then the Common Factor
n
Theorem permits us to write:
Ν
Β � Α � � ��Βi
n m i�1
n�m
� Α m ��Βi
m
where �� n
m � and Β �Β1 � Β1 �Β2�
Β2 � � �ΒΝ � ΒΝ
n n�m m n�m m
n�m m
We can visualize how the formula operates by writing Α and Β as simple products and then
m k
exchanging the Βi with the Αi in all the essentially different ways possible whilst always
retaining the original ordering. To make this more concrete, suppose n is 5 and m is 2:
2001 4 5
�Β1 � Β2 � Β3 � Β4 � Β5��Α1 � Α2 �
�Α1 � Α2 � Β3 � Β4 � Β5���Β1 � Β2�
��Α1 � Β2 � Α2 � Β4 � Β5���Β1 � Β3�
��Α1 � Β2 � Β3 � Α2 � Β5���Β1 � Β4�
��Α1 � Β2 � Β3 � Β4 � Α2���Β1 � Β5�
��Β1 � Α1 � Α2 � Β4 � Β5���Β2 � Β3�
��Β1 � Α1 � Β3 � Α2 � Β5���Β2 � Β4�
��Β1 � Α1 � Β3 � Β4 � Α2���Β2 � Β5�
��Β1 � Β2 � Α1 � Α2 � Β5���Β3 � Β4�
��Β1 � Β2 � Α1 � Β4 � Α2���Β3 � Β5�
��Β1 � Β2 � Β3 � Α1 � Α2���Β4 � Β5�