Grassmann Algebra

TheRegressiveProduct.nb 20
Example: The decomposition of a 1-element
The special case of the regressive product of a 1-element Β with an n-element Α enables us to
n
decompose Β directly in terms of the factors of Α. The Common Factor Theorem gives:
n
where:
n
Α n � Β� �
i�1
�Αi � Β��Αi
n�1
Α n �Α1 � Α2 � � � Αn � ��1� n�i ��Α1 � Α2 � � � ���� i � � � Αn��Αi
The symbol ���� i means that the ith factor is missing from the product. Substituting in the
Common Factor Theorem gives:
n
Α n � Β� �
n
i�1
��1� n�i ���Α1 � Α2 � � � ���� i � � � Αn��Β��Αi
� � �Α1 � Α2 � � � Αi�1 � Β � Αi�1 � � � Αn��Αi
i�1
Hence the decomposition formula becomes:
n
�Α1 � Α2 � � � Αn��Β�
� �Α1 � Α2 � � � Αi�1 � Β � Αi�1 � � � Αn��Αi
i�1
Writing this out in full shows that we can expand the expression simply by interchanging Β
successively with each of the factors of Α1 � Α2 � � � Αn , and summing the results.
�Α1 � Α2 � � � Αn��Β��Β � Α2 � � � Αn��Α1 �
�Α1 � Β � � � Αn��Α2 � � � �Α1 � Α2 � � � Β��Αn
We can make the result express more explicitly the decomposition of Β in terms of the Αi by
'dividing through' by Α n .
n
�
�� ��
�
Α1 � Α2 � � � Αi�1 � Β � Αi�1 � � � Αn �
�������������������������������� �������������������������������� ��������������� ���Αi
Α1 � Α2 � � � Αn
�
i�1
Here the quotient of two n-elements has been defined previously as the quotient of their scalar
coefficients.
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