Grassmann Algebra

TheRegressiveProduct.nb 44
Here the denominators are the same in each term but are expressed this way to show the
positioning of the factor Α in the numerator.
The symmetric expansion of a 1-element in terms of another basis
For the case m = 1, Β � Α may be expanded by the Common Factor Theorem to give:
n
�Β1 � Β2 � � � Βn��Α� �
Ν
���1�
i�1
n�i ��Β1 � � � ���� i � � � Βn � Α��Βi
or, in terms of the mnemonic expansion in the previous section:
�Β1 � Β2 � Β3 � � � Βn��Α
� �Α � Β2 � Β3 � � � Βn��Β1
��Β1 � Α � Β3 � � � Βn��Β2
��Β1 � Β2 � Α � � � Βn��Β3
� �
��Β1 � Β2 � Β3 � � � Α��Βn
Putting Α equal to Β0 , this may be written more symmetrically as:
n
� ���1� i ��Β0 � Β1 � � � ���� i � � � Βn��Βi � 0
i�0
Example: A relationship between four 1-elements which span a 3-space
Suppose Β0 , Β1 , Β2 , Β3 are four dependent 1-elements which span a 3-space, then formula
3.50 reduces to the identity:
�Β1 � Β2 � Β3��Β0 � �Β0 � Β2 � Β3��Β1
��Β0 � Β1 � Β3��Β2 � �Β0 � Β1 � Β2��Β3 � 0
Product formulae leading to scalar results
A formula particularly useful in its interior product form to be derived later is obtained by
application of the Common Factor Theorem to the product �Α1 � � � Αm ���Β1 � � � Βm �
n�1 n�1
where the Αi are 1-elements and the Βi are (nÐ1)-elements.
n�1
2001 4 5
3.52
3.53