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Grassmann Algebra

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TheRegressiveProduct.nb 20<br />

Example: The decomposition of a 1-element<br />

The special case of the regressive product of a 1-element Β with an n-element Α enables us to<br />

n<br />

decompose Β directly in terms of the factors of Α. The Common Factor Theorem gives:<br />

n<br />

where:<br />

n<br />

Α n � Β� �<br />

i�1<br />

�Αi � Β��Αi<br />

n�1<br />

Α n �Α1 � Α2 � � � Αn � ��1� n�i ��Α1 � Α2 � � � ���� i � � � Αn��Αi<br />

The symbol ���� i means that the ith factor is missing from the product. Substituting in the<br />

Common Factor Theorem gives:<br />

n<br />

Α n � Β� �<br />

n<br />

i�1<br />

��1� n�i ���Α1 � Α2 � � � ���� i � � � Αn��Β��Αi<br />

� � �Α1 � Α2 � � � Αi�1 � Β � Αi�1 � � � Αn��Αi<br />

i�1<br />

Hence the decomposition formula becomes:<br />

n<br />

�Α1 � Α2 � � � Αn��Β�<br />

� �Α1 � Α2 � � � Αi�1 � Β � Αi�1 � � � Αn��Αi<br />

i�1<br />

Writing this out in full shows that we can expand the expression simply by interchanging Β<br />

successively with each of the factors of Α1 � Α2 � � � Αn , and summing the results.<br />

�Α1 � Α2 � � � Αn��Β��Β � Α2 � � � Αn��Α1 �<br />

�Α1 � Β � � � Αn��Α2 � � � �Α1 � Α2 � � � Β��Αn<br />

We can make the result express more explicitly the decomposition of Β in terms of the Αi by<br />

'dividing through' by Α n .<br />

n<br />

�<br />

�� ��<br />

�<br />

Α1 � Α2 � � � Αi�1 � Β � Αi�1 � � � Αn �<br />

�������������������������������� �������������������������������� ��������������� ���Αi<br />

Α1 � Α2 � � � Αn<br />

�<br />

i�1<br />

Here the quotient of two n-elements has been defined previously as the quotient of their scalar<br />

coefficients.<br />

2001 4 5<br />

3.30<br />

3.31<br />

3.32

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