Grassmann Algebra

TheComplement.nb 42
Α m �Α1 � Α2 � � � Αm
Take any other nÐm 1-elements Αm�1 , Αm�2 , �, Αn such that Α1 , Α2 , �, Αm , Αm�1 , �, Αn
form an independent set, and thus a basis of � 1 . Then by equation 5.34 we have:
�����
Α
���������������������������� �������
�Α1 � Α2 � � � Αm � gΑ �Α
m
m�1 � Α m�2 � � � Α n
where gΑ is the determinant of the metric tensor in the Α basis.
The complement of Α1 � Α2 � � � Αm is thus a scalar multiple of Α m�1 � Α m�2 � � � Α n . Since
this is evidently simple, the assertion is proven.
5.12 Summary
In this chapter we have shown that by defining the complement operation on a basis of � as
1
����� n
��������� ����� �����
ei � � �j�1 gij�ej , and accepting the complement axiom Α � Β � Αm � Βk as the
����� m k
mechanism for extending the complement to higher grade elements, the requirement that
����� �����
Α ��Αm is true, constrains gij to be symmetric and � to have the value �
m
1
�������� ����� ,where g is
g
the determinant of the gij .
The metric tensor gij was introduced as a mapping between the two linear spaces � 1 and �
n�1 .
To this point has not yet been related to notions of interior, inner or scalar products. This will be
addressed in the next chapter, where we will also see that the constraint � �� 1
�������� ����� is
g
equivalent to requiring that the magnitude of the unit n-element is unity.
Once we have constrained gij to be symmetric, we are able to introduce the standard notion of
a reciprocal basis. The formulae for complements of basis elements in any of the � m then become
much simpler to express.
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