Grassmann Algebra

TheExteriorProduct.nb 37
a3 a5 � a2 a6 � a1 a8 � 0, a4 a5 � a2 a7 � a1 a9 � 0,
a4 a6 � a3 a7 � a1 a10 � 0,
a4 a8 � a3 a9 � a2 a10 � 0, a7 a8 � a6 a9 � a5 a10 � 0
2.11 Exterior Division
The definition of an exterior quotient
It will be seen in Chapter 4: Geometric Interpretations that one way of defining geometric
entities like lines and planes is by the exterior quotient of two interpreted elements. Such a
quotient does not yield a unique element. Indeed this is why it is useful for defining geometric
entities, for they are composed of sets of elements.
The exterior quotient of a simple (m+k)-element Α
m�k by another simple element Β k
contained in Α
m�k is defined to be the most general m-element contained in Α
m�k (Γ m
the exterior product of the quotient Γ m
Γ �
m
with the denominator Β k
Α
m�k
��������
Βk
� � m k
Note the convention adopted for the order of the factors.
� Α
m�k
which is
yields the numerator Α
m�k .
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, say) such that
In Chapter 9: Exploring Grassmann Algebras we shall generalize this definition of quotient to
define the general division of Grassmann numbers.
Division by a 1-element
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Consider the quotient of a simple (m+1)-element Α by a 1-element Β. Since Α contains Β we
m�1 m�1
can write it as Α1 � Α2 � � � Αm � Β. However, we could also have written this numerator in
the more general form:
�Α1 � t1�Β���Α2 � t2�Β��� ��Αm � tm �Β��Β
where the ti are arbitrary scalars. It is in this more general form that the numerator must be
written before Β can be 'divided out'. Thus the quotient may be written:
2001 4 5
Α1 � Α2 � � � Αm � Β
�������������������������������� ���������� � �Α1 � t1�Β���Α2 � t2�Β��� ��Αm � tm �Β�
Β
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