Grassmann Algebra

TheRegressiveProduct.nb 26
e1 ����� � e2 ����� � 1� ��e1 � e2 ������������� �
or, by writing ��1� � 1 as:
n
e1 ����� � e2 �����
1
� ����� �� e1 � e2
�
������������� �
Taking the regressive product of this equation with e3 ����� gives:
e1 ����� � e2 ����� � e3 �����
1
�����
� �1� ��e1 � e2 � e3
1
� ����� �� e1 � e2
�
������������� ��e3 ����� �
1
��������������������
� � ��������
�������������������� �
� 2 �� e1 � e2 � e3
Continuing this process, we arrive finally at the result:
e1 ����� � e2 ����� � � � em �����
1
� ������������
�m�1 �� e1 � e2 � � � em
���������������������������� �
A special case which we will have occasion to use in Chapter 5 is where the result reduces to a
1-element.
��1� j�1�e1 ����� � e2 ����� � � � ���� j � � � en �����
� 1
������������
� n�2 ���1�n�1 �ej
In sum: The regressive product of cobasis elements of basis 1-elements is congruent to the
cobasis element of their exterior product.
In fact, this formula is just an instance of a more general result which says that: The regressive
product of cobasis elements of any grade is congruent to the cobasis element of their exterior
product. We will discuss a result very similar to this in more detail after we have defined the
complement of an element in Chapter 5.
3.8 Factorization of Simple Elements
Factorization using the regressive product
3.35
3.36
The Common Factor Axiom asserts that in an n-space, the regressive product of a simple melement
with a simple (nÐm+1)-element will give either zero, or a 1-element belonging to them
both, and hence a factor of the m-element. If m such factors can be obtained which are
independent, their product will therefore constitute (apart from a scalar factor easily determined)
a factorization of the m-element.
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