Grassmann Algebra

TheComplement.nb 35
��������� ����
� � Α � Αm
m
The complement of the complement of an m-vector
5.43
We can now use this relationship to relate the complement of the complement of an m-vector in
the n-plane to the complement of the complement in the vector subspace.
����� m ����
Α � ��1� �� � Αm
m
����� ����� �����������
m
Α � ��1� �� �
����
m
Αm � ��1�
���� �Αm
m
� ���
����� ����� m ���� Α � ��1� �Αm
m
� ���
The complement of a bound element
5.44
We now come to the point where we can determine formulae which express the complement of
a general bound element in an n-plane in terms of the origin and complements in the vector
subspace of the n-plane. We have already introduced a specific case of this in Section 5.9 above.
In Chapter 6: The Interior Product, we will depict the concepts graphically.
Consider a bound m-vector P � Α m in an n-plane.
P � Α m � �� � x��Α m � � � Α m � x � Α m
The complement of this bound m-vector is:
��������� ��������� ���������
P � Α � � � Αm � x � Αm
m
�
���
Α
� m�1
� ��1� �� �
���������
x �
�
Αm
m
���� � Α � � � Αm � x
m
� ���������
���������
P � Α ��� � Αm � x
m
� ��������� �
���
Α
�
m
P � � � x
The simplest bound element is the point. Putting Α m � 1 in the formula above gives:
�����
P ��� � x
� �
� 1
The complement of a bound vector is:
2001 4 5
P � � � x
5.45
5.46