Grassmann Algebra

TheComplement.nb 12
ei
m
�����
� ej �∆ij
m
These forms will be the basis for the definition of the inner product in the next chapter.
5.22
We note that the concept of cobasis which we introduced in Chapter 2: The Exterior Product,
despite its formal similarity to the Euclidean complement, is only a notational convenience. We
do not define it for linear combinations of elements as the definition of a complement requires.
5.5 Complementary Interlude
Alternative forms for complements
As a consequence of the complement axiom and the fact that multiplication by a scalar is
equivalent to exterior multiplication (see Section [2.4]) we can write the complement of a scalar,
an m-element, and a scalar multiple of an m-element in several alternative forms.
Alternative forms for the complement of a scalar
From the linearity axiom 2
����� ������ �����
we have a Α � a�Αm , hence:
m
���� ������ ���� ����
a �� a1�
a� 1 �� a � 1
The complement of a scalar can then be expressed in any of the following forms:
���� ��������� ���� ���� ���� ���� ���� ����
a � 1 � a � 1 � a � 1 � a � 1 a � a � 1 � a 1
Alternative forms for the complement of an m-element
����� ��������� ����
Α � 1 � Αm � 1 �
�����
Αm � 1 �
�����
Αm � 1
�����
Αm
m
Alternative forms for the complement of a scalar multiple of an m-element
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������
a Α �
���������
a � Αm �
����
a �
�����
Αm � a �
�����
Αm � a
�����
Αm
m
5.23
5.24
5.25