Grassmann Algebra

TheComplement.nb 4
5.2 Axioms for the Complement
The grade of a complement
� 1
����� : The complement of an m-element is an (nÐm)-element.
Α �� �
�����
Α � �
m m m n�m
The grade of the complement of an element is the complementary grade of the element.
The linearity of the complement operation
� 2
����� : The complement operation is linear.
��������������� ������ ������ ����� �����
a�Α � b�Β � a�Αm � b�Βk � a� Αm � b� Βk
m k
For scalars a and b, the complement of a sum of elements (perhaps of different grades) is the
sum of the complements of the elements. The complement of a scalar multiple of an element is
the scalar multiple of the complement of the element.
The complement axiom
� 3
����� : The complement of a product is the dual product of the complements.
��������� ����� �����
Α � Β � Αm � Βk
m k
���������
Α � Β �
����� �����
Αm � Βk
m k
Note that for the terms on each side of the expression 5.3 to be non-zero we require m+k £ n,
while in expression 5.4 we require m+k ³ n.
Expressions 5.3 and 5.4 are duals of each other. We call these dual expressions the complement
axiom. Note its enticing similarity to de Morgan's law in Boolean algebra.
2001 4 5
5.1
5.2
5.3
5.4