Grassmann Algebra

Introduction.nb 21
Calculating interior products
We can use the definition to calculate interior products. In what follows we calculate the interior
products of representative basis elements of a 3-space with Euclidean metric. As expected, the
scalar products e1 ���� e1 and e1 ���� e2 turn out to be 1 and 0 respectively.
e1 ���� e1 � e1 � e1
����� � e1 ��e2 � e3� � �e1 � e2 � e3��1 � 1 � � 1 � 1
e1 ���� e2 � e1 � e2
����� � e1 ��e3 � e1� � �e1 � e3 � e1��1 � 0 � 1 � 0
Inner products of identical basis 2-elements are unity:
�����������������
�e1 � e2� ���� �e1 � e2� � �e1 � e2���e1 � e2� �
�e1 � e2��e3 � �e1 � e2 � e3��1 � 1 � � 1 � 1
Inner products of non-identical basis 2-elements are zero:
�����������������
�e1 � e2� ���� �e2 � e3� � �e1 � e2���e2 � e3� �
�e1 � e2��e1 � �e1 � e2 � e3��1 � 1 � � 1 � 1
If a basis 2-element contains a given basis 1-element, then their interior product is not zero:
�e1 � e2� ���� e1 � �e1 � e2��e1
����� �
�e1 � e2���e2 � e3� � �e1 � e2 � e3��e2 � 1 � � e2 � e2
If a basis 2-element does not contain a given basis 1-element, then their interior product is zero:
�e1 � e2� ���� e3 � �e1 � e2��e3
����� � �e1 � e2���e1 � e2� � 0
Expanding interior products
To expand interior products, we use the Interior Common Factor Theorem. This theorem shows
how an interior product of a simple element Α with another, not necessarily simple element of
m
equal or lower grade Β, may be expressed as a linear combination of the Ν (= �
k
m
�) essentially
k
different factors Αi (of grade mÐk) of the simple element of higher degree.
m�k
Α m �Α1
k
Α m ���� Β k
Ν
� �
i�1
�Αi
k
���� Β��Αi k m�k
� Α1 �Α2�
Α2 � � �ΑΝ�
ΑΝ
m�k k m�k
k m�k
For example, the Interior Common Factor Theorem may be used to prove a relationship
involving the interior product of a 1-element x with the exterior product of two factors, each of
which may not be simple. This relationship and the special cases that derive from it find
application throughout the explorations in the rest of this book.
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