Grassmann Algebra

TheInteriorProduct.nb 7
If x is a 1-element and Α is a simple m-element, then x and Α may be said to be linearly
m m
dependent if and only if Α � x � 0, that is, x is contained in (the space of) Α. m m
Similarly, x and Α are said to be orthogonal if and only if
�����
Α � x � 0, that is, x is contained in
m m
�����
Α .
m
By taking the complement of
�����
Α � x � 0, we can see that
�����
Αm � x � 0 if and only if
m
Α � x
m ����� � 0. Thus it may also be said that a 1-element x is orthogonal to a simple element Α if
m
and only if their interior product is zero, that is, Α ���� x � 0.
m
If x � x1 � x2 � � � xk then Α and x are said to be totally orthogonal if and only if
k m k
Α ���� xi � 0 for all xi contained in x. m k
However, for Α m ���� �x1 � x2 � � � xk� to be zero it is only necessary that one of the xi be
orthogonal to Α m . To show this, suppose it to be (without loss of generality) x1 . Then by
formula 6.5 we can write Α ���� �x1 � x2 � � � xk� as �Α ���� x1� ���� �x2 � � � xk�, whence it
m m
becomes immediately clear that if Α ���� x1 is zero then so is the whole product Α ���� x. m m k
Just as the vanishing of the exterior product of two simple elements Α and x implies only that
m k
they have some 1-element in common, so the vanishing of their interior product implies only
�����
that Α and xk (m ³ k) have some 1-element in common, and conversely. That is, there is a 1-
m
element x such that the following implications hold.
Α m � x k � 0 � �Α m � x � 0, x k � x � 0�
Α ���� x � 0 � �
�����
Α � x � 0, xk � x � 0�
m k m
Α m ���� x k � 0 � �Α m ���� x � 0, x k � x � 0�
Α ���� x � 0 � Α���� x � 0 �
�����
Α � x � 0 xk�x�0 m k m m
� The interior product of a vector with a simple bivector
Suppose x is a vector and Α � Β is a simple bivector. The interior product of the bivector Α � Β
with the vector x is the vector Ξ.
Ξ��Α � Β� ���� x;
To explore interior products, we can use the GrassmannAlgebra function
ToScalarProducts. Applying ToScalarProducts to expand Ξ in terms of Α and Β
gives:
2001 4 5
6.18