Grassmann Algebra

TheRegressiveProduct.nb 37
� A 4-element in a 5-space:
�5; X� CreateBasisForm�4, a�
a1 e1 � e2 � e3 � e4 � a2 e1 � e2 � e3 � e5 �
a3 e1 � e2 � e4 � e5 � a4 e1 � e3 � e4 � e5 � a5 e2 � e3 � e4 � e5
SimpleQ�X�
True
For non-simple m-elements
If SimpleQ is applied to an element which is not simple, the element will simply be returned.
SimpleQ�x � y � u � v�
False
For contingently simple m-elements
If SimpleQ is applied to an element which may be simple conditional on the values taken by
some of its scalar coefficients, the conditional result will be returned.
� A 2-element in a 4-space:
�4; X� CreateBasisForm�2, a�
a1 e1 � e2 � a2 e1 � e3 � a3 e1 � e4 � a4 e2 � e3 � a5 e2 � e4 � a6 e3 � e4
SimpleQ�X�
a3 a4 � a2 a5
������������������������������
a1
� a6 �� 0
If we are able to assert the condition required, that a3 a4 �a2 a5
a1
indeed simple.
SimpleQ�X �. a6 �� a3 a4 � a2 a5
������������������������������ �
True
� A 2-element in a 5-space:
2001 4 5
a1
�5; X� CreateBasisForm�2, a�
���������������������
� a6 � 0, then the 2-element is
a1 e1 � e2 � a2 e1 � e3 � a3 e1 � e4 � a4 e1 � e5 � a5 e2 � e3 �
a6 e2 � e4 � a7 e2 � e5 � a8 e3 � e4 � a9 e3 � e5 � a10 e4 � e5