Grassmann Algebra

TheExteriorProduct.nb 34
Evaluating this with the GrassmannSimplify (�) function gives the required relationship
between x and y from the coefficient of the product of the introduced basis elements.
�� �xCx � yCy��Cw � Cz� � ��C0 � Cw � Cz� ��Simplify
�63 � 27 x � 29 y� e1 � e2 � e3 �� 0
Had we had a fourth equation, say x - 3y + z � 7, and wanted to solve for x, we can
incorporate the new information into the above formulation and re-evaluate:
�� x �Cx � e4��Cw ��Cy � 3�e4���Cz � e4�� �
���C0 � 7�e4��Cw ��Cy � 3�e4���Cz � e4�� �� Simplify
0 �� �30 � 41 x� e1 � e2 � e3 � e4
Or we can redefine the Ci .
Cw � e1 � 2�e2 � e3;
Cx ��2�e1 � e3 � e4;
Cy � 3�e1 � 7�e2 � e3 � 3�e4;
Cz � 4�e1 � 5�e2 � e3 � e4;
C0 � 2�e1 � 9�e2 � 8�e3 � 7�e4;
In this case, since we will be using the product Cw � Cy � Cz twice, it makes sense to calculate
it just once:
Cwyz � ��Cw � Cy � Cz�
�29 e1 � e2 � e3 � 38 e1 � e2 � e4 � 11 e1 � e3 � e4 � 16 e2 � e3 � e4
The final result is then obtained from the formula:
x � ��C0 � Cwyz�
�����������������������������
��Cx � Cwyz�
x �� � 30
�������
41
2.10 Simplicity
The concept of simplicity
�� Simplify
An important concept in the Grassmann algebra is that of simplicity. Earlier in the chapter we
introduced the concept informally. Now we will discuss it in a little more detail.
An element is simple if it is the exterior product of 1-elements. We extend this definition to
scalars by defining all scalars to be simple. Clearly also, since any n-element always reduces to
a product of 1-elements, all n-elements are simple. Thus we see immediately that all 0-elements,
1-elements, and n-elements are simple. In the next section we show that all (nÐ1)-elements are
also simple.
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