Grassmann Algebra

TheExteriorProduct.nb 8
Note that whereas several congruent elements are linearly dependent, several linearly dependent
elements are not necessarily congruent.
When we have one element equal to a scalar multiple of another, Α � a Β say, we may
m m
sometimes take the liberty of writing the scalar multiple as a quotient of the two elements:
a �
Α m
����
Βm
These notions will be particularly useful in the discussion of unions and intersections in the next
chapter, and on applications to geometry.
The associativity of the exterior product
The exterior product is associative in all groups of (adjacent) factors. For example:
Hence
�x1 � x2��x3 � x4 � x1 ��x2 � x3��x4 � �x1 � x2 � x3��x4 � �
�Α � Β��Γ m k p
Thus the brackets may be omitted altogether.
�Α� m ����Β
�k
�
� ��� p�
From this associativity together with the anti-symmetric property of 1-elements it may be shown
that the exterior product is anti-symmetric in all (1-element) factors. That is, a transposition of
any two 1-element factors changes the sign of the product. For example:
x1 � x2 � x3 � x4 ��x3 � x2 � x1 � x4 � x3 � x4 � x1 � x2 � �
Furthermore, from the nilpotency axiom, a product with two identical 1-element factors is zero.
For example:
x1 � x2 � x3 � x2 � 0
Example: Non-simple elements are not generally nilpotent
It should be noted that for simple elements Α, Α � Α � 0, but that non-simple elements do not
m m m
necessarily possess this property, as the following example shows.
Suppose � 1 is of dimension 4 with basis e1 , e2 , e3 , e4 . Then the following exterior product of
identical elements is not zero:
2001 4 5
�e1 � e2 � e3 � e4���e1 � e2 � e3 � e4� � 2�e1 � e2 � e3 � e4
2.6