Grassmann Algebra

TheExteriorProduct.nb 19
Indexing basis elements of exterior linear spaces
Just as we can denote the basis elements of � 1 by indexed symbols ei , (i : 1, �, n), so too can
we denote the basis elements of � by indexed symbols ei , (i : 1, �, �
m m
n
�). This will enable us
m
to denote the basis elements of an exterior linear space without the 'orgy of indices' which
would accrue if we denoted the elements explicitly.
For example, suppose we have a basis �e1, e2, e3, e4�, then the standard bases are given
by:
BasisTable�4�
� BASIS
�
0
1
�
l
�e1, e2, e3, e4�
�
2
�e1 � e2, e1� e3, e1� e4, e2� e3, e2� e4, e3� e4�
� 3 �e1 � e2 � e3, e1 � e2 � e4, e1 � e3 � e4, e2 � e3 � e4�
� 4
e1 � e2 � e3 � e4
We can then denote these basis elements more compactly as:
�e1
2
,e2
2
,e3
2
,e4
2
,e5
2
,e6�
�
2
�e1 � e2, e1 � e3, e1 � e4, e2 � e3, e2 � e4, e3 � e4�
�e1
3
,e2
3
,e3
3
e 4 � e1 � e2 � e3 � e4
2.6 Cobases
Definition of a cobasis
,e4�
� �e1 � e2 � e3, e1� e2 � e4, e1� e3 � e4, e2� e3 � e4�
3
The notion of cobasis of a basis will be conceptually and notationally useful in our development
of later concepts. The cobasis of a basis of � m is simply the basis of �
n�m which has been ordered
in a special way relative to the basis of � m .
Let �e1, e2, �, en� be a basis for � 1 . The cobasis element associated with a basis element
ei is denoted ei ����� and is defined as the product of the remaining basis elements such that the
exterior product of a basis element with its cobasis element is equal to the basis element of �. n
2001 4 5