Grassmann Algebra

TheComplement.nb 33
All the formulae for complements in the vector subspace still hold. In an n-plane (which is of
dimension n+1), the vector subspace has dimension n, and hence the complement operation for
the n-plane and its vector subspace will not be the same. We will denote the complement
operation in the vector subspace by using an overvector � � instead of an overbar ����
����� .
Such a complement is called a free complement because it is a complement in a vector space in
which all elements are 'free', that is, not bound through points.
The constant � will be the same in both spaces and still equate to the inverse of the square root
of the determinant g of the metric tensor. Hence it is easy to show from the metric tensor above
that:
����� 1
1 � ����������
�����
g
� 1
1 � ����������
�����
g
�� � e1 � e2 � � � en
�e1 � e2 � � � en
����� �
1 � � � 1
����� �
� � 1
5.34
5.35
5.36
5.37
In this interpretation 1
����� is called the unit n-plane, while 1 � is called the unit n-vector. The unit nplane
is the unit n-vector bound through the origin.
The complement of an m-vector
If we define ei � as the cobasis element of ei in the vector basis of the vector subspace (note
that this is denoted by an 'underbracket' rather than an 'underbar'), then the formula for the
complement of a basis vector in the vector subspace is:
n
ei � ��� � 1
����������
�����
��
g
j�1
gij� ej
�
5.38
In the n-plane, the complement ei
����� of a basis vector ei is given by the metric of the n-plane as:
n
����� 1
ei � ����������
�����
��
g
j�1
gij���� � ej
� �
Hence by using formula 5.31 we have that:
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