Grassmann Algebra

GeometricInterpretations.nb 12
A particularly symmetrical formula results from the expansion of this product reducing it to a
form no longer showing preference for P0 .
�P1 � P0���P2 � P0��� ��Pm � P0� �
m
� ��1�
i�0
i�P0 � P1 � � � ���� i � � � Pm
Here ���� i denotes deletion of the factor Pi from the product.
4.5 Decomposition into Components
The shadow
In Chapter 3 we developed a formula which expressed a p-element as a sum of components, the
element being decomposed with respect to a pair of elements Α and Β which together span the
m n�m
whole space. The first and last terms of this decomposition were given as:
x p �
�Α m � x p �� Β
n�m
����������������������������
Α m � Β
n�m
� � �
Α ��x� Β �
m p n�m
����������������������������
Α � Β
m n�m
Grassmann called the last term the shadow of x on Α excluding Β .
p m n�m
It can be seen that the first term can be rearranged as the shadow of x on Β excluding Α. p n�m
m
�Α m � x p �� Β
n�m
����������������������������
Α m � Β
n�m
�
Β ��x� Α� n�m p m
����������������������������
Β � Α
n�m m
If p = 1, the decomposition formula reduces to the sum of just two components, xΑ and xΒ ,
where xΑ lies in Α and xΒ lies in Β .
m n�m
Α ��x� Β � Β ��x� Α� m n�m n�m m
x � xΑ � xΒ � ���������������������������� � ����������������������������
Α � Β
Β � Α
m n�m
n�m m
We now explore this decomposition with a number of geometric examples, beginning with the
simplest case of decomposing a point on a line into a sum of two weighted points in the line.
2001 4 5
4.3
4.4