Grassmann Algebra

TheInteriorProduct.nb 10
����������� �����
Α � Β � Αm
m m
����� ����� �����
m��n�m�
� Βm � ��1�
�����
�Αm � Βm
� Β m
�
�����
Α � Βm ���� Α
m
m
From these two results we see that the order of the factors in an inner product is immaterial, that
is, the inner product is symmetric.
Α m ���� Β m
� Β m
���� Α m
6.24
This symmetry, is of course, a property that we would expect of an inner product. However it is
interesting to remember that the major result which permits this symmetry is that the
complement operation has been required to satisfy the requirement of formula 5.30, that is, that
the complement of the complement of an element should be, apart from a possible sign, equal to
the element itself.
We have already shown in ���� 10 of the previous section that:
Α m ���� Β k
� ��1� �n�m���m�k� � k
����� �����
���� Αm
Putting k = m, and using the above result on the symmetry of the inner product shows that the
inner product of two elements of the same grade is also equal to the inner product of their
complements.
The scalar product
Α m ���� Β m
����� �����
� Α ���� Βm
m
A scalar product is simply an interior or inner product of 1-elements. If x and y are 1-elements,
then x����y is a scalar product.
Example: Orthogonalizing a set of 1-elements
Suppose we have a set of independent 1-elements Α1 , Α2 , Α3 , �, and we wish to create an
orthogonal set e1 , e2 , e3 , � spanning the same space.
We begin by choosing one of the elements, Α1 say, arbitrarily and setting this to be the first
element e1 in the orthogonal set to be created.
e1 �Α1
To create a second element e2 orthogonal to e1 within the space concerned, we choose a
second element of the space, Α2 say, and form the interior product.
e2 � �e1 � Α2� ���� e1
We can see that e2 is orthogonal to e1 by taking their interior product and using formula 6.21
to see that the product is zero:
2001 4 5
e2 ���� e1 � ��e1 � Α2� ���� e1� ���� e1 � �e1 � Α2� ���� �e1 � e1� � 0
6.25