Grassmann Algebra

TheComplement.nb 3
an m-element and an (nÐm)-element is a scalar. Thus there is the opportunity of defining for
each m-element a corresponding 'co-m-element' of grade nÐm such that the regressive product of
these two elements gives a scalar. We will see that this scalar measures the square of the
'magnitude' of the m-element or the (nÐm)-element, and corresponds to the inner product of
either of them with itself. We will also see that the notion of orthogonality is defined by the
correspondence between m-elements and their 'co-m-elements'. But most importantly, the
definition of this inner product as a regressive product of an m-element with a 'co-m-element' is
immediately generalizable to elements of arbitrary grade, thus permitting a theory of interior
products to be developed which is consistent with the exterior and regressive product axioms
and which, via the notion of 'co-m-element', leads to explicit and easily derived formulae
between elements of arbitrary (and possibly different) grade.
The foundation of the notion of measure or metric then is the notion of 'co-m-element'. In this
book we use the term complement rather than 'co-m-element'. In this chapter we will develop the
notion of complement in preparation for the development of the notions of interior product and
orthogonality in the next chapter.
The complement of an element is denoted with a horizontal bar over the element. For example
�����
Α, x+y and x�y are denoted Α ,
���������
x � y and
���������
x � y.
m m
Finally, it should be noted that the term 'complement' may be used either to refer to an operation
(the operation of taking the complement of an element), or to the element itself (which is the
result of the operation).
Historical Note
Grassmann introduced the notion of complement (ErgŠnzung) into the Ausdehnungslehre of
1862 [Grassmann 1862]. He denoted the complement of an element x by preceding it with a
vertical bar, viz |x. For mnemonic reasons which will become apparent later, the notation for the
complement used in this book is rather the horizontal bar: x
����� . In discussing the complement,
Grassmann defines the product of the n basis elements (the basis n-element) to be unity. That is
�e1�e2�� �en� � 1 or, in the present notation, e1 � e2 � � � en � 1. Since Grassmann
discussed only the Euclidean complement (equivalent to imposing a Euclidean metric
gij �∆ij ), this statement in the present notation is equivalent to 1
����� � 1. The introduction of
such an identity, however, destroys the essential duality between � and � which requires
m n�m
rather the identity 1
����� �����
� 1. In current terminology, equating e1 � e2 � � � en to 1 is
equivalent to equating n-elements (or pseudo-scalars) and scalars. All other writers in the
Grassmannian tradition (for example, Hyde, Whitehead and Forder) followed Grassmann's
approach. This enabled them to use the same notation for both the progressive and regressive
products. While being an attractive approach in a Euclidean system, it is not tenable for general
metric spaces. The tenets upon which the Ausdehnungslehre are based are so geometrically
fundamental however, that it is readily extended to more general metrics.
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