Grassmann Algebra

TheRegressiveProduct.nb 2
3.9 Product Formulae for Regressive Products
The Product Formula
The General Product Formula
� Exploring the General Product Formula
Decomposition formulae
Expressing an element in terms of another basis
Product formulae leading to scalar results
3.1 Introduction
Since the linear spaces � an � are of the same dimension �
m n�m n n
� = � � and hence are
m n � m
isomorphic, the opportunity exists to define a product operation (called the regressive product)
dual to the exterior product such that theorems involving exterior products of m-elements have
duals involving regressive products of (nÐm)-elements.
Very roughly speaking, if the exterior product is associated with the notion of union, then the
regressive product is associated with the notion of intersection.
The regressive product appears to be unused in the recent literature. Grassmann's original
development did not distinguish notationally between the exterior and regressive product
operations. Instead he capitalized on the inherent duality and used a notation which, depending
on the grade of the elements in the product, could only sensibly be interpreted by one or the
other operation. This was a very elegant idea, but its subtlety may have been one of the reasons
the notion has become lost. (See the historical notes in Section 3.3.)
However, since the regressive product is a simple dual operation to the exterior product, an
enticing and powerful symmetry is lost by ignoring it. We will find that its 'intersection'
properties are a useful conceptual and algorithmic addition to non-metric geometry, and that its
algebraic properties enable a firm foundation for the development of metric geometry. Some of
the results which are proven in metric spaces via the inner and cross products can also be proven
using the exterior and regressive products, thus showing that the result is independent of
whether or not the space has a metric.
The approach adopted in this book is to distinguish between the two product operations by using
different notations, just as Boolean algebra has its dual operations of union and intersection. We
will find that this approach does not detract from the elegance of the results. We will also find
that differentiating the two operations explicitly enhances the simplicity and power of the
derivation of results.
Since the commonly accepted modern notation for the exterior product operation is the 'wedge'
symbol �, we will denote the regressive product operation by a 'vee' symbol �. Note however
that this (unfortunately) does not correspond to the Boolean algebra usage of the 'vee' for union
and the 'wedge' for intersection.
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