Grassmann Algebra

TheComplement.nb 2
� Converting to complement form
� Simplifying complements
� Creating tables and palettes of complements of basis elements
5.9 Geometric Interpretations
The Euclidean complement in a vector 2-space
� The Euclidean complement in a plane
� The Euclidean complement in a vector 3-space
5.10 Complements in a vector subspace of a multiplane
Metrics in a multiplane
The complement of an m-vector
The complement of an element bound through the origin
The complement of the complement of an m-vector
The complement of a bound element
� Calculating with free complements
Example: The complement of a screw
5.11 Reciprocal Bases
Contravariant and covariant bases
The complement of a basis element
The complement of a cobasis element
The complement of a complement of a basis element
The exterior product of basis elements
The regressive product of basis elements
The complement of a simple element is simple
5.12 Summary
5.1 Introduction
Up to this point various linear spaces and the dual exterior and regressive product operations
have been introduced. The elements of these spaces were incommensurable unless they were
congruent, that is, nowhere was there involved the concept of measure or magnitude of an
element by which it could be compared with any other element. Thus the subject of the last
chapter on Geometric Interpretations was explicitly non-metric geometry; or, to put it another
way, it was what geometry is before the ability to compare or measure is added.
The question then arises, how do we associate a measure with the elements of a Grassmann
algebra in a consistent way? Of course, we already know the approach that has developed over
the last century, that of defining a metric tensor. In this book we will indeed define a metric
tensor, but we will take an approach which develops from the concepts of the exterior product
and the duality operations which are the foundations of the Grassmann algebra. This will enable
us to see how the metric tensor on the underlying linear space generates metric tensors on the
exterior linear spaces of higher grade; the implications of the symmetry of the metric tensor; and
how consistently to generalize the notion of inner product to elements of arbitrary grade.
One of the consequences of the anti-symmetry of the exterior product of 1-elements is that the
exterior linear space of m-elements has the same dimension as the exterior linear space of (nÐm)elements.
We have already seen this property evidenced in the notions of duality and cobasis
element. And one of the consequences of the notion of duality is that the regressive product of
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