Grassmann Algebra

Introduction.nb 9
�a�Α m ��Β k
�� Α m ��a�Β k
� � a��Α � Β �
m k
¥ An exterior product is anti-commutative whenever the grades of the factors are both odd.
Α m � Β k
� ��1� mk  k
¥ The exterior product is both left and right distributive under addition.
�Α m �Β m
��à r
� Α m � Γ r
�Β� Γ
m r
1.3 The Regressive Product
Α m ��Β r
� Α m
�Γ� � Α�Β �Α� Γ
r m r m r
The regressive product as a dual product to the exterior product
1.5
One of Grassmann's major contributions, which appears to be all but lost to current
mathematics, is the regressive product. The regressive product is the real foundation for the
theory of the inner and scalar products (and their generalization, the interior product). Yet the
regressive product is often ignored and the inner product defined as a new construct independent
of the regressive product. This approach not only has potential for inconsistencies, but also fails
to capitalize on the wealth of results available from the natural duality between the exterior and
regressive products. The approach adopted in this book follows Grassmann's original concept.
The regressive product is a simple dual operation to the exterior product and an enticing and
powerful symmetry is lost by ignoring it, particularly in the development of metric results
involving complements and interior products.
The underlying beauty of the Ausdehnungslehre is due to this symmetry, which in turn is due to
the fact that linear spaces of m-elements and linear spaces of (nÐm)-elements have the same
dimension. This too is the key to the duality of the exterior and regressive products. For
example, the exterior product of m 1-elements is an m-element. The dual to this is that the
regressive product of m (nÐ1)-elements is an (nÐm)-element. This duality has the same form as
that in a Boolean algebra: if the exterior product corresponds to a type of 'union' then the
regressive product corresponds to a type of 'intersection'.
It is this duality that permits the definition of complement, and hence interior, inner and scalar
products defined in later chapters. To underscore this duality it is proposed to adopt here the �
('vee') for the regressive product operation.
2001 4 5
1.6
1.7