Grassmann Algebra

Introduction.nb 17
Graphic showing the intersection of two lines.
To verify that P does indeed lie on both the lines L1 and L2 , we only need to carry out the
straightforward verification that the products P � L1 and P � L2 are both zero.
Although this approach in this simple case is certainly more complex than the standard algebraic
approach, its interest lies in the fact that it is immediately generalizable to intersections of any
geometric objects in spaces of any number of dimensions.
1.5 The Complement
The complement as a correspondence between spaces
The Grassmann algebra has a duality in its structure which not only gives it a certain elegance,
but is also the basis of its power. We have already introduced the regressive product as the dual
product operation to the exterior product. In this section we extend the notion of duality to
define the complement of an element. The notions of orthogonality and interior, inner and scalar
products are all based on the complement.
Consider a linear space of dimension n with basis e1 , e2 , É, en . The set of all the essentially
different m-element products of these basis elements forms the basis of another linear space, but
this time of dimension � n
�. For example, when n is 3, the linear space of 2-elements has three
m
elements in its basis: e1 � e2 , e1 � e3 , e2 � e3 .
The anti-symmetric nature of the exterior product means that there are just as many basis
elements in the linear space of (nÐm)-elements as there are in the linear space of m-elements.
Because these linear spaces have the same dimension, we can set up a correspondence between
m-elements and (nÐm)-elements. That is, given any m-element, we can determine its
corresponding (nÐm)-element. The (nÐm)-element is called the complement of the m-element.
Normally this correspondence is set up between basis elements and extended to all other
elements by linearity.
The Euclidean complement
Suppose we have a three-dimensional linear space with basis e1 , e2 , e3 . We define the
Euclidean complement of each of the basis elements as the basis 2-element whose exterior
product with the basis element gives the basis 3-element e1 � e2 � e3 . We denote the
complement of an element by placing a 'bar' over it. Thus:
�����
e1 � e2 � e3 � e1 � e1
����� � e1 � e2 � e3
�����
e2 � e3 � e1 � e2 � e2
����� � e1 � e2 � e3
�����
e3 � e1 � e2 � e3 � e3
����� � e1 � e2 � e3
The Euclidean complement is the simplest type of complement and defines a Euclidean metric,
that is, where the basis elements are mutually orthonormal. This was the only type of
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