Grassmann Algebra

TheExteriorProduct.nb 2
� Calculation of determinants using minors and cofactors
Transformations of cobases
2.9 Solution of Linear Equations
Grassmann's approach to solving linear equations
� Example solution
2.10 Simplicity
The concept of simplicity
All (nÐ1)-elements are simple
� Conditions for simplicity of a 2-element in a 4-space
� Conditions for simplicity of a 2-element in a 5-space
2.11 Exterior division
The definition of an exterior quotient
Division by a 1-element
Division by a k-element
2.1 Introduction
The exterior product is the natural fundamental product operation for elements of a linear space.
Although it is not a closed operation (that is, the product of two elements is not itself an element
of the same linear space), the products it generates form a series of new linear spaces, the
totality of which can be used to define a true algebra, which is closed.
The exterior product is naturally and fundamentally connected with the notion of linear
dependence. Several 1-elements are linearly dependent if and only if their exterior product is
zero. All the properties of determinants flow naturally from the simple axioms of the exterior
product. The notion of 'exterior' is equivalent to the notion of linear independence, since
elements which are truly exterior to one another (that is, not lying in the same space) will have a
non-zero product. This product has a straightforward geometric interpretation as the multidimensional
equivalent to the 1-elements from which it is constructed. And if the space
possesses a metric, its measure or magnitude may be interpreted as a length, area, volume, or
hyper-volume according to the grade of the product.
However, the exterior product does not require the linear space to possess a metric. This is in
direct contrast to the three-dimensional vector calculus in which the vector (or cross) product
does require a metric, since it is defined as the vector orthogonal to its two factors, and
orthogonality is a metric concept. Some of the results which use the cross product can equally
well be cast in terms of the exterior product, thus avoiding unnecessary assumptions.
We start the chapter with an informal discussion of the exterior product, and then collect the
axioms for exterior linear spaces together more formally into a set of 12 axioms which combine
those of the underlying field and underlying linear space with the specific properties of the
exterior product. In later chapters we will derive equivalent sets for the regressive and interior
products, to which we will often refer.
Next, we pin down the linear space notions that we have introduced axiomatically by
introducing a basis onto the primary linear space and showing how this can induce a basis onto
each of the other linear spaces generated by the exterior product. A constantly useful partner to
a basis element of any of these exterior linear spaces is its cobasis element. The exterior product
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