Grassmann Algebra

TheExteriorProduct.nb 4
This may be proved from the distributivity and nilpotency axioms since:
�x � y���x � y� � 0
� x � x � x � y � y � x � y � y � 0
� x � y � y � x � 0
� x � y ��y � x
An element of � will be called a 2-element (of grade 2) and is denoted by a kernel letter with a
2
'2' written below. For example:
Α 2 � x � y � z � w � �
A simple 2-element is the exterior product of two 1-elements.
It is important to note the distinction between a 2-element and a simple 2-element (a distinction
of no consequence in �, where all elements are simple). A 2-element is in general a sum of
1
simple 2-elements, and is not generally expressible as the product of two 1-elements (except
where � is of dimension n � 3). The structure of �, whilst still that of a linear space, is thus
1 2
richer than that of �, that is, it has more properties.
1
Example: 2-elements in a three-dimensional space are simple
By way of example, the following shows that when � is of dimension 3, every element of � is
1 2
simple.
Suppose that � has basis e1 , e2 , e3 . Then a basis of � is the set of all essentially different
1 2
(linearly independent) products of basis elements of � taken two at a time: e1 � e2 , e2 � e3 ,
1
e3 � e1 . (The product e1 � e2 is not considered essentially different from e2 � e1 in view of
the anti-symmetry property).
Let a general 2-element Α 2 be expressed in terms of this basis as:
Α 2 � ae1 � e2 � be2 � e3 � ce3 � e1
Without loss of generality, suppose a ¹ 0. Then Α can be recast in the form below, thus proving
2
the proposition.
Α � a�
2 ���e1
�
�
b ���� �e3
a ���
��e2 �
�
c
���� �e3�
a
Historical Note
In the Ausdehnungslehre of 1844 Grassmann denoted the exterior product of two symbols Α and
Β by a simple concatenation viz. Α Β; whilst in the Ausdehnungslehre of 1862 he enclosed them
in square brackets viz. [Α Β]. This notation has survived in the three-dimensional vector
calculus as the 'box' product [Α Β Γ] used for the triple scalar product. Modern usage denotes
the exterior product operation by the wedge �, thus Α�Β. Amongst other writers, Whitehead [ ]
used the 1844 version whilst Forder [ ] and Cartan [ ] followed the 1862 version.
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