Grassmann Algebra

TheRegressiveProduct.nb 7
The unit n-element
� The unit n-element is congruent to any basis n-element.
The duality algorithm has generated a unit n-element 1 which acts as the multiplicative identity
n
for the regressive product, just as the unit scalar 1 (or 1) acts as the multiplicative identity for
0
the exterior product (axioms �8 and �8).
We have already seen that any basis of � n contains only one element. If the basis of � 1 is
�e1, e2, �, en�, then the single basis element of � is congruent to e1 � e2 � � � en . If
n
the basis of � is changed by an arbitrary (non-singular) linear transformation, then the basis of
1
� n changes by a scalar factor which is the determinant of the transformation. Any basis of � n may
therefore be expressed as a scalar multiple of some given basis, say e1 � e2 � � � en . Hence
we can therefore also express 1 n as some scalar multiple � of e1 � e2 � � � en .
1 n � � �e1 � e2 � � � en
3.10
Defining 1 any more specifically than this is normally done by imposing a metric onto the
n
space. This we do in Chapter 5: The Complement, and Chapter 6: The Interior Product. It turns
out then that 1 is the n-element whose measure (magnitude, volume) is unity.
n
On the other hand, for geometric application in spaces without a metric, for example the
calculation of intersections of lines, planes, and hyperplanes, it is inconsequential that we only
know 1 up to congruence, because we will see that if an element defines a geometric entity then
n
any element congruent to it will define the same geometric entity.
� The unit n-element is idempotent under the regressive product.
By putting Α m equal to 1 n in the regressive product axiom �8 we have immediately that:
� n-elements allow a sort of associativity.
1 n � 1 n � 1 n
By letting Α n � a�1 n we can show that Α n allows a sort of associativity with the exterior product.
2001 4 5
�Α � Β��Γ n k p
�Α� n ����Β
�k
�
� ��� p�
3.11
3.12