Grassmann Algebra

TheRegressiveProduct.nb 10
1. Replace each product operation by its dual operation, and the grades of elements and spaces
by their complementary grades.
2. Replace arbitrary grades m with nÐm', k with nÐk'. Drop the primes.
Since this algorithm applies to both sets of axioms, it also applies to any theorem. Thus to each
theorem involving exterior or regressive products corresponds a dual theorem obtained by
applying the algorithm. We call this the Duality Principle.
The Duality Principle
To every theorem involving exterior and regressive products, a dual theorem may be
obtained by:
1. Replacing each product operation by its dual operation, and the grades of elements and
spaces by their complementary grades.
2. Replacing arbitrary grades m with nÐm', k with nÐk', then dropping the primes.
Example 1
The following theorem says that the exterior product of two elements is zero if the sum of their
grades is greater than the dimension of the linear space.
�Α m � Β k
� 0, m � k � n � 0�
The dual theorem states that the regressive product of two elements is zero if the sum of their
grades is less than the dimension of the linear space.
�Α m � Β k
� 0, n � �k � m� � 0�
We can recover the original theorem as the dual of this one.
Example 2
This theorem below says that the exterior product of an element with itself is zero if it is of odd
grade.
�Α m � Α m � 0, m � �OddIntegers��
The dual theorem states that the regressive product of an element with itself is zero if its
complementary grade is odd.
�Α m � Α m � 0, �n � m� � �OddIntegers��
Again, we can recover the original theorem by calculating the dual of this one.
Example 3
The following theorem that states that the exterior product of unity with itself any number of
times remains unity.
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