Grassmann Algebra

TheExteriorProduct.nb 26
������a e1 � ge7���ce2 � de6���ae2 � he7�� �
���b e3 � fe6���fe3 � ce6�� �
���e e5���ee4 � de8���ge5 � be8���
abce 2 �bc� f 2 � he1 � e2 � e3 � e4 � e5 � e6 � e7 � e8
Historical Note
Grassmann applied his Ausdehnungslehre to the theory of determinants and linear equations
quite early in his work. Later, Cauchy published his technique of 'algebraic keys' which
essentially duplicated Grassmann's results. To claim priority, Grassmann was led to publish
his only paper in French, obviously directed at Cauchy: 'Sur les diffŽrents genres de
multiplication' ('On different types of multiplication') [26]. For a complete treatise on the
theory of determinants from a Grassmannian viewpoint see R. F. Scott [51].
2.8 Cofactors
Cofactors from exterior products
The cofactor is an important concept in the usual approach to determinants. One often calculates
a determinant by summing the products of the elements of a row by their corresponding
cofactors. Cofactors divided by the determinant form the elements of an inverse matrix. In the
subsequent development of the Grassmann algebra, particularly the development of the
complement, we will find it useful to see how cofactors arise from exterior products.
Consider the product of n 1-elements introduced in the previous section:
Α1 � Α2 � � � Αn � �a11�e1 � a12�e2 � � � a1�n�en�
� �a21�e1 � a22�e2 � � � a2�n�en�
� � � �
� �an1�e1 � an2�e2 � � � ann�en�
Omitting the first factor Α1 :
Α2 � � � Αn � �a21�e1 � a22�e2 � � � a2�n�en�
� � � �
� �an1�e1 � an2�e2 � � � ann�en�
and multiplying out the remaining nÐ1 factors results in an expression of the form:
Α2 � � � Αn � a11 ������ �e2 � e3 � � � en �
a12 ������ ��� e1 � e3 � � � en� � � � a1�n ������� ���1�n�1 �e1 � e2 � � � en�1
Here, the signs of the basis (nÐ1)-elements have been specifically chosen so that they
correspond to the cobasis elements of e1 to en . The underscored scalar coefficients have yet to
be interpreted. Thus we can write:
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