Grassmann Algebra

TheExteriorProduct.nb 23
That is, the cobasis element of a cobasis element of an element is (apart from a possible sign)
equal to the element itself. We shall find this formula useful as we develop general formulae for
the complement and interior products in later chapters.
2.7 Determinants
Determinants from exterior products
All the properties of determinants follow naturally from the properties of the exterior product.
The determinant:
��������
a11
a21
D � �
�������
�
� an1
a12
a22
�
an2
�
�
�
�
a1�n
�������� a2�n
�
� ������� ann �
may be calculated by considering each of its rows (or columns) as a 1-element. Here, we
consider rows. Development by columns may be obtained mutatis mutandis. Introduce an
arbitrary basis ei and let
Αi � ai1�e1 � ai2�e2 � � � ain�en
Then form the exterior product of all the Αi . We can arrange the Αi in rows to portray the
effect of an array:
Α1 � Α2 � � � Αn � �a11�e1 � a12�e2 � � � a1�n�en�
� �a21�e1 � a22�e2 � � � a2�n�en�
� � � �
� �an1�e1 � an2�e2 � � � ann�en�
The determinant D is then the coefficient of the resulting n-element:
Α1 � Α2 � � � Αn � De1 � e2 � � � en
Because � n is of dimension 1, we can express this uniquely as:
Properties of determinants
D � Α1 � Α2 � � � Αn
�����������������������������������
e1 � e2 � � � en
All the well-known properties of determinants proceed directly from the properties of the
exterior product.
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2.28