Grassmann Algebra

TheExteriorProduct.nb 32
2.9 Solution of Linear Equations
Grassmann's approach to solving linear equations
Because of its encapsulation of the properties of linear independence, Grassmann was able to
use the exterior product to present a theory and formulae for the solution of linear equations
well before anyone else.
Suppose m independent equations in n (m £ n) unknowns xi :
a11�x1 � a12�x2 � � � a1�n�xn � a1
a21�x1 � a22�x2 � � � a2�n�xn � a2
� � � � �
am1�x1 � am2�x2 � � � amn�xn � am
Multiply these equations by e1 , e2 , �, em respectively and define
Ci � a1�i�e1 � a2�i�e2 � � � ami�em
C0 � a1�e1 � a2�e2 � � � am �em
The Ci and C0 are therefore 1-elements in a linear space of dimension m.
Adding the resulting equations then gives the system in the form:
x1�C1 � x2�C2 � � � xn�Cn � C0
To obtain an equation from which the unknowns xi have been eliminated, it is only necessary
to multiply the linear system through by the corresponding Ci .
If m = n and a solution for xi exists, it is obtained by eliminating x1 , �, xi�1 , xi�1 , �, xn ;
that is, by multiplying the linear system through by C1 � � � Ci�1 � Ci�1 � � � Cn :
xi Ci � C1 � � � Ci�1 � Ci�1 � � � Cn � C0 � C1 � � � Ci�1 � Ci�1 � � � Cn
Solving for x i gives:
xi � C0 ��C1 � � � Ci�1 � Ci�1 � � � Cn�
�������������������������������� �������������������������������� �������������
Ci ��C1 � � � Ci�1 � Ci�1 � � � Cn�
In this form we only have to calculate the (nÐ1)-element C1 � � � Ci�1 � Ci�1 � � � Cn once.
An alternative form more reminiscent of Cramer's Rule is:
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2.31
2.32