Grassmann Algebra

TheExteriorProduct.nb 31
Verification of the calculations
Finally, we verify that these two results are the same and correspond to that given by the inbuilt
Mathematica function Det.
Expand�D1� � Expand�D2� � Det �� D0
True
Transformations of cobases
In this section we show that if a basis of the underlying linear space is transformed by a
transformation whose components are aij , then its cobasis is transformed by the induced
transformation whose components are the cofactors of the aij . For simplicity in what follows,
we use Einstein's summation convention in which a summation over repeated indices is
understood.
Let aij be a transformation on the basis ej to give the new basis ���� i . That is, ���� i � aij�ej .
Let aij be the corresponding transformation on the cobasis ej to give the new cobasis ����
� ����� ����� i .
That is, ����� ���� i
� aij ej .
� �����
Now take the exterior product of these two equations.
���� i �
�����
���� k � �aip�ep���akj ej�
� aip� akj �ep � ej
� ����� � �����
But the product of a basis element and its cobasis element is equal to the n-element of that basis.
That is, ���� i ������ ���� k �∆ik����� and ep � ej �∆pj�e . Substituting in the previous equation gives:
n ����� n
∆ik����� n � aip� akj
� �∆pj e n
Using the properties of the Kronecker delta we can simplify the right side to give:
∆ik����� n � aij� akj
� e n
We can now substitute ���� � Dewhere D � Det�aij � is the determinant of the
n n
transformation.
∆ik�D � aij� akj
�
This is precisely the relationship derived in the previous section for the expansion of a
determinant in terms of cofactors. Hence we have shown that akj � akj � ������ .
���� i � aij�ej �
�����
���� i
� aij
������ ej
�����
In sum: If a basis of the underlying linear space is transformed by a transformation whose
components are aij , then its cobasis is transformed by the induced transformation whose
components are the cofactors aij
������ of the aij .
2001 4 5
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