Grassmann Algebra

TheComplement.nb 29
ComplementPalette
� BASIS COMPLEMENT
� 0
� l
� l
1
e1
e2
� 2 e1 � e2
e2 g11 ���������������������������
���������������������������������
2
�g11 g22 �g 12
e2 g12 ���������������������������
���������������������������������
2
�g11 g22
�g 12
e1 �e2 ���������������������������
���������������������������������
�g2 12 �g11 g22 �
�
e1 g12 ���������������������������
���������������������������������
2
�g11 g22 �g 12
e1 g22 ���������������������������
���������������������������������
2
�g11 g22
�g 12
���������������������������������
2 �g12 � g11 g22
5.9 Geometric Interpretations
The Euclidean complement in a vector 2-space
Consider a vector x in a 2-dimensional vector space expressed in terms of basis vectors e1 and
e2 .
x � ae1 � be2
Since this is a Euclidean vector space, we can depict the basis vectors at right angles to each
other. But note that since it is a vector space, we do not depict an origin.
Graphic of x and two orthogonal basis vectors at right angles to each other.
The complement of x is given by:
�����
x � ae1
����� � be2
����� � ae2 � be1
Remember, the Euclidean complement of a basis element is its cobasis element, and a basis
element and its cobasis element are defined by their exterior product being the basis n-element,
in this case e1 � e2 .
It is clear from simple geometry that x and x
����� are at right angles to each other, thus verifying
our geometric interpretation of the algebraic notion of orthogonality: a simple element and its
complement are orthogonal.
Taking the complement of x
����� gives -x:
����� �����
x � ae2
����� � be1
����� ��ae1 � be2 ��x
Or, we could have used the complement of a complement axiom:
����� ����� 1��2�1�
x � ��1� �x ��x
Continuing to take complements we find that we eventually return to the original element.
����� �����
x
�
��x
�����
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