An alternating multilinear form on a real vector space
is a multilinear form
![F:V tensor ... tensor V->R](/images/equations/AlternatingMultilinearForm/NumberedEquation1.svg) |
(1)
|
such that
![F(x_1,...,x_i,x_(i+1),...,x_n)=-F(x_1,...,x_(i+1),x_i,...,x_n)](/images/equations/AlternatingMultilinearForm/NumberedEquation2.svg) |
(2)
|
for any index
. For example,
![F((a_1,a_2,a_3),(b_1,b_2,b_3),(c_1,c_2,c_3))
=a_1b_2c_3-a_1b_3c_2+a_2b_3c_1-a_2b_1c_3+a_3b_1c_2-a_3b_2c_1](/images/equations/AlternatingMultilinearForm/NumberedEquation3.svg) |
(3)
|
is an alternating form on
.
An alternating multilinear form is defined on a module in a similar way, by replacing
with the ring.
See also
Dual Vector Space,
Exterior Algebra,
Module,
Multilinear
Form,
Vector Space
This entry contributed by Todd
Rowland
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Cite this as:
Rowland, Todd. "Alternating Multilinear Form." From MathWorld--A Wolfram Web Resource, created
by Eric W. Weisstein. https://mathworld.wolfram.com/AlternatingMultilinearForm.html
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