Alternating Multilinear Form
An alternating multilinear form on a real vector space V is a multilinear form
| F:V tensor ... tensor V->R |
(1)
|
such that
| F(x_1,...,x_i,x_(i+1),...,x_n)=-F(x_1,...,x_(i+1),x_i,...,x_n) |
(2)
|
for any index i. 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 |
(3)
|
is an alternating form on R^3.
An alternating multilinear form is defined on a module in a similar way, by replacing R with the ring.
See also
Dual Vector Space, Exterior Algebra, Module, Multilinear Form, Vector SpaceThis entry contributed by Todd Rowland
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Rowland, Todd. "Alternating Multilinear Form." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/AlternatingMultilinearForm.html