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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 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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