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Multilinear Form

A multilinear form on a vector space V(F) over a field F is a map

f:V(F)×...×V(F)->F
(1)

such that

c·f(u_1,...,u_i,...,u_n)=f(u_1,...,c·u_i,...,u_n)
(2)

and

f(u_1,...,u_i,...,u_n)+f(u_1,...,u_i^',...,u_n) =f(u_1,...,u_i+u_i^',...,u_n)
(3)

for every c in F and any indexes i,j.

For example, the determinant of a square matrix of degree n is an n-linear form for the columns or rows of a matrix.

See also

Bilinear Form, Vector Space

This entry contributed by Marek Pomp

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Cite this as:

Weisstein, Eric W., with contributions by Marek Pomp. "Multilinear Form." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MultilinearForm.html

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