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Bilinear Form


A bilinear form on a real vector space is a function

 b:V×V->R

that satisfies the following axioms for any scalar alpha and any choice of vectors v,w,v_1,v_2,w_1, and w_2.

1. b(alphav,w)=b(v,alphaw)=alphab(v,w)

2. b(v_1+v_2,w)=b(v_1,w)+b(v_2,w)

3. b(v,w_1+w_2)=b(v,w_1)+b(v,w_2).

For example, the function b((x_1,x_2),(y_1,y_2))=x_1y_2+x_2y_1 is a bilinear form on R^2.

On a complex vector space, a bilinear form takes values in the complex numbers. In fact, a bilinear form can take values in any vector space, since the axioms make sense as long as vector addition and scalar multiplication are defined.


See also

Bilinear Function, Multilinear Form, Symmetric Bilinear Form, Vector Space

This entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd. "Bilinear Form." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/BilinearForm.html

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