: Mathematics: bilinear form

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bilinear form

In mathematics, a bilinear form is a bilinear map V × V → K on a vector space V (the elements of which are called vectors) over a field K (the elements of which are called scalars). In other words, a bilinear form is a function B : V × V → K that is linear in each argument separately:
- $B (u + v, w) = B (u, w) + B (v, w)$ and $B (λ u, v) = λB (u, v)$
- $B (u, v + w) = B (u, v) + B (u, w)$ and $B (u, λ v) = λB (u, v)$
The dot product on ${\\displaystyle \\mathbb {R} ^{n}}$ ${\\displaystyle \\mathbb {R} ^{n}}$ is an example of a bilinear form. The definition of a bilinear form can be extended to include modules over a ring, with linear maps replaced by module homomorphisms. When K is the field of complex numbers C, one is often more interested in sesquilinear forms, which are similar to bilinear forms but are conjugate linear in one argument.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Bilinear_form)

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