Bilinear form
Scalar-valued bilinear function / From Wikipedia, the free encyclopedia
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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(λu, v) = λB(u, v)
- B(u, v + w) = B(u, v) + B(u, w) and B(u, λv) = λB(u, v)
The dot product on is an example of a bilinear form.[1]
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.