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Symmetric bilinear space


In mathematics, more specifically in abstract algebra and linear algebra, a bilinear form on a vector space V is a bilinear map V × VK, where K is the field of scalars. In other words, a bilinear form is a function B : V × VK that is linear in each argument separately:

The definition of a bilinear form can be extended to include modules over a commutative 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.

Let VKn be an n-dimensional vector space with basis {e1, ..., en}. Define the n × n matrix A by Aij = B(ei, ej). If the n × 1 matrix x represents a vector v with respect to this basis, and analogously, y represents w, then:

Suppose {f1, ..., fn} is another basis for V, such that:

where S ∈ GL(n, K). Now the new matrix representation for the bilinear form is given by: STAS.

Every bilinear form B on V defines a pair of linear maps from V to its dual space V. Define B1, B2: VV by


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