Concept

# Bilinear form

Summary
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 \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. Coordinate representation Let V be an n-dimensional vector space with basis {e1, …, en}. The n × n matrix A, defined by Aij = B(ei,
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