Bilinear form

In mathematics, a bilinear form is a bilinear map $V \times V \to K$ 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 $B : V \times V \to K$ 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 $\mathbb {R} ^{n}$ 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.

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