bilinear form
Noun
Meaning
bilinear form (plural bilinear forms)
(linear algebra) A function of two arguments from the same vector space which maps onto a field of scalars, which acts like a linear form with respect to either one of its arguments when the other one is held constant.
Synonyms
Examples
An almost symplectic manifold is a differentiable manifold equipped with a smoothly varying non-degenerate skew-symmetric bilinear form on each tangent space, i.e., a nondegenerate 2- form ω, called the symplectic form.
As above, we let (V, g) be an n -dimensional complex vector space equipped with a nondegenerate bilinear form.
For example, a bilinear form is the same thing as a (0, 2) -tensor; an inner product is an example of a (0, 2) -tensor, but not all (0, 2) -tensors are inner products.
For v, w in R 4 introduce the degenerate bilinear form : This degenerate scalar product projects distance measurements in R 4 onto the R 3 hyperplane.
Suppose that U has even dimension and a non-singular bilinear form with discriminant d, and suppose that V is another vector space with a quadratic form.
So, an inner product on a real vector space is a positive-definite symmetric bilinear form.