# Bilinear Form

In mathematics, a bilinear form on a vector space V is a bilinear mapping V × VF, where F is the field of scalars. That is, a bilinear form is a function B: V × VF which is linear in each argument separately:

• B(u + v, w) = B(u, w) + B(v, w)
• B(u, v + w) = B(u, v) + B(u, w)
• Bu, v) = B(u, λv) = λB(u, v)

The definition of a bilinear form can easily be extended to include modules over a commutative ring, with linear maps replaced by module homomorphisms. When F 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.

### Other articles related to "bilinear form, form, bilinear forms, forms, bilinear":

Pseudo-Euclidean Space - Geometry - Symmetric Bilinear Form
... The quadratic form q gives rise to a symmetric bilinear form defined as follows The quadratic form can be expressed in terms of the bilinear form ... Some authors use the terms "inner product" or "dot product" for the bilinear form, but these terms have a different meaning in this encyclopedia ...
An Application of Céa's Lemma
... one obtains the equivalent problem for all in with (here the bilinear form is given by the same expression as the inner product, this is not always ... exists a constant dependent only on the bilinear form such that for all in To explicitly calculate the error between and consider the function in that has the same values as at the nodes of the partition (so is ... is a different constant from the above (it depends only on the bilinear form, which implicitly depends on the interval ) ...
Coercive Function - Coercive Operators and Forms
... A bilinear form is called coercive if there exists a constant such that for all in It follows from the Riesz representation theorem that any symmetric ( for all in ), continuous ( for all in and some ... given a coercive operator self-adjoint operator the bilinear form defined as above is coercive ... The definitions of coercivity for functions, operators, and bilinear forms are closely related and compatible ...
Symmetric Bilinear Form
... A symmetric bilinear form is a bilinear form on a vector space that is symmetric ... Symmetric bilinear forms are of great importance in the study of orthogonal polarity and quadrics ... They are also more briefly referred to as just symmetric forms when "bilinear" is understood ...
Bilinear Form - On Normed Vector Spaces
... Definition A bilinear form on a normed vector space is bounded, if there is a constant C such that for all u, v ∈ V Definition A bilinear form on a normed vector space is elliptic, or coercive, if there is a ...

### Famous quotes containing the word form:

‘A thing is called by a certain name because it instantiates a certain universal’ is obviously circular when particularized, but it looks imposing when left in this general form. And it looks imposing in this general form largely because of the inveterate philosophical habit of treating the shadows cast by words and sentences as if they were separately identifiable. Universals, like facts and propositions, are such shadows.
David Pears (b. 1921)