bilinear form: meaning, definition, pronunciation and examples

Yes, on real vector spaces, an inner product is precisely a symmetric, positive-definite bilinear form. On complex vector spaces, inner products are 'sesquilinear' (linear in one argument, conjugate-linear in the other), not bilinear.

A linear map has one vector input and one output (vector or scalar). A bilinear form has two vector inputs and one scalar output, and it is linear in each input when the other is held fixed.

Yes. While often defined on V × V, it can be a map from V × W to a field F, where V and W are potentially different vector spaces. This is sometimes called a 'pairing'.

Given a basis for the vector space, a bilinear form is uniquely represented by a square matrix B such that B(u, v) = u^T * M * v, where u and v are coordinate column vectors relative to that basis, and M is the matrix of the form.

A function that is linear in each of its two arguments separately. In mathematics, it maps two vectors from a vector space (or two vectors from possibly different vector spaces) to a scalar, and it satisfies linearity properties when fixing one argument and varying the other.

Bilinear form is usually technical/formal in register.

Bilinear form: in British English it is pronounced /ˌbaɪˈlɪn.i.ə ˌfɔːm/, and in American English it is pronounced /ˌbaɪˈlɪn.i.ɚ ˌfɔːrm/. Tap the audio buttons above to hear it.