digplanet beta 1: Athena
Share digplanet:

Agriculture

Applied sciences

Arts

Belief

Business

Chronology

Culture

Education

Environment

Geography

Health

History

Humanities

Language

Law

Life

Mathematics

Nature

People

Politics

Science

Society

Technology

In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example.

Definition[edit]

Vector spaces[edit]

Let V, W and X be three vector spaces over the same base field F. A bilinear map is a function

B : V × WX

such that for any w in W the map

vB(v, w)

is a linear map from V to X, and for any v in V the map

wB(v, w)

is a linear map from W to X.

In other words, when we hold the first entry of the bilinear map fixed while letting the second entry vary, the result is a linear operator, and similarly for when we hold the second entry fixed.

If V = W and we have B(v, w) = B(w, v) for all v, w in V, then we say that B is symmetric.

The case where X is the base field F, and we have a bilinear form, is particularly useful (see for example scalar product, inner product and quadratic form).

Modules[edit]

The definition works without any changes if instead of vector spaces over a field F, we use modules over a commutative ring R. It generalizes to n-ary functions, where the proper term is multilinear.

For non-commutative rings R and S, a left R-module M and a right S-module N, a bilinear map is a map B : M × NT with T an (R, S)-bimodule, and for which any n in N, mB(m, n) is an R-module homomorphism, and for any m in M, nB(m, n) is an S-module homomorphism. This satisfies

B(rm, n) = rB(m, n)
B(m, ns) = B(m, n) ⋅ s

for all m in M, n in N, r in R and s in S, as well as B being additive in each argument.

Properties[edit]

A first immediate consequence of the definition is that B(v, w) = 0X whenever v = 0V or w = 0W. This may be seen by writing the zero vector 0X as 0 ⋅ 0X (and similarly for 0W) and moving the scalar 0 "outside", in front of B, by linearity.

The set L(V, W; X) of all bilinear maps is a linear subspace of the space (viz. vector space, module) of all maps from V × W into X.

A matrix M determines a bilinear map into the real by means of a real bilinear form (v, w) ↦ vMw, then associates of this are taken to the other three possibilities using duality and the musical isomorphism

If V, W, X are finite-dimensional, then so is L(V, W; X). For X = F, i.e. bilinear forms, the dimension of this space is dim V × dim W (while the space L(V × W; F) of linear forms is of dimension dim V + dim W). To see this, choose a basis for V and W; then each bilinear map can be uniquely represented by the matrix B(ei, fj), and vice versa. Now, if X is a space of higher dimension, we obviously have dim L(V, W; X) = dim V × dim W × dim X.

Examples[edit]

  • Matrix multiplication is a bilinear map M(m, n) × M(n, p) → M(m, p).
  • If a vector space V over the real numbers R carries an inner product, then the inner product is a bilinear map V × VR.
  • In general, for a vector space V over a field F, a bilinear form on V is the same as a bilinear map V × VF.
  • If V is a vector space with dual space V, then the application operator, b(f, v) = f(v) is a bilinear map from V × V to the base field.
  • Let V and W be vector spaces over the same base field F. If f is a member of V and g a member of W, then b(v, w) = f(v)g(w) defines a bilinear map V × WF.
  • The cross product in R3 is a bilinear map R3 × R3R3.
  • Let B : V × WX be a bilinear map, and L : UW be a linear map, then (v, u) ↦ B(v, Lu) is a bilinear map on V × U.

See also[edit]

External links[edit]


Original courtesy of Wikipedia: http://en.wikipedia.org/wiki/Bilinear_map — Please support Wikipedia.
This page uses Creative Commons Licensed content from Wikipedia. A portion of the proceeds from advertising on Digplanet goes to supporting Wikipedia.
478 videos foundNext > 

3rd BIU Winter School on Cryptography:The Basics of Pairings - Dan Boneh

The 3rd Bar-Ilan Winter School on Cryptography: Bilinear Pairings in Cryptography, which was held between February 4th - 7th, 2013. The event's program: ...

Self-bilinear Map on Unknown Order Groups from Indistinguishability Obfuscation and Its Applications

Self-bilinear Map on Unknown Order Groups from Indistinguishability Obfuscation and Its Applications by Goichiro Hanaoka, Noboru Kunihiro, Takashi ...

Unit-3 Bilinear Transformation - Mathematics

http://www.gurug.net Unit-3 Bilinear Transformation - Mathematics.

CCA Proxy Re-Encryption without Bilinear Maps in the Standard Model.

Talk at pkc 2010. Authors: Toshihide Matsuda, Ryo Nishimaki, Keisuke Tanaka. See http://www.iacr.org/cryptodb/data/paper.php?pubkey=23414.

Bilinear transform: introduction (#0000)

The problem with the impulse invariance / matched z method is that they map frequencies from analogue linearly. This means that the mapping works only up to ...

Bilinear form

Bilinear form In mathematics, more specifically in abstract algebra and linear algebra, a bilinear form on a vector space V is a bilinear map V × V → K, where K is ...

Tutorial: Bilinear Transform

The bilinear transform maps a continuous-time filter to a discrete-time filter free of aliasing distortion. The "Bilinear Transform Visualizer" LabVIEW application VI ...

bilinear 2x2 image

Test of bilinear mapping in Image data structure generation.

29 - Bilinear transform

In this lecture, we discuss how to transform an analog filter into a digital filter This lecture is adapted from the ECE 410: Digital Signal Processing course notes ...

The Bilinear Transform

http://AllSignalProcessing.com for more great signal processing content, including concept/screenshot files, quizzes, MATLAB and data files. Definition and ...

478 videos foundNext > 

We're sorry, but there's no news about "Bilinear map" right now.

Loading

Oops, we seem to be having trouble contacting Twitter

Support Wikipedia

A portion of the proceeds from advertising on Digplanet goes to supporting Wikipedia. Please add your support for Wikipedia!

Searchlight Group

Digplanet also receives support from Searchlight Group. Visit Searchlight