dual homomorphism


Let U,V be vector spacesMathworldPlanetmath over a field 𝕂, and T:UV be a homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath (a linear map) between them. Letting U*,V* denote the corresponding dual spacesPlanetmathPlanetmath, we define the dual homomorphism T*:V*U*, to be the linear mapping with action


We can also characterize T* as the adjointPlanetmathPlanetmathPlanetmath of T relative to the natural evaluation bracket between linear forms and vectors:


To be more precise T* is characterized by the condition


If U and V are finite dimensional, we can also characterize the dualizing operationMathworldPlanetmath as the compositionMathworldPlanetmathPlanetmath of the following canonical isomorphisms:


Category theory perspective.

The dualizing operation behaves contravariantly with respect to composition, i.e.


for all vector space homomorphisms S,T with suitably matched domains. Furthermore, the dual of the identityPlanetmathPlanetmathPlanetmathPlanetmath homomorphism is the identity homomorphism of the dual space. Thus, using the languagePlanetmathPlanetmath of category theoryMathworldPlanetmathPlanetmathPlanetmathPlanetmath, the dualizing operation can be characterized as the homomorphism action of the contravariant, dual-space functorMathworldPlanetmath.

Relation to the matrix transpose.

The above properties closely mirror the algebraic properties of the matrix transpose operation. Indeed, T* is sometimes referred to as the transposeMathworldPlanetmath of T, because at the level of matrices the dual homomorphism is calculated by taking the transpose.

To be more precise, suppose that U and V are finite-dimensional, and let MMatn,m(𝕂) be the matrix of T relative to some fixed bases of U and V. Then, the dual homomorphism T* is represented as the transposed matrix MtMatm,n(𝕂) relative to the corresponding dual bases of U*,V*.

Title dual homomorphism
Canonical name DualHomomorphism
Date of creation 2013-03-22 12:29:33
Last modified on 2013-03-22 12:29:33
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 10
Author rmilson (146)
Entry type Definition
Classification msc 15A72
Classification msc 15A04
Synonym adjoint homomorphism
Synonym adjoint
Related topic LinearTransformation
Related topic DualSpace
Related topic DoubleDualEmbedding