Let be vector spaces over a field , and be a homomorphism (a linear map) between them. Letting denote the corresponding dual spaces, we define the dual homomorphism , to be the linear mapping with action
To be more precise is characterized by the condition
Category theory perspective.
The dualizing operation behaves contravariantly with respect to composition, i.e.
for all vector space homomorphisms with suitably matched domains. Furthermore, the dual of the identity homomorphism is the identity homomorphism of the dual space. Thus, using the language of category theory, the dualizing operation can be characterized as the homomorphism action of the contravariant, dual-space functor.
Relation to the matrix transpose.
The above properties closely mirror the algebraic properties of the matrix transpose operation. Indeed, is sometimes referred to as the transpose of , because at the level of matrices the dual homomorphism is calculated by taking the transpose.
To be more precise, suppose that and are finite-dimensional, and let be the matrix of relative to some fixed bases of and . Then, the dual homomorphism is represented as the transposed matrix relative to the corresponding dual bases of .
|Date of creation||2013-03-22 12:29:33|
|Last modified on||2013-03-22 12:29:33|
|Last modified by||rmilson (146)|