Transposed application

In mathematics, the concept of transposed application raises of the Linear algebra. With all Linear application U between two vector spaces E and F is associated the transposed application ${} ^t u$ defined by

\ forall \ ell \ in F^*, \ qquad {} ^t U (\ ell) = \ to elect \ circ u

It is a linear application of the dual Espace

F^*

of F in the dual space of E .

By employing the notation of the Hook of duality, the definition of the transposed application can be rewritten in the form

\ forall X \ in E, \ forall \ ell \ in F^*, \ qquad \ langle {} ^t U (\ ell), X \ rangle= \ langle \ ell, U (X) \ rangle

The application which has a linear application associates its transposed is called the transposition . Using the bilinearity of the hook, one shows that the application of transposition itself is a linear application of $L (E, F)$ in $L (F^*, E^*)$ .

The application of transposition also has properties with respect to the law produced. When v and U are respectively linear of E in F and of F in a third vector space G ,

{} ^t (U \ circ v) = {} ^tv \ circ {} ^t u

In particular if U is an isomorphism of vector spaces, then the reverse of transposed of U equal to is transposed of the reverse of U .

See too

dual Space
transposed Matrix
assistant Application

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