Linear functionals on vector spaces and modules

[mkoeppe]

We propose to add a method dual to modules M as an abbreviation for Hom(M, M.base_ring()).

For vector spaces, we create a specialized class so that it prints in a more concise way than this:

sage: Hom(QQ^2, QQ)
Set of Morphisms (Linear Transformations) from Vector space of dimension 2 over Rational Field to Rational Field

and has more functionality to make linear algebra less awkward.

sage: _.an_element()
NotImplementedError: please implement _an_element_ for Set of Morphisms (Linear Transformations) from Vector space of dimension 2 over Rational Field to Rational Field

FiniteRankFreeModule already has this functionality. The new interface should be compatible with it.

sage: Q3 = FiniteRankFreeModule(QQ, 3)
sage: Q3.dual()
Dual of the 3-dimensional vector space over the Rational Field
sage: l = Q3.linear_form(); l
Linear form on the 3-dimensional vector space over the Rational Field

Related (more general):

• Meta-ticket: Duals of modules, algebras, coalgebras, hopf algebras, etc. #13372: add functionality for duals of algebras, coalgebras, hopf algebras, etc.

CC: @tscrim @simonbrandhorst @egourgoulhon

Component: categories

Issue created by migration from https://trac.sagemath.org/ticket/30091

[mkoeppe]

comment:3

Setting new milestone based on a cursory review of ticket status, priority, and last modification date.