FiniteRankFreeModule_abstract.isomorphism_with_fixed_basis: Move here…

… from FiniteRankFreeModule, rewrite using Family

src/sage/tensor/modules/finite_rank_free_module.py

@@ -538,6 +538,7 @@ class :class:`~sage.modules.free_module.FreeModule_generic`
 from sage.categories.rings import Rings
 from sage.misc.cachefunc import cached_method
 from sage.rings.integer import Integer
+from sage.sets.family import Family, TrivialFamily
 from sage.structure.parent import Parent
 from sage.structure.unique_representation import UniqueRepresentation
 from sage.tensor.modules.free_module_element import FiniteRankFreeModuleElement
@@ -729,6 +730,129 @@ def is_submodule(self, other):
         """
         return self == other or self.ambient_module() == other
 
+    def isomorphism_with_fixed_basis(self, basis=None, codomain=None):
+        r"""
+        Construct the canonical isomorphism from the free module ``self``
+        to a free module in which ``basis`` of ``self`` is mapped to the
+        distinguished basis of ``codomain``.
+
+        INPUT:
+
+        - ``basis`` -- (default: ``None``) the basis of ``self`` which
+          should be mapped to the distinguished basis on ``codomain``;
+          if ``None``, the default basis is assumed.
+        - ``codomain`` -- (default: ``None``) the codomain of the
+          isomorphism represented by a free module within the category
+          :class:`~sage.categories.modules_with_basis.ModulesWithBasis` with
+          the same rank and base ring as ``self``; if ``None`` a free module
+          represented by
+          :class:`~sage.combinat.free_module.CombinatorialFreeModule` is
+          constructed
+
+        OUTPUT:
+
+        - a module morphism represented by
+          :class:`~sage.modules.with_basis.morphism.ModuleMorphismFromFunction`
+
+        EXAMPLES::
+
+            sage: V = FiniteRankFreeModule(QQ, 3, start_index=1); V
+            3-dimensional vector space over the Rational Field
+            sage: basis = e = V.basis("e"); basis
+            Basis (e_1,e_2,e_3) on the 3-dimensional vector space over the
+             Rational Field
+            sage: phi_e = V.isomorphism_with_fixed_basis(basis); phi_e
+            Generic morphism:
+              From: 3-dimensional vector space over the Rational Field
+              To:   Free module generated by {1, 2, 3} over Rational Field
+            sage: phi_e.codomain().category()
+            Category of finite dimensional vector spaces with basis over
+             Rational Field
+            sage: phi_e(e[1] + 2 * e[2])
+            e[1] + 2*e[2]
+
+            sage: abc = V.basis(['a', 'b', 'c'], symbol_dual=['d', 'e', 'f']); abc
+            Basis (a,b,c) on the 3-dimensional vector space over the Rational Field
+            sage: phi_abc = V.isomorphism_with_fixed_basis(abc); phi_abc
+            Generic morphism:
+            From: 3-dimensional vector space over the Rational Field
+            To:   Free module generated by {1, 2, 3} over Rational Field
+            sage: phi_abc(abc[1] + 2 * abc[2])
+            B[1] + 2*B[2]
+
+        Providing a codomain::
+
+            sage: W = CombinatorialFreeModule(QQ, ['a', 'b', 'c'])
+            sage: phi_eW = V.isomorphism_with_fixed_basis(basis, codomain=W); phi_eW
+            Generic morphism:
+            From: 3-dimensional vector space over the Rational Field
+            To:   Free module generated by {'a', 'b', 'c'} over Rational Field
+            sage: phi_eW(e[1] + 2 * e[2])
+            B['a'] + 2*B['b']
+
+        TESTS::
+
+            sage: V = FiniteRankFreeModule(QQ, 3); V
+            3-dimensional vector space over the Rational Field
+            sage: e = V.basis("e")
+            sage: V.isomorphism_with_fixed_basis(e, codomain=QQ^42)
+            Traceback (most recent call last):
+            ...
+            ValueError: domain and codomain must have the same rank
+            sage: V.isomorphism_with_fixed_basis(e, codomain=RR^3)
+            Traceback (most recent call last):
+            ...
+            ValueError: domain and codomain must have the same base ring
+        """
+        base_ring = self.base_ring()
+        if basis is None:
+            basis = self.default_basis()
+        if codomain is None:
+            from sage.combinat.free_module import CombinatorialFreeModule
+            if isinstance(basis._symbol, str):
+                prefix = basis._symbol
+            else:
+                prefix = None
+            codomain = CombinatorialFreeModule(base_ring, basis.keys(),
+                                               prefix=prefix)
+        else:
+            if codomain.rank() != self.rank():
+                raise ValueError("domain and codomain must have the same rank")
+            if codomain.base_ring() != base_ring:
+                raise ValueError("domain and codomain must have the same "
+                                 "base ring")
+
+        codomain_basis = Family(codomain.basis())
+        if isinstance(codomain_basis, TrivialFamily):
+            # assume that codomain basis keys are to be ignored
+            key_pairs = enumerate(basis_keys())
+        else:
+            # assume that the keys of the codomain should be used
+            key_pairs = zip(codomain_basis.keys(), basis.keys())
+
+        def _isomorphism(x):
+            r"""
+            Concrete isomorphism from ``self`` to ``codomain``.
+            """
+            return codomain.sum(x[basis, domain_key] * codomain_basis[codomain_key]
+                                for codomain_key, domain_key in key_pairs)
+
+        return self.module_morphism(function=_isomorphism, codomain=codomain)
+
+    def _test_isomorphism_with_fixed_basis(self, **options):
+        r"""
+        Test that the method ``isomorphism_with_fixed_basis`` works correctly.
+
+        EXAMPLES::
+
+            sage: M = FiniteRankFreeModule(ZZ, 3, name='M')
+            sage: M._test_isomorphism_with_fixed_basis()
+        """
+        tester = self._tester(**options)
+        basis = self.basis('test')
+        morphism = self.isomorphism_with_fixed_basis(basis)
+        tester.assertEqual(morphism.codomain().rank(), self.rank())
+
 
 class FiniteRankFreeModule(FiniteRankFreeModule_abstract):
     r"""
@@ -2713,123 +2837,6 @@ def hom(self, codomain, matrix_rep, bases=None, name=None,
         return homset(matrix_rep, bases=bases, name=name,
                       latex_name=latex_name)
 
-    def isomorphism_with_fixed_basis(self, basis=None, codomain=None):
-        r"""
-        Construct the canonical isomorphism from the free module ``self``
-        to a free module in which ``basis`` of ``self`` is mapped to the
-        distinguished basis of ``codomain``.
-
-        INPUT:
-
-        - ``basis`` -- (default: ``None``) the basis of ``self`` which
-          should be mapped to the distinguished basis on ``codomain``;
-          if ``None``, the default basis is assumed.
-        - ``codomain`` -- (default: ``None``) the codomain of the
-          isomorphism represented by a free module within the category
-          :class:`~sage.categories.modules_with_basis.ModulesWithBasis` with
-          the same rank and base ring as ``self``; if ``None`` a free module
-          represented by
-          :class:`~sage.combinat.free_module.CombinatorialFreeModule` is
-          constructed
-
-        OUTPUT:
-
-        - a module morphism represented by
-          :class:`~sage.modules.with_basis.morphism.ModuleMorphismFromFunction`
-
-        EXAMPLES::
-
-            sage: V = FiniteRankFreeModule(QQ, 3, start_index=1); V
-            3-dimensional vector space over the Rational Field
-            sage: basis = e = V.basis("e"); basis
-            Basis (e_1,e_2,e_3) on the 3-dimensional vector space over the
-             Rational Field
-            sage: phi_e = V.isomorphism_with_fixed_basis(basis); phi_e
-            Generic morphism:
-              From: 3-dimensional vector space over the Rational Field
-              To:   Free module generated by {1, 2, 3} over Rational Field
-            sage: phi_e.codomain().category()
-            Category of finite dimensional vector spaces with basis over
-             Rational Field
-            sage: phi_e(e[1] + 2 * e[2])
-            e[1] + 2*e[2]
-
-            sage: abc = V.basis(['a', 'b', 'c'], symbol_dual=['d', 'e', 'f']); abc
-            Basis (a,b,c) on the 3-dimensional vector space over the Rational Field
-            sage: phi_abc = V.isomorphism_with_fixed_basis(abc); phi_abc
-            Generic morphism:
-            From: 3-dimensional vector space over the Rational Field
-            To:   Free module generated by {1, 2, 3} over Rational Field
-            sage: phi_abc(abc[1] + 2 * abc[2])
-            B[1] + 2*B[2]
-
-        Providing a codomain::
-
-            sage: W = CombinatorialFreeModule(QQ, ['a', 'b', 'c'])
-            sage: phi_eW = V.isomorphism_with_fixed_basis(basis, codomain=W); phi_eW
-            Generic morphism:
-            From: 3-dimensional vector space over the Rational Field
-            To:   Free module generated by {'a', 'b', 'c'} over Rational Field
-            sage: phi_eW(e[1] + 2 * e[2])
-            B['a'] + 2*B['b']
-
-        TESTS::
-
-            sage: V = FiniteRankFreeModule(QQ, 3); V
-            3-dimensional vector space over the Rational Field
-            sage: e = V.basis("e")
-            sage: V.isomorphism_with_fixed_basis(e, codomain=QQ^42)
-            Traceback (most recent call last):
-            ...
-            ValueError: domain and codomain must have the same rank
-            sage: V.isomorphism_with_fixed_basis(e, codomain=RR^3)
-            Traceback (most recent call last):
-            ...
-            ValueError: domain and codomain must have the same base ring
-        """
-        base_ring = self.base_ring()
-        if basis is None:
-            basis = self.default_basis()
-        if codomain is None:
-            from sage.combinat.free_module import CombinatorialFreeModule
-            if isinstance(basis._symbol, str):
-                prefix = basis._symbol
-            else:
-                prefix = None
-            codomain = CombinatorialFreeModule(base_ring, list(self.irange()),
-                                               prefix=prefix)
-        else:
-            if codomain.rank() != self.rank():
-                raise ValueError("domain and codomain must have the same rank")
-            if codomain.base_ring() != base_ring:
-                raise ValueError("domain and codomain must have the same "
-                                 "base ring")
-
-        codomain_basis = list(codomain.basis())
-
-        def _isomorphism(x):
-            r"""
-            Concrete isomorphism from ``self`` to ``codomain``.
-            """
-            return codomain.sum(x[basis, i] * codomain_basis[i - self._sindex]
-                                for i in self.irange())
-
-        return self.module_morphism(function=_isomorphism, codomain=codomain)
-
-    def _test_isomorphism_with_fixed_basis(self, **options):
-        r"""
-        Test that the method ``isomorphism_with_fixed_basis`` works correctly.
-
-        EXAMPLES::
-
-            sage: M = FiniteRankFreeModule(ZZ, 3, name='M')
-            sage: M._test_isomorphism_with_fixed_basis()
-        """
-        tester = self._tester(**options)
-        basis = self.basis('test')
-        morphism = self.isomorphism_with_fixed_basis(basis)
-        tester.assertEqual(morphism.codomain().rank(), self.rank())
-
     def endomorphism(self, matrix_rep, basis=None, name=None, latex_name=None):
         r"""
         Construct an endomorphism of the free module ``self``.