Package sage :: Package modules :: Module free_module :: Class FreeModule_ambient_domain

Class FreeModule_ambient_domain

                         object --+                                
                                  |                                
   structure.sage_object.SageObject --+                            
                                      |                            
                structure.parent.Parent --+                        
                                          |                        
       structure.parent_base.ParentWithBase --+                    
                                              |                    
           structure.parent_gens.ParentWithGens --+                
                                                  |                
structure.parent_gens.ParentWithAdditiveAbelianGens --+            
                                                      |            
                                          module.Module --+        
                                                          |        
                                         FreeModule_generic --+    
                                                              |    
                                             FreeModule_ambient --+
                                                                  |
                                                                 FreeModule_ambient_domain

Known Subclasses:: FreeModule_ambient_pid


Ambient free module over an integral domain.

Instance Methods

[hide private]

__init__(self, base_ring, rank, sparse=False, inner_product_matrix=['4ti2-20061025', 'R-2.6.0', 'atlas-3.7.37', 'atlas-3.8.1', 'a...)
EXAMPLES:... source code

_repr_(self)

source code

base_field(self)
Return the fraction field of the base ring of self.

source code

ambient_vector_space(self)
Returns the ambient vector space, which is this free module tensored with its fraction field.

source code

coordinate_vector(self, v, check=True)
Write $v$ in terms of the standard basis for self and return the resulting coeffcients in a vector over the fraction field of the base ring.

source code

vector_space(self, base_field=['4ti2-20061025', 'R-2.6.0', 'atlas-3.7.37', 'atlas-3.8.1', 'a...)
Returns the vector space obtained from self by tensoring with the fraction field of the base ring and extending to the field. source code

Inherited from FreeModule_ambient: __cmp__, __hash__, _latex_, ambient_module, basis, change_ring, echelon_coordinate_vector, echelon_coordinates, echelonized_basis, echelonized_basis_matrix, is_ambient, linear_combination_of_basis

Inherited from FreeModule_ambient (private): _dense_module, _sparse_module

Inherited from FreeModule_generic: __call__, __contains__, __iter__, __len__, _magma_init_, base_extend, basis_matrix, category, construction, coordinate_module, coordinates, degree, dense_module, dimension, direct_sum, discriminant, element_class, free_module, gen, gram_matrix, has_user_basis, inner_product_matrix, is_dense, is_finite, is_full, is_sparse, is_submodule, matrix, ngens, nonembedded_free_module, random_element, rank, sparse_module, uses_ambient_inner_product, zero_vector

Inherited from FreeModule_generic (private): _an_element_impl, _coerce_impl, _has_coerce_map_from_space, _inner_product_is_dot_product

Inherited from module.Module: __new__, endomorphism_ring, is_atomic_repr

Inherited from structure.parent_gens.ParentWithAdditiveAbelianGens: generator_orders

Inherited from structure.parent_gens.ParentWithGens: __getitem__, __getslice__, __getstate__, __setstate__, _is_valid_homomorphism_, gens, gens_dict, hom, inject_variables, injvar, latex_name, latex_variable_names, list, objgen, objgens, variable_name, variable_names

Inherited from structure.parent_gens.ParentWithGens (private): _assign_names, _first_ngens

Inherited from structure.parent_base.ParentWithBase: Hom, base, base_extend_canonical, base_extend_canonical_sym, base_extend_recursive, base_ring

Inherited from structure.parent.Parent: _coerce_, coerce_map_from, coerce_map_from_impl, get_action, get_action_impl, has_coerce_map_from, has_coerce_map_from_impl, init_coerce

Inherited from structure.parent.Parent (private): _an_element, _coerce_self, _coerce_try

Inherited from structure.sage_object.SageObject: __repr__, _axiom_, _axiom_init_, _gap_, _gap_init_, _gp_, _gp_init_, _interface_, _interface_init_, _interface_is_cached_, _kash_, _kash_init_, _macaulay2_, _macaulay2_init_, _magma_, _maple_, _maple_init_, _mathematica_, _mathematica_init_, _maxima_, _maxima_init_, _octave_, _octave_init_, _pari_, _pari_init_, _r_init_, _sage_, _singular_, _singular_init_, db, dump, dumps, plot, rename, reset_name, save, version

Inherited from object: __delattr__, __getattribute__, __reduce__, __reduce_ex__, __setattr__, __str__

Properties

[hide private]

Inherited from structure.parent_gens.ParentWithAdditiveAbelianGens (private): _generator_orders

Inherited from structure.parent_gens.ParentWithGens (private): _gens, _gens_dict, _latex_names, _list, _names

Inherited from structure.parent.Parent (private): _has_coerce_map_from

Inherited from object: __class__

Method Details

[hide private]

init(self, base_ring, rank, sparse=False, inner_product_matrix=`['4ti2-20061025',` `'R-2.6.0',` `'atlas-3.7.37',` `'atlas-3.8.1',` `'a...`)
(Constructor)

source code


EXAMPLES:
    sage: FreeModule(PolynomialRing(GF(5),'x'), 3)
    Ambient free module of rank 3 over the principal ideal domain 
    Univariate Polynomial Ring in x over Finite Field of size 5

Overrides: FreeModule_ambient.__init__

_repr_(self)

source code

Overrides: FreeModule_ambient._repr_

base_field(self)

source code


Return the fraction field of the base ring of self.

EXAMPLES:
    sage: M = ZZ^3;  M.base_field()
    Rational Field
    sage: M = PolynomialRing(GF(5),'x')^3;  M.base_field()
    Fraction Field of Univariate Polynomial Ring in x over Finite Field of size 5

ambient_vector_space(self)

source code


Returns the ambient vector space, which is this free module tensored
with its fraction field.

EXAMPLES:
    sage: M = ZZ^3;  M.ambient_vector_space()
    Vector space of dimension 3 over Rational Field

coordinate_vector(self, v, check=True)

source code


Write $v$ in terms of the standard basis for self and return the
resulting coeffcients in a vector over the fraction field of the
base ring.

INPUT:
    v -- vector
    check -- bool (default: True); if True, also verify that v is really in self.
OUTPUT:
    list        

Returns a vector c such that if B is the basis for self, then 
        sum c[i] B[i] = v
If v is not in self, raises an ArithmeticError exception.

EXAMPLES:
    sage: V = ZZ^3
    sage: v = V.coordinate_vector([1,5,9]); v
    (1, 5, 9)
    sage: v.parent()
    Vector space of dimension 3 over Rational Field

Overrides: FreeModule_ambient.coordinate_vector

vector_space(self, base_field=`['4ti2-20061025',` `'R-2.6.0',` `'atlas-3.7.37',` `'atlas-3.8.1',` `'a...`)

source code


Returns the vector space obtained from self by tensoring with
the fraction field of the base ring and extending to the field.

EXAMPLES:
    sage: M = ZZ^3;  M.vector_space()
    Vector space of dimension 3 over Rational Field

Class FreeModule_ambient_domain

__init__(self, base_ring, rank, sparse=False, inner_product_matrix=['4ti2-20061025', 'R-2.6.0', 'atlas-3.7.37', 'atlas-3.8.1', 'a...) (Constructor)

_repr_(self)

base_field(self)

ambient_vector_space(self)

coordinate_vector(self, v, check=True)

vector_space(self, base_field=['4ti2-20061025', 'R-2.6.0', 'atlas-3.7.37', 'atlas-3.8.1', 'a...)

init(self, base_ring, rank, sparse=False, inner_product_matrix=`['4ti2-20061025',` `'R-2.6.0',` `'atlas-3.7.37',` `'atlas-3.8.1',` `'a...`)
(Constructor)

vector_space(self, base_field=`['4ti2-20061025',` `'R-2.6.0',` `'atlas-3.7.37',` `'atlas-3.8.1',` `'a...`)