Free module bases¶

The class FreeModuleBasis implements bases on a free module \(M\) of finite rank over a commutative ring, while the class FreeModuleCoBasis implements the dual bases (i.e. bases of the dual module \(M^*\)).

AUTHORS:

Eric Gourgoulhon, Michal Bejger (2014-2015): initial version
Travis Scrimshaw (2016): ABC Basis_abstract and list functionality for bases (Issue #20770)
Eric Gourgoulhon (2018): some refactoring and more functionalities in the choice of symbols for basis elements (Issue #24792)

REFERENCES:

Chap. 10 of R. Godement : Algebra [God1968]
Chap. 3 of S. Lang : Algebra [Lan2002]

class sage.tensor.modules.free_module_basis.Basis_abstract(fmodule, symbol, latex_symbol, indices, latex_indices)[source]¶

Bases: UniqueRepresentation, AbstractFamily

Abstract base class for (dual) bases of free modules.

A basis is an AbstractFamily, hence like collections.abc.Mapping subclasses such as dict, it is an associative Container, providing methods keys(), values(), and items(). Thus, e[i] returns the element of the basis e indexed by the key i. However, in contrast to Mapping subclasses, not the keys() but the values() are considered the elements.

EXAMPLES:

Sage

sage: M = FiniteRankFreeModule(ZZ, 3, name='M', start_index=1)
sage: e = M.basis('e'); e
Basis (e_1,e_2,e_3) on the Rank-3 free module M over the Integer Ring
sage: list(e)
[Element e_1 of the Rank-3 free module M over the Integer Ring,
Element e_2 of the Rank-3 free module M over the Integer Ring,
Element e_3 of the Rank-3 free module M over the Integer Ring]
sage: e.category()
Category of facade finite enumerated sets
sage: list(e.keys())
[1, 2, 3]
sage: list(e.values())
[Element e_1 of the Rank-3 free module M over the Integer Ring,
Element e_2 of the Rank-3 free module M over the Integer Ring,
Element e_3 of the Rank-3 free module M over the Integer Ring]
sage: list(e.items())
[(1, Element e_1 of the Rank-3 free module M over the Integer Ring),
(2, Element e_2 of the Rank-3 free module M over the Integer Ring),
(3, Element e_3 of the Rank-3 free module M over the Integer Ring)]

Python

>>> from sage.all import *
>>> M = FiniteRankFreeModule(ZZ, Integer(3), name='M', start_index=Integer(1))
>>> e = M.basis('e'); e
Basis (e_1,e_2,e_3) on the Rank-3 free module M over the Integer Ring
>>> list(e)
[Element e_1 of the Rank-3 free module M over the Integer Ring,
Element e_2 of the Rank-3 free module M over the Integer Ring,
Element e_3 of the Rank-3 free module M over the Integer Ring]
>>> e.category()
Category of facade finite enumerated sets
>>> list(e.keys())
[1, 2, 3]
>>> list(e.values())
[Element e_1 of the Rank-3 free module M over the Integer Ring,
Element e_2 of the Rank-3 free module M over the Integer Ring,
Element e_3 of the Rank-3 free module M over the Integer Ring]
>>> list(e.items())
[(1, Element e_1 of the Rank-3 free module M over the Integer Ring),
(2, Element e_2 of the Rank-3 free module M over the Integer Ring),
(3, Element e_3 of the Rank-3 free module M over the Integer Ring)]

Sage Live

M = FiniteRankFreeModule(ZZ, 3, name='M', start_index=1)
e = M.basis('e'); e
list(e)
e.category()
list(e.keys())
list(e.values())
list(e.items())

cardinality()[source]¶

Return the basis length, i.e. the rank of the free module.

EXAMPLES:

Sage

sage: M = FiniteRankFreeModule(ZZ, 3, name='M')
sage: e = M.basis('e')
sage: e.cardinality()
3

Python

>>> from sage.all import *
>>> M = FiniteRankFreeModule(ZZ, Integer(3), name='M')
>>> e = M.basis('e')
>>> e.cardinality()
3

Sage Live

M = FiniteRankFreeModule(ZZ, 3, name='M')
e = M.basis('e')
e.cardinality()

free_module()[source]¶

Return the free module of self.

EXAMPLES:

Sage

sage: M = FiniteRankFreeModule(QQ, 2, name='M', start_index=1)
sage: e = M.basis('e')
sage: e.free_module() is M
True

Python

>>> from sage.all import *
>>> M = FiniteRankFreeModule(QQ, Integer(2), name='M', start_index=Integer(1))
>>> e = M.basis('e')
>>> e.free_module() is M
True

Sage Live

M = FiniteRankFreeModule(QQ, 2, name='M', start_index=1)
e = M.basis('e')
e.free_module() is M

keys()[source]¶

Return the keys (indices) of the family.

EXAMPLES:

Sage

sage: M = FiniteRankFreeModule(ZZ, 3, name='M')
sage: e = M.basis('e')
sage: list(e.keys())
[0, 1, 2]

Python

>>> from sage.all import *
>>> M = FiniteRankFreeModule(ZZ, Integer(3), name='M')
>>> e = M.basis('e')
>>> list(e.keys())
[0, 1, 2]

Sage Live

M = FiniteRankFreeModule(ZZ, 3, name='M')
e = M.basis('e')
list(e.keys())

set_name(symbol, latex_symbol=None, indices=None, latex_indices=None, index_position='down')[source]¶

Set (or change) the text name and LaTeX name of self.

INPUT:

symbol – either a string, to be used as a common base for the symbols of the elements of self, or a list of strings, representing the individual symbols of the elements of self
latex_symbol – (default: None) either a string, to be used as a common base for the LaTeX symbols of the elements of self, or a list of strings, representing the individual LaTeX symbols of the elements of self; if None, symbol is used in place of latex_symbol
indices – (default: None; used only if symbol is a single string) tuple of strings representing the indices labelling the elements of self; if None, the indices will be generated as integers within the range declared on the free module on which self is defined
latex_indices – (default: None) list of strings representing the indices for the LaTeX symbols of the elements of self; if None, indices is used instead
index_position – (default: 'down') determines the position of the indices labelling the individual elements of self; can be either 'down' or 'up'

EXAMPLES:

Sage

sage: M = FiniteRankFreeModule(ZZ, 3, name='M')
sage: e = M.basis('e'); e
Basis (e_0,e_1,e_2) on the Rank-3 free module M over the Integer Ring
sage: e.set_name('f'); e
Basis (f_0,f_1,f_2) on the Rank-3 free module M over the Integer Ring
sage: e.set_name(['a', 'b', 'c']); e
Basis (a,b,c) on the Rank-3 free module M over the Integer Ring
sage: e.set_name('e', indices=['x', 'y', 'z']); e
Basis (e_x,e_y,e_z) on the Rank-3 free module M over the Integer Ring
sage: e.set_name('e', index_position='up'); e
Basis (e^0,e^1,e^2) on the Rank-3 free module M over the Integer Ring
sage: latex(e)
\left(e^{0},e^{1},e^{2}\right)
sage: e.set_name('e', latex_symbol=r'\epsilon'); e
Basis (e_0,e_1,e_2) on the Rank-3 free module M over the Integer Ring
sage: latex(e)
\left(\epsilon_{0},\epsilon_{1},\epsilon_{2}\right)
sage: e.set_name('e', latex_symbol=[r'\alpha', r'\beta', r'\gamma'])
sage: latex(e)
\left(\alpha,\beta,\gamma\right)
sage: e.set_name('e', latex_symbol='E',
....:            latex_indices=[r'\alpha', r'\beta', r'\gamma'])
sage: latex(e)
\left(E_{\alpha},E_{\beta},E_{\gamma}\right)
sage: e.set_name('e') # back to the default

Python

>>> from sage.all import *
>>> M = FiniteRankFreeModule(ZZ, Integer(3), name='M')
>>> e = M.basis('e'); e
Basis (e_0,e_1,e_2) on the Rank-3 free module M over the Integer Ring
>>> e.set_name('f'); e
Basis (f_0,f_1,f_2) on the Rank-3 free module M over the Integer Ring
>>> e.set_name(['a', 'b', 'c']); e
Basis (a,b,c) on the Rank-3 free module M over the Integer Ring
>>> e.set_name('e', indices=['x', 'y', 'z']); e
Basis (e_x,e_y,e_z) on the Rank-3 free module M over the Integer Ring
>>> e.set_name('e', index_position='up'); e
Basis (e^0,e^1,e^2) on the Rank-3 free module M over the Integer Ring
>>> latex(e)
\left(e^{0},e^{1},e^{2}\right)
>>> e.set_name('e', latex_symbol=r'\epsilon'); e
Basis (e_0,e_1,e_2) on the Rank-3 free module M over the Integer Ring
>>> latex(e)
\left(\epsilon_{0},\epsilon_{1},\epsilon_{2}\right)
>>> e.set_name('e', latex_symbol=[r'\alpha', r'\beta', r'\gamma'])
>>> latex(e)
\left(\alpha,\beta,\gamma\right)
>>> e.set_name('e', latex_symbol='E',
...            latex_indices=[r'\alpha', r'\beta', r'\gamma'])
>>> latex(e)
\left(E_{\alpha},E_{\beta},E_{\gamma}\right)
>>> e.set_name('e') # back to the default

Sage Live

M = FiniteRankFreeModule(ZZ, 3, name='M')
e = M.basis('e'); e
e.set_name('f'); e
e.set_name(['a', 'b', 'c']); e
e.set_name('e', indices=['x', 'y', 'z']); e
e.set_name('e', index_position='up'); e
latex(e)
e.set_name('e', latex_symbol=r'\epsilon'); e
latex(e)
e.set_name('e', latex_symbol=[r'\alpha', r'\beta', r'\gamma'])
latex(e)
e.set_name('e', latex_symbol='E',
           latex_indices=[r'\alpha', r'\beta', r'\gamma'])
latex(e)
e.set_name('e') # back to the default

values()[source]¶

Return the basis elements of self.

EXAMPLES:

Sage

sage: M = FiniteRankFreeModule(ZZ, 3, name='M')
sage: e = M.basis('e')
sage: list(e.values())
[Element e_0 of the Rank-3 free module M over the Integer Ring,
 Element e_1 of the Rank-3 free module M over the Integer Ring,
 Element e_2 of the Rank-3 free module M over the Integer Ring]

Python

>>> from sage.all import *
>>> M = FiniteRankFreeModule(ZZ, Integer(3), name='M')
>>> e = M.basis('e')
>>> list(e.values())
[Element e_0 of the Rank-3 free module M over the Integer Ring,
 Element e_1 of the Rank-3 free module M over the Integer Ring,
 Element e_2 of the Rank-3 free module M over the Integer Ring]

Sage Live

M = FiniteRankFreeModule(ZZ, 3, name='M')
e = M.basis('e')
list(e.values())

class sage.tensor.modules.free_module_basis.FreeModuleBasis(fmodule, symbol, latex_symbol=None, indices=None, latex_indices=None, symbol_dual=None, latex_symbol_dual=None)[source]¶

Bases: Basis_abstract

Basis of a free module over a commutative ring \(R\).

INPUT:

fmodule – free module \(M\) (as an instance of FiniteRankFreeModule)
symbol – either a string, to be used as a common base for the symbols of the elements of the basis, or a tuple of strings, representing the individual symbols of the elements of the basis
latex_symbol – (default: None) either a string, to be used as a common base for the LaTeX symbols of the elements of the basis, or a tuple of strings, representing the individual LaTeX symbols of the elements of the basis; if None, symbol is used in place of latex_symbol
indices – (default: None; used only if symbol is a single string) tuple of strings representing the indices labelling the elements of the basis; if None, the indices will be generated as integers within the range declared on fmodule
latex_indices – (default: None) tuple of strings representing the indices for the LaTeX symbols of the elements of the basis; if None, indices is used instead
symbol_dual – (default: None) same as symbol but for the dual basis; if None, symbol must be a string and is used for the common base of the symbols of the elements of the dual basis
latex_symbol_dual – (default: None) same as latex_symbol but for the dual basis

EXAMPLES:

A basis on a rank-3 free module over \(\ZZ\):

Sage

sage: M0 = FiniteRankFreeModule(ZZ, 3, name='M_0')
sage: from sage.tensor.modules.free_module_basis import FreeModuleBasis
sage: e = FreeModuleBasis(M0, 'e') ; e
Basis (e_0,e_1,e_2) on the Rank-3 free module M_0 over the Integer Ring

Python

>>> from sage.all import *
>>> M0 = FiniteRankFreeModule(ZZ, Integer(3), name='M_0')
>>> from sage.tensor.modules.free_module_basis import FreeModuleBasis
>>> e = FreeModuleBasis(M0, 'e') ; e
Basis (e_0,e_1,e_2) on the Rank-3 free module M_0 over the Integer Ring

Sage Live

M0 = FiniteRankFreeModule(ZZ, 3, name='M_0')
from sage.tensor.modules.free_module_basis import FreeModuleBasis
e = FreeModuleBasis(M0, 'e') ; e

Instead of importing FreeModuleBasis in the global name space, it is recommended to use the module’s method basis():

Sage

sage: M = FiniteRankFreeModule(ZZ, 3, name='M')
sage: e = M.basis('e') ; e
Basis (e_0,e_1,e_2) on the Rank-3 free module M over the Integer Ring

Python

>>> from sage.all import *
>>> M = FiniteRankFreeModule(ZZ, Integer(3), name='M')
>>> e = M.basis('e') ; e
Basis (e_0,e_1,e_2) on the Rank-3 free module M over the Integer Ring

Sage Live

M = FiniteRankFreeModule(ZZ, 3, name='M')
e = M.basis('e') ; e

The individual elements constituting the basis are accessed via the square bracket operator:

Sage

sage: e[0]
Element e_0 of the Rank-3 free module M over the Integer Ring
sage: e[0] in M
True

Python

>>> from sage.all import *
>>> e[Integer(0)]
Element e_0 of the Rank-3 free module M over the Integer Ring
>>> e[Integer(0)] in M
True

Sage Live

e[0]
e[0] in M

The slice operator : can be used to access to more than one element:

Sage

sage: e[0:2]
(Element e_0 of the Rank-3 free module M over the Integer Ring,
 Element e_1 of the Rank-3 free module M over the Integer Ring)
sage: e[:]
(Element e_0 of the Rank-3 free module M over the Integer Ring,
 Element e_1 of the Rank-3 free module M over the Integer Ring,
 Element e_2 of the Rank-3 free module M over the Integer Ring)

Python

>>> from sage.all import *
>>> e[Integer(0):Integer(2)]
(Element e_0 of the Rank-3 free module M over the Integer Ring,
 Element e_1 of the Rank-3 free module M over the Integer Ring)
>>> e[:]
(Element e_0 of the Rank-3 free module M over the Integer Ring,
 Element e_1 of the Rank-3 free module M over the Integer Ring,
 Element e_2 of the Rank-3 free module M over the Integer Ring)

Sage Live

e[0:2]
e[:]

The LaTeX symbol can be set explicitly:

Sage

sage: latex(e)
\left(e_{0},e_{1},e_{2}\right)
sage: eps = M.basis('eps', latex_symbol=r'\epsilon') ; eps
Basis (eps_0,eps_1,eps_2) on the Rank-3 free module M over the Integer
 Ring
sage: latex(eps)
\left(\epsilon_{0},\epsilon_{1},\epsilon_{2}\right)

Python

>>> from sage.all import *
>>> latex(e)
\left(e_{0},e_{1},e_{2}\right)
>>> eps = M.basis('eps', latex_symbol=r'\epsilon') ; eps
Basis (eps_0,eps_1,eps_2) on the Rank-3 free module M over the Integer
 Ring
>>> latex(eps)
\left(\epsilon_{0},\epsilon_{1},\epsilon_{2}\right)

Sage Live

latex(e)
eps = M.basis('eps', latex_symbol=r'\epsilon') ; eps
latex(eps)

The individual elements of the basis are labelled according the parameter start_index provided at the free module construction:

Sage

sage: M = FiniteRankFreeModule(ZZ, 3, name='M', start_index=1)
sage: e = M.basis('e') ; e
Basis (e_1,e_2,e_3) on the Rank-3 free module M over the Integer Ring
sage: e[1]
Element e_1 of the Rank-3 free module M over the Integer Ring

Python

>>> from sage.all import *
>>> M = FiniteRankFreeModule(ZZ, Integer(3), name='M', start_index=Integer(1))
>>> e = M.basis('e') ; e
Basis (e_1,e_2,e_3) on the Rank-3 free module M over the Integer Ring
>>> e[Integer(1)]
Element e_1 of the Rank-3 free module M over the Integer Ring

Sage Live

M = FiniteRankFreeModule(ZZ, 3, name='M', start_index=1)
e = M.basis('e') ; e
e[1]

It is also possible to fully customize the labels, via the argument indices:

Sage

sage: f = M.basis('f', indices=('x', 'y', 'z')); f
Basis (f_x,f_y,f_z) on the Rank-3 free module M over the Integer Ring
sage: f[1]
Element f_x of the Rank-3 free module M over the Integer Ring

Python

>>> from sage.all import *
>>> f = M.basis('f', indices=('x', 'y', 'z')); f
Basis (f_x,f_y,f_z) on the Rank-3 free module M over the Integer Ring
>>> f[Integer(1)]
Element f_x of the Rank-3 free module M over the Integer Ring

Sage Live

f = M.basis('f', indices=('x', 'y', 'z')); f
f[1]

The symbol of each element of the basis can also be freely chosen, by providing a tuple of symbols as the first argument of basis; it is then mandatory to specify some symbols for the dual basis as well:

Sage

sage: g = M.basis(('a', 'b', 'c'), symbol_dual=('A', 'B', 'C')); g
Basis (a,b,c) on the Rank-3 free module M over the Integer Ring
sage: g[1]
Element a of the Rank-3 free module M over the Integer Ring
sage: g.dual_basis()[1]
Linear form A on the Rank-3 free module M over the Integer Ring

Python

>>> from sage.all import *
>>> g = M.basis(('a', 'b', 'c'), symbol_dual=('A', 'B', 'C')); g
Basis (a,b,c) on the Rank-3 free module M over the Integer Ring
>>> g[Integer(1)]
Element a of the Rank-3 free module M over the Integer Ring
>>> g.dual_basis()[Integer(1)]
Linear form A on the Rank-3 free module M over the Integer Ring

Sage Live

g = M.basis(('a', 'b', 'c'), symbol_dual=('A', 'B', 'C')); g
g[1]
g.dual_basis()[1]

dual_basis()[source]¶

Return the basis dual to self.

OUTPUT:

instance of FreeModuleCoBasis representing the dual of self

EXAMPLES:

Dual basis on a rank-3 free module:

Sage

sage: M = FiniteRankFreeModule(ZZ, 3, name='M', start_index=1)
sage: e = M.basis('e') ; e
Basis (e_1,e_2,e_3) on the Rank-3 free module M over the Integer Ring
sage: f = e.dual_basis() ; f
Dual basis (e^1,e^2,e^3) on the Rank-3 free module M over the Integer Ring

Python

>>> from sage.all import *
>>> M = FiniteRankFreeModule(ZZ, Integer(3), name='M', start_index=Integer(1))
>>> e = M.basis('e') ; e
Basis (e_1,e_2,e_3) on the Rank-3 free module M over the Integer Ring
>>> f = e.dual_basis() ; f
Dual basis (e^1,e^2,e^3) on the Rank-3 free module M over the Integer Ring

Sage Live

M = FiniteRankFreeModule(ZZ, 3, name='M', start_index=1)
e = M.basis('e') ; e
f = e.dual_basis() ; f

Let us check that the elements of f are elements of the dual of M:

Sage

sage: f[1] in M.dual()
True
sage: f[1]
Linear form e^1 on the Rank-3 free module M over the Integer Ring

Python

>>> from sage.all import *
>>> f[Integer(1)] in M.dual()
True
>>> f[Integer(1)]
Linear form e^1 on the Rank-3 free module M over the Integer Ring

Sage Live

f[1] in M.dual()
f[1]

and that f is indeed the dual of e:

Sage

sage: f[1](e[1]), f[1](e[2]), f[1](e[3])
(1, 0, 0)
sage: f[2](e[1]), f[2](e[2]), f[2](e[3])
(0, 1, 0)
sage: f[3](e[1]), f[3](e[2]), f[3](e[3])
(0, 0, 1)

Python

>>> from sage.all import *
>>> f[Integer(1)](e[Integer(1)]), f[Integer(1)](e[Integer(2)]), f[Integer(1)](e[Integer(3)])
(1, 0, 0)
>>> f[Integer(2)](e[Integer(1)]), f[Integer(2)](e[Integer(2)]), f[Integer(2)](e[Integer(3)])
(0, 1, 0)
>>> f[Integer(3)](e[Integer(1)]), f[Integer(3)](e[Integer(2)]), f[Integer(3)](e[Integer(3)])
(0, 0, 1)

Sage Live

f[1](e[1]), f[1](e[2]), f[1](e[3])
f[2](e[1]), f[2](e[2]), f[2](e[3])
f[3](e[1]), f[3](e[2]), f[3](e[3])

module()[source]¶

Return the free module on which the basis is defined.

OUTPUT:

instance of FiniteRankFreeModule representing the free module of which self is a basis

EXAMPLES:

Sage

sage: M = FiniteRankFreeModule(ZZ, 3, name='M')
sage: e = M.basis('e')
sage: e.module()
Rank-3 free module M over the Integer Ring
sage: e.module() is M
True

Python

>>> from sage.all import *
>>> M = FiniteRankFreeModule(ZZ, Integer(3), name='M')
>>> e = M.basis('e')
>>> e.module()
Rank-3 free module M over the Integer Ring
>>> e.module() is M
True

Sage Live

M = FiniteRankFreeModule(ZZ, 3, name='M')
e = M.basis('e')
e.module()
e.module() is M

new_basis(change_of_basis, symbol, latex_symbol=None, indices=None, latex_indices=None, symbol_dual=None, latex_symbol_dual=None)[source]¶

Define a new module basis from self.

The new basis is defined by means of a module automorphism.

INPUT:

change_of_basis – instance of FreeModuleAutomorphism describing the automorphism \(P\) that relates the current basis \((e_i)\) (described by self) to the new basis \((n_i)\) according to \(n_i = P(e_i)\)
symbol – either a string, to be used as a common base for the symbols of the elements of the basis, or a tuple of strings, representing the individual symbols of the elements of the basis
latex_symbol – (default: None) either a string, to be used as a common base for the LaTeX symbols of the elements of the basis, or a tuple of strings, representing the individual LaTeX symbols of the elements of the basis; if None, symbol is used in place of latex_symbol
indices – (default: None; used only if symbol is a single string) tuple of strings representing the indices labelling the elements of the basis; if None, the indices will be generated as integers within the range declared on the free module on which self is defined
latex_indices – (default: None) tuple of strings representing the indices for the LaTeX symbols of the elements of the basis; if None, indices is used instead
symbol_dual – (default: None) same as symbol but for the dual basis; if None, symbol must be a string and is used for the common base of the symbols of the elements of the dual basis
latex_symbol_dual – (default: None) same as latex_symbol but for the dual basis

OUTPUT:

the new basis \((n_i)\), as an instance of FreeModuleBasis

EXAMPLES:

Change of basis on a vector space of dimension 2:

Sage

sage: M = FiniteRankFreeModule(QQ, 2, name='M', start_index=1)
sage: e = M.basis('e')
sage: a = M.automorphism()
sage: a[:] = [[1, 2], [-1, 3]]
sage: f = e.new_basis(a, 'f') ; f
Basis (f_1,f_2) on the 2-dimensional vector space M over the
 Rational Field
sage: f[1].display()
f_1 = e_1 - e_2
sage: f[2].display()
f_2 = 2 e_1 + 3 e_2
sage: e[1].display(f)
e_1 = 3/5 f_1 + 1/5 f_2
sage: e[2].display(f)
e_2 = -2/5 f_1 + 1/5 f_2

Python

>>> from sage.all import *
>>> M = FiniteRankFreeModule(QQ, Integer(2), name='M', start_index=Integer(1))
>>> e = M.basis('e')
>>> a = M.automorphism()
>>> a[:] = [[Integer(1), Integer(2)], [-Integer(1), Integer(3)]]
>>> f = e.new_basis(a, 'f') ; f
Basis (f_1,f_2) on the 2-dimensional vector space M over the
 Rational Field
>>> f[Integer(1)].display()
f_1 = e_1 - e_2
>>> f[Integer(2)].display()
f_2 = 2 e_1 + 3 e_2
>>> e[Integer(1)].display(f)
e_1 = 3/5 f_1 + 1/5 f_2
>>> e[Integer(2)].display(f)
e_2 = -2/5 f_1 + 1/5 f_2

Sage Live

M = FiniteRankFreeModule(QQ, 2, name='M', start_index=1)
e = M.basis('e')
a = M.automorphism()
a[:] = [[1, 2], [-1, 3]]
f = e.new_basis(a, 'f') ; f
f[1].display()
f[2].display()
e[1].display(f)
e[2].display(f)

Use of some keyword arguments:

Sage

sage: b = e.new_basis(a, 'b', indices=('x', 'y'),
....:                 symbol_dual=('A', 'B'))
sage: b
Basis (b_x,b_y) on the 2-dimensional vector space M over the
 Rational Field
sage: b.dual_basis()
Dual basis (A,B) on the 2-dimensional vector space M over the
 Rational Field

Python

>>> from sage.all import *
>>> b = e.new_basis(a, 'b', indices=('x', 'y'),
...                 symbol_dual=('A', 'B'))
>>> b
Basis (b_x,b_y) on the 2-dimensional vector space M over the
 Rational Field
>>> b.dual_basis()
Dual basis (A,B) on the 2-dimensional vector space M over the
 Rational Field

Sage Live

b = e.new_basis(a, 'b', indices=('x', 'y'),
                symbol_dual=('A', 'B'))
b
b.dual_basis()

class sage.tensor.modules.free_module_basis.FreeModuleCoBasis(basis, symbol, latex_symbol=None, indices=None, latex_indices=None)[source]¶

Bases: Basis_abstract

Dual basis of a free module over a commutative ring.

INPUT:

basis – basis of a free module \(M\) of which self is the dual (must be an instance of FreeModuleBasis)
symbol – either a string, to be used as a common base for the symbols of the elements of the cobasis, or a tuple of strings, representing the individual symbols of the elements of the cobasis
latex_symbol – (default: None) either a string, to be used as a common base for the LaTeX symbols of the elements of the cobasis, or a tuple of strings, representing the individual LaTeX symbols of the elements of the cobasis; if None, symbol is used in place of latex_symbol
indices – (default: None; used only if symbol is a single string) tuple of strings representing the indices labelling the elements of the cobasis; if None, the indices will be generated as integers within the range declared on the free module on which the cobasis is defined
latex_indices – (default: None) tuple of strings representing the indices for the LaTeX symbols of the elements of the cobasis; if None, indices is used instead

EXAMPLES:

Dual basis on a rank-3 free module:

Sage

sage: M = FiniteRankFreeModule(ZZ, 3, name='M', start_index=1)
sage: e = M.basis('e') ; e
Basis (e_1,e_2,e_3) on the Rank-3 free module M over the Integer Ring
sage: from sage.tensor.modules.free_module_basis import FreeModuleCoBasis
sage: f = FreeModuleCoBasis(e, 'f') ; f
Dual basis (f^1,f^2,f^3) on the Rank-3 free module M over the Integer Ring

Python

>>> from sage.all import *
>>> M = FiniteRankFreeModule(ZZ, Integer(3), name='M', start_index=Integer(1))
>>> e = M.basis('e') ; e
Basis (e_1,e_2,e_3) on the Rank-3 free module M over the Integer Ring
>>> from sage.tensor.modules.free_module_basis import FreeModuleCoBasis
>>> f = FreeModuleCoBasis(e, 'f') ; f
Dual basis (f^1,f^2,f^3) on the Rank-3 free module M over the Integer Ring

Sage Live

M = FiniteRankFreeModule(ZZ, 3, name='M', start_index=1)
e = M.basis('e') ; e
from sage.tensor.modules.free_module_basis import FreeModuleCoBasis
f = FreeModuleCoBasis(e, 'f') ; f

Instead of importing FreeModuleCoBasis in the global name space, it is recommended to use the method dual_basis() of the basis e:

Sage

sage: f = e.dual_basis() ; f
Dual basis (e^1,e^2,e^3) on the Rank-3 free module M over the Integer Ring

Python

>>> from sage.all import *
>>> f = e.dual_basis() ; f
Dual basis (e^1,e^2,e^3) on the Rank-3 free module M over the Integer Ring

Sage Live

f = e.dual_basis() ; f

Let us check that the elements of f are in the dual of M:

Sage

sage: f[1]
Linear form e^1 on the Rank-3 free module M over the Integer Ring
sage: f[1] in M.dual()
True

Python

>>> from sage.all import *
>>> f[Integer(1)]
Linear form e^1 on the Rank-3 free module M over the Integer Ring
>>> f[Integer(1)] in M.dual()
True

Sage Live

f[1]
f[1] in M.dual()

and that f is indeed the dual of e:

Sage

sage: f[1](e[1]), f[1](e[2]), f[1](e[3])
(1, 0, 0)
sage: f[2](e[1]), f[2](e[2]), f[2](e[3])
(0, 1, 0)
sage: f[3](e[1]), f[3](e[2]), f[3](e[3])
(0, 0, 1)

Python

>>> from sage.all import *
>>> f[Integer(1)](e[Integer(1)]), f[Integer(1)](e[Integer(2)]), f[Integer(1)](e[Integer(3)])
(1, 0, 0)
>>> f[Integer(2)](e[Integer(1)]), f[Integer(2)](e[Integer(2)]), f[Integer(2)](e[Integer(3)])
(0, 1, 0)
>>> f[Integer(3)](e[Integer(1)]), f[Integer(3)](e[Integer(2)]), f[Integer(3)](e[Integer(3)])
(0, 0, 1)

Sage Live

f[1](e[1]), f[1](e[2]), f[1](e[3])
f[2](e[1]), f[2](e[2]), f[2](e[3])
f[3](e[1]), f[3](e[2]), f[3](e[3])