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:
- 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 likecollections.abc.Mapping
subclasses such asdict
, it is an associativeContainer
, providing methodskeys()
,values()
, anditems()
. Thus,e[i]
returns the element of the basise
indexed by the keyi
. However, in contrast toMapping
subclasses, not thekeys()
but thevalues()
are considered the elements.EXAMPLES:
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)]
>>> 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)]
- cardinality()[source]¶
Return the basis length, i.e. the rank of the free module.
EXAMPLES:
sage: M = FiniteRankFreeModule(ZZ, 3, name='M') sage: e = M.basis('e') sage: e.cardinality() 3
>>> from sage.all import * >>> M = FiniteRankFreeModule(ZZ, Integer(3), name='M') >>> e = M.basis('e') >>> e.cardinality() 3
- free_module()[source]¶
Return the free module of
self
.EXAMPLES:
sage: M = FiniteRankFreeModule(QQ, 2, name='M', start_index=1) sage: e = M.basis('e') sage: e.free_module() is M True
>>> 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
- keys()[source]¶
Return the keys (indices) of the family.
EXAMPLES:
sage: M = FiniteRankFreeModule(ZZ, 3, name='M') sage: e = M.basis('e') sage: list(e.keys()) [0, 1, 2]
>>> from sage.all import * >>> M = FiniteRankFreeModule(ZZ, Integer(3), name='M') >>> e = M.basis('e') >>> list(e.keys()) [0, 1, 2]
- 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 ofself
, or a list of strings, representing the individual symbols of the elements ofself
latex_symbol
– (default:None
) either a string, to be used as a common base for the LaTeX symbols of the elements ofself
, or a list of strings, representing the individual LaTeX symbols of the elements ofself
; ifNone
,symbol
is used in place oflatex_symbol
indices
– (default:None
; used only ifsymbol
is a single string) tuple of strings representing the indices labelling the elements ofself
; ifNone
, the indices will be generated as integers within the range declared on the free module on whichself
is definedlatex_indices
– (default:None
) list of strings representing the indices for the LaTeX symbols of the elements ofself
; ifNone
,indices
is used insteadindex_position
– (default:'down'
) determines the position of the indices labelling the individual elements ofself
; can be either'down'
or'up'
EXAMPLES:
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
>>> 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
- values()[source]¶
Return the basis elements of
self
.EXAMPLES:
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]
>>> 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]
- 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 ofFiniteRankFreeModule
)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 basislatex_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; ifNone
,symbol
is used in place oflatex_symbol
indices
– (default:None
; used only ifsymbol
is a single string) tuple of strings representing the indices labelling the elements of the basis; ifNone
, the indices will be generated as integers within the range declared onfmodule
latex_indices
– (default:None
) tuple of strings representing the indices for the LaTeX symbols of the elements of the basis; ifNone
,indices
is used insteadsymbol_dual
– (default:None
) same assymbol
but for the dual basis; ifNone
,symbol
must be a string and is used for the common base of the symbols of the elements of the dual basislatex_symbol_dual
– (default:None
) same aslatex_symbol
but for the dual basis
EXAMPLES:
A basis on a rank-3 free module over \(\ZZ\):
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
>>> 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
Instead of importing
FreeModuleBasis
in the global name space, it is recommended to use the module’s methodbasis()
: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
>>> 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
The individual elements constituting the basis are accessed via the square bracket operator:
sage: e[0] Element e_0 of the Rank-3 free module M over the Integer Ring sage: e[0] in M True
>>> 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
The slice operator
:
can be used to access to more than one element: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)
>>> 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)
The LaTeX symbol can be set explicitly:
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)
>>> 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)
The individual elements of the basis are labelled according the parameter
start_index
provided at the free module construction: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
>>> 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
It is also possible to fully customize the labels, via the argument
indices
: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
>>> 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
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: 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
>>> 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
- dual_basis()[source]¶
Return the basis dual to
self
.OUTPUT:
instance of
FreeModuleCoBasis
representing the dual ofself
EXAMPLES:
Dual basis on a rank-3 free module:
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
>>> 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
Let us check that the elements of f are elements of the dual of M:
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
>>> 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
and that f is indeed the dual of e:
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)
>>> 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)
- module()[source]¶
Return the free module on which the basis is defined.
OUTPUT:
instance of
FiniteRankFreeModule
representing the free module of whichself
is a basis
EXAMPLES:
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
>>> 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
- 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 ofFreeModuleAutomorphism
describing the automorphism \(P\) that relates the current basis \((e_i)\) (described byself
) 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 basislatex_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; ifNone
,symbol
is used in place oflatex_symbol
indices
– (default:None
; used only ifsymbol
is a single string) tuple of strings representing the indices labelling the elements of the basis; ifNone
, the indices will be generated as integers within the range declared on the free module on whichself
is definedlatex_indices
– (default:None
) tuple of strings representing the indices for the LaTeX symbols of the elements of the basis; ifNone
,indices
is used insteadsymbol_dual
– (default:None
) same assymbol
but for the dual basis; ifNone
,symbol
must be a string and is used for the common base of the symbols of the elements of the dual basislatex_symbol_dual
– (default:None
) same aslatex_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: 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
>>> 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
Use of some keyword arguments:
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
>>> 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
- 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 whichself
is the dual (must be an instance ofFreeModuleBasis
)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 cobasislatex_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; ifNone
,symbol
is used in place oflatex_symbol
indices
– (default:None
; used only ifsymbol
is a single string) tuple of strings representing the indices labelling the elements of the cobasis; ifNone
, the indices will be generated as integers within the range declared on the free module on which the cobasis is definedlatex_indices
– (default:None
) tuple of strings representing the indices for the LaTeX symbols of the elements of the cobasis; ifNone
,indices
is used instead
EXAMPLES:
Dual basis on a rank-3 free module:
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
>>> 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
Instead of importing
FreeModuleCoBasis
in the global name space, it is recommended to use the methoddual_basis()
of the basise
: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
>>> 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
Let us check that the elements of
f
are in the dual ofM
: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
>>> 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
and that
f
is indeed the dual ofe
: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)
>>> 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)