Free modules#
Sage supports computation with free modules over an arbitrary commutative ring. Nontrivial functionality is available over \(\ZZ\), fields, and some principal ideal domains (e.g. \(\QQ[x]\) and rings of integers of number fields). All free modules over an integral domain are equipped with an embedding in an ambient vector space and an inner product, which you can specify and change.
Create the free module of rank \(n\) over an arbitrary commutative ring \(R\) using
the command FreeModule(R,n)
. Equivalently, R^n
also creates that free
module.
The following example illustrates the creation of both a vector space and a
free module over the integers and a submodule of it. Use the functions
FreeModule
, span
and member functions of free modules to create free
modules. Do not use the FreeModule_xxx constructors directly.
EXAMPLES:
sage: V = VectorSpace(QQ,3)
sage: W = V.subspace([[1,2,7], [1,1,0]])
sage: W
Vector space of degree 3 and dimension 2 over Rational Field
Basis matrix:
[ 1 0 -7]
[ 0 1 7]
sage: C = VectorSpaces(FiniteField(7))
sage: C
Category of vector spaces over Finite Field of size 7
sage: C(W)
Vector space of degree 3 and dimension 2 over Finite Field of size 7
Basis matrix:
[1 0 0]
[0 1 0]
sage: M = ZZ^3
sage: C = VectorSpaces(FiniteField(7))
sage: C(M)
Vector space of dimension 3 over Finite Field of size 7
sage: W = M.submodule([[1,2,7], [8,8,0]])
sage: C(W)
Vector space of degree 3 and dimension 2 over Finite Field of size 7
Basis matrix:
[1 0 0]
[0 1 0]
We illustrate the exponent notation for creation of free modules.
sage: ZZ^4
Ambient free module of rank 4 over the principal ideal domain Integer Ring
sage: QQ^2
Vector space of dimension 2 over Rational Field
sage: RR^3
Vector space of dimension 3 over Real Field with 53 bits of precision
Base ring:
sage: R.<x,y> = QQ[]
sage: M = FreeModule(R,2)
sage: M.base_ring()
Multivariate Polynomial Ring in x, y over Rational Field
sage: VectorSpace(QQ, 10).base_ring()
Rational Field
Enumeration of \(\ZZ^n\) happens in order of increasing \(1\)-norm primarily and increasing \(\infty\)-norm secondarily:
sage: print([v for _,v in zip(range(31), ZZ^3)])
[(0, 0, 0),
(1, 0, 0), (-1, 0, 0), (0, 1, 0), (0, -1, 0), (0, 0, 1), (0, 0, -1),
(1, 1, 0), (-1, 1, 0), (1, -1, 0), (-1, -1, 0), (1, 0, 1), (-1, 0, 1), (1, 0, -1), (-1, 0, -1), (0, 1, 1), (0, -1, 1), (0, 1, -1), (0, -1, -1),
(2, 0, 0), (-2, 0, 0), (0, 2, 0), (0, -2, 0), (0, 0, 2), (0, 0, -2),
(1, 1, 1), (-1, 1, 1), (1, -1, 1), (-1, -1, 1), (1, 1, -1), ...]
For other infinite enumerated base rings (i.e., rings which
are objects of the category InfiniteEnumeratedSets
),
a free module of rank \(r\) is enumerated by applying
FreeModule_ambient.linear_combination_of_basis()
to all vectors in \(\ZZ^r\), enumerated in the way shown above.
AUTHORS:
William Stein (2005, 2007)
David Kohel (2007, 2008)
Niles Johnson (2010-08): (trac ticket #3893)
random_element()
should pass on*args
and**kwds
.Simon King (2010-12): (trac ticket #8800) fixed a bug in
denominator()
.Simon King (2010-12), Peter Bruin (June 2014): (trac ticket #10513) new coercion model and category framework.
- class sage.modules.free_module.ComplexDoubleVectorSpace_class(n)#
Bases:
FreeModule_ambient_field
- coordinates(v)#
- class sage.modules.free_module.EchelonMatrixKey(obj)#
Bases:
object
A total ordering on free modules for sorting.
This class orders modules by their ambient spaces, then by dimension, then in order by their echelon matrices. If a function returns a list of free modules, this can be used to sort the output and thus render it deterministic.
INPUT:
obj
– a free module
EXAMPLES:
sage: V = span([[1,2,3], [5,6,7], [8,9,10]], QQ) sage: W = span([[5,6,7], [8,9,10]], QQ) sage: X = span([[5,6,7]], ZZ).scale(1/11) sage: Y = CC^3 sage: Z = ZZ^2 sage: modules = [V,W,X,Y,Z] sage: modules_sorted = [Z,X,V,W,Y] sage: from sage.modules.free_module import EchelonMatrixKey sage: modules.sort(key=EchelonMatrixKey) sage: modules == modules_sorted True
- sage.modules.free_module.FreeModule(base_ring, rank_or_basis_keys, sparse, inner_product_matrix, with_basis=None, rank=False, basis_keys=None, **args)#
Create a free module over the given commutative
base_ring
FreeModule
can be called with the following positional arguments:FreeModule(base_ring, rank, ...)
FreeModule(base_ring, basis_keys, ...)
INPUT:
base_ring
– a commutative ringrank
– a nonnegative integerbasis_keys
– a finite or enumerated family of arbitrary objectssparse
– boolean (defaultFalse
)inner_product_matrix
– the inner product matrix (defaultNone
)with_basis
– either"standard"
(the default), in which case a free module with the standard basis as the distinguished basis is created; orNone
, in which case a free module without distinguished basis is created.further options may be accepted by various implementation classes
OUTPUT: a free module
This factory function creates instances of various specialized classes depending on the input. Not all combinations of options are implemented.
If the parameter
basis_keys
is provided, it must be a finite or enumerated family of objects, and an instance ofCombinatorialFreeModule
is created.EXAMPLES:
sage: CombinatorialFreeModule(QQ, ['a','b','c']) Free module generated by {'a', 'b', 'c'} over Rational Field
It has a distinguished standard basis that is indexed by the provided
basis_keys
. See the documentation ofCombinatorialFreeModule
for more examples and details, including itsUniqueRepresentation
semantics.If the parameter
with_basis
is set toNone
, then a free module of the givenrank
without distinguished basis is created. It is represented by an instance ofFiniteRankFreeModule
.EXAMPLES:
sage: FiniteRankFreeModule(ZZ, 3, name='M') Rank-3 free module M over the Integer Ring
See the documentation of
FiniteRankFreeModule
for more options, examples, and details.If
rank
is provided and the optionwith_basis
is left at its default value,"standard"
, then a free ambient module with distinguished standard basis indexed byrange(rank)
is created. There is only one dense and one sparse free ambient module of givenrank
overbase_ring
.EXAMPLES:
sage: FreeModule(Integers(8), 10) Ambient free module of rank 10 over Ring of integers modulo 8
The remainder of this documentation discusses this case of free ambient modules.
EXAMPLES:
First we illustrate creating free modules over various base fields. The base field affects the free module that is created. For example, free modules over a field are vector spaces, and free modules over a principal ideal domain are special in that more functionality is available for them than for completely general free modules.
sage: FreeModule(QQ,10) Vector space of dimension 10 over Rational Field sage: FreeModule(ZZ,10) Ambient free module of rank 10 over the principal ideal domain Integer Ring sage: FreeModule(FiniteField(5),10) Vector space of dimension 10 over Finite Field of size 5 sage: FreeModule(Integers(7),10) Vector space of dimension 10 over Ring of integers modulo 7 sage: FreeModule(PolynomialRing(QQ,'x'),5) Ambient free module of rank 5 over the principal ideal domain Univariate Polynomial Ring in x over Rational Field sage: FreeModule(PolynomialRing(ZZ,'x'),5) Ambient free module of rank 5 over the integral domain Univariate Polynomial Ring in x over Integer Ring
Of course we can make rank 0 free modules:
sage: FreeModule(RealField(100),0) Vector space of dimension 0 over Real Field with 100 bits of precision
Next we create a free module with sparse representation of elements. Functionality with sparse modules is identical to dense modules, but they may use less memory and arithmetic may be faster (or slower!).
sage: M = FreeModule(ZZ,200,sparse=True) sage: M.is_sparse() True sage: type(M.0) <class 'sage.modules.free_module_element.FreeModuleElement_generic_sparse'>
The default is dense.
sage: M = ZZ^200 sage: type(M.0) <class 'sage.modules.vector_integer_dense.Vector_integer_dense'>
Note that matrices associated in some way to sparse free modules are sparse by default:
sage: M = FreeModule(Integers(8), 2) sage: A = M.basis_matrix() sage: A.is_sparse() False sage: Ms = FreeModule(Integers(8), 2, sparse=True) sage: M == Ms # as mathematical objects they are equal True sage: Ms.basis_matrix().is_sparse() True
We can also specify an inner product matrix, which is used when computing inner products of elements.
sage: A = MatrixSpace(ZZ,2)([[1,0],[0,-1]]) sage: M = FreeModule(ZZ,2,inner_product_matrix=A) sage: v, w = M.gens() sage: v.inner_product(w) 0 sage: v.inner_product(v) 1 sage: w.inner_product(w) -1 sage: (v+2*w).inner_product(w) -2
You can also specify the inner product matrix by giving anything that coerces to an appropriate matrix. This is only useful if the inner product matrix takes values in the base ring.
sage: FreeModule(ZZ,2,inner_product_matrix=1).inner_product_matrix() [1 0] [0 1] sage: FreeModule(ZZ,2,inner_product_matrix=[1,2,3,4]).inner_product_matrix() [1 2] [3 4] sage: FreeModule(ZZ,2,inner_product_matrix=[[1,2],[3,4]]).inner_product_matrix() [1 2] [3 4]
Todo
Refactor modules such that it only counts what category the base ring belongs to, but not what is its Python class.
EXAMPLES:
sage: FreeModule(QQ, ['a', 'b', 'c']) Free module generated by {'a', 'b', 'c'} over Rational Field sage: _.category() Category of finite dimensional vector spaces with basis over Rational Field sage: FreeModule(QQ, 3, with_basis=None) 3-dimensional vector space over the Rational Field sage: _.category() Category of finite dimensional vector spaces over Rational Field sage: FreeModule(QQ, [1, 2, 3, 4], with_basis=None) 4-dimensional vector space over the Rational Field sage: _.category() Category of finite dimensional vector spaces over Rational Field
- class sage.modules.free_module.FreeModuleFactory#
Bases:
UniqueFactory
Factory class for the finite-dimensional free modules with standard basis
- create_key(base_ring, rank, sparse=False, inner_product_matrix=None)#
- create_object(version, key)#
- class sage.modules.free_module.FreeModule_ambient(base_ring, rank, sparse=False, coordinate_ring=None, category=None)#
Bases:
FreeModule_generic
Ambient free module over a commutative ring.
- ambient_module()#
Return
self
, sinceself
is ambient.EXAMPLES:
sage: A = QQ^5; A.ambient_module() Vector space of dimension 5 over Rational Field sage: A = ZZ^5; A.ambient_module() Ambient free module of rank 5 over the principal ideal domain Integer Ring
- basis()#
Return a basis for this ambient free module.
OUTPUT:
Sequence
- an immutable sequence with universe this ambient free module
EXAMPLES:
sage: A = ZZ^3; B = A.basis(); B [ (1, 0, 0), (0, 1, 0), (0, 0, 1) ] sage: B.universe() Ambient free module of rank 3 over the principal ideal domain Integer Ring
- change_ring(R)#
Return the ambient free module over
R
of the same rank asself
.This also preserves the sparsity.
EXAMPLES:
sage: A = ZZ^3; A.change_ring(QQ) Vector space of dimension 3 over Rational Field sage: A = ZZ^3; A.change_ring(GF(5)) Vector space of dimension 3 over Finite Field of size 5
For ambient modules any change of rings is defined:
sage: A = GF(5)**3; A.change_ring(QQ) Vector space of dimension 3 over Rational Field
- coordinate_vector(v, check=True)#
Write \(v\) in terms of the standard basis for
self
and return the resulting coefficients in a vector over the fraction field of the base ring.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, raise an
ArithmeticError
exception.EXAMPLES:
sage: V = Integers(16)^3 sage: v = V.coordinate_vector([1,5,9]); v (1, 5, 9) sage: v.parent() Ambient free module of rank 3 over Ring of integers modulo 16
- echelon_coordinate_vector(v, check=True)#
Same as
self.coordinate_vector(v)
, sinceself
is an ambient free module.INPUT:
v
- vectorcheck
- boolean (default:True
); if True, also verify that \(v\) is really inself
.
OUTPUT: list
EXAMPLES:
sage: V = QQ^4 sage: v = V([-1/2,1/2,-1/2,1/2]) sage: v (-1/2, 1/2, -1/2, 1/2) sage: V.coordinate_vector(v) (-1/2, 1/2, -1/2, 1/2) sage: V.echelon_coordinate_vector(v) (-1/2, 1/2, -1/2, 1/2) sage: W = V.submodule_with_basis([[1/2,1/2,1/2,1/2],[1,0,1,0]]) sage: W.coordinate_vector(v) (1, -1) sage: W.echelon_coordinate_vector(v) (-1/2, 1/2)
- echelon_coordinates(v, check=True)#
Returns the coordinate vector of v in terms of the echelon basis for self.
EXAMPLES:
sage: U = VectorSpace(QQ,3) sage: [ U.coordinates(v) for v in U.basis() ] [[1, 0, 0], [0, 1, 0], [0, 0, 1]] sage: [ U.echelon_coordinates(v) for v in U.basis() ] [[1, 0, 0], [0, 1, 0], [0, 0, 1]] sage: V = U.submodule([[1,1,0],[0,1,1]]) sage: V Vector space of degree 3 and dimension 2 over Rational Field Basis matrix: [ 1 0 -1] [ 0 1 1] sage: [ V.coordinates(v) for v in V.basis() ] [[1, 0], [0, 1]] sage: [ V.echelon_coordinates(v) for v in V.basis() ] [[1, 0], [0, 1]] sage: W = U.submodule_with_basis([[1,1,0],[0,1,1]]) sage: W Vector space of degree 3 and dimension 2 over Rational Field User basis matrix: [1 1 0] [0 1 1] sage: [ W.coordinates(v) for v in W.basis() ] [[1, 0], [0, 1]] sage: [ W.echelon_coordinates(v) for v in W.basis() ] [[1, 1], [0, 1]]
- echelonized_basis()#
Return a basis for this ambient free module in echelon form.
EXAMPLES:
sage: A = ZZ^3; A.echelonized_basis() [ (1, 0, 0), (0, 1, 0), (0, 0, 1) ]
- echelonized_basis_matrix()#
The echelonized basis matrix of self.
EXAMPLES:
sage: V = ZZ^4 sage: W = V.submodule([ V.gen(i)-V.gen(0) for i in range(1,4) ]) sage: W.basis_matrix() [ 1 0 0 -1] [ 0 1 0 -1] [ 0 0 1 -1] sage: W.echelonized_basis_matrix() [ 1 0 0 -1] [ 0 1 0 -1] [ 0 0 1 -1] sage: U = V.submodule_with_basis([ V.gen(i)-V.gen(0) for i in range(1,4) ]) sage: U.basis_matrix() [-1 1 0 0] [-1 0 1 0] [-1 0 0 1] sage: U.echelonized_basis_matrix() [ 1 0 0 -1] [ 0 1 0 -1] [ 0 0 1 -1]
- gen(i=0)#
Return the \(i\)-th generator for
self
.Here \(i\) is between 0 and rank - 1, inclusive.
INPUT:
\(i\) – an integer (default 0)
OUTPUT: \(i\)-th basis vector for
self
.EXAMPLES:
sage: n = 5 sage: V = QQ^n sage: B = [V.gen(i) for i in range(n)] sage: B [(1, 0, 0, 0, 0), (0, 1, 0, 0, 0), (0, 0, 1, 0, 0), (0, 0, 0, 1, 0), (0, 0, 0, 0, 1)] sage: V.gens() == tuple(B) True
- is_ambient()#
Return
True
since this module is an ambient module.EXAMPLES:
sage: A = QQ^5; A.is_ambient() True sage: A = (QQ^5).span([[1,2,3,4,5]]); A.is_ambient() False
- linear_combination_of_basis(v)#
Return the linear combination of the basis for
self
obtained from the elements of the list v.INPUT:
v
- list
EXAMPLES:
sage: V = span([[1,2,3], [4,5,6]], ZZ) sage: V Free module of degree 3 and rank 2 over Integer Ring Echelon basis matrix: [1 2 3] [0 3 6] sage: V.linear_combination_of_basis([1,1]) (1, 5, 9)
This should raise an error if the resulting element is not in self:
sage: W = span([[2,4]], ZZ) sage: W.linear_combination_of_basis([1/2]) Traceback (most recent call last): ... TypeError: element [1, 2] is not in free module
- random_element(prob=1.0, *args, **kwds)#
Returns a random element of self.
INPUT:
prob
- float. Each coefficient will be set to zero withprobability \(1-prob\). Otherwise coefficients will be chosen randomly from base ring (and may be zero).
*args, **kwds
- passed on to random_element function of basering.
EXAMPLES:
sage: M = FreeModule(ZZ, 3) sage: M.random_element().parent() is M True
Passes extra positional or keyword arguments through:
sage: all(i in range(5, 10) for i in M.random_element(1.0, 5, 10)) True
sage: M = FreeModule(ZZ, 16) sage: M.random_element().parent() is M True sage: def add_sample(**kwds): ....: global total, zeros ....: v = M.random_element(**kwds) ....: total += M.rank() ....: zeros += sum(i == 0 for i in v) sage: total = 0 sage: zeros = 0 sage: add_sample() sage: expected = 1/5 sage: while abs(zeros/total - expected) > 0.01: ....: add_sample() sage: total = 0 sage: zeros = 0 sage: add_sample(prob=0.3) sage: expected = 1/5 * 3/10 + 7/10 sage: while abs(zeros/total - expected) > 0.01: ....: add_sample(prob=0.3) sage: total = 0 sage: zeros = 0 sage: add_sample(prob=0.7) sage: expected = 1/5 * 7/10 + 3/10 sage: while abs(zeros/total - expected) > 0.01: ....: add_sample(prob=0.7)
- class sage.modules.free_module.FreeModule_ambient_domain(base_ring, rank, sparse=False, coordinate_ring=None, category=None)#
Bases:
FreeModule_generic_domain
,FreeModule_ambient
Ambient free module over an integral domain.
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
- ambient_vector_space()#
Return the ambient vector space, which is this free module tensored with its fraction field.
EXAMPLES:
sage: M = ZZ^3 sage: V = M.ambient_vector_space(); V Vector space of dimension 3 over Rational Field
If an inner product on the module is specified, then this is preserved on the ambient vector space.
sage: N = FreeModule(ZZ,4,inner_product_matrix=1) sage: U = N.ambient_vector_space() sage: U Ambient quadratic space of dimension 4 over Rational Field Inner product matrix: [1 0 0 0] [0 1 0 0] [0 0 1 0] [0 0 0 1] sage: P = N.submodule_with_basis([[1,-1,0,0],[0,1,-1,0],[0,0,1,-1]]) sage: P.gram_matrix() [ 2 -1 0] [-1 2 -1] [ 0 -1 2] sage: U == N.ambient_vector_space() True sage: U == V False
- coordinate_vector(v, check=True)#
Write \(v\) in terms of the standard basis for
self
and return the resulting coefficients in a vector over the fraction field of the base ring.INPUT:
v
– vectorcheck
– boolean (default:True
); ifTrue
, also verify that\(v\) is really in
self
.
OUTPUT: list
The output is 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, raise 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
- vector_space(base_field=None)#
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 sage.modules.free_module.FreeModule_ambient_field(base_field, dimension, sparse=False, category=None)#
Bases:
FreeModule_generic_field
,FreeModule_ambient_pid
- ambient_vector_space()#
Returns
self
as the ambient vector space.EXAMPLES:
sage: M = QQ^3 sage: M.ambient_vector_space() Vector space of dimension 3 over Rational Field
- base_field()#
Returns the base field of this vector space.
EXAMPLES:
sage: M = QQ^3 sage: M.base_field() Rational Field
- class sage.modules.free_module.FreeModule_ambient_pid(base_ring, rank, sparse=False, coordinate_ring=None, category=None)#
Bases:
FreeModule_generic_pid
,FreeModule_ambient_domain
Ambient free module over a principal ideal domain.
- class sage.modules.free_module.FreeModule_generic(base_ring, rank, degree, sparse=False, coordinate_ring=None, category=None)#
Bases:
Module_free_ambient
Base class for all free modules.
INPUT:
base_ring
– a commutative ringrank
– a non-negative integerdegree
– a non-negative integersparse
– boolean (default:False
)coordinate_ring
– a ring containingbase_ring
(default: equal tobase_ring
)category
– category (default:None
)
If
base_ring
is a field, then the default category is the category of finite-dimensional vector spaces over that field; otherwise it is the category of finite-dimensional free modules over that ring. In addition, the category is intersected with the category of finite enumerated sets if the ring is finite or the rank is 0.EXAMPLES:
sage: PolynomialRing(QQ,3,'x')^3 Ambient free module of rank 3 over the integral domain Multivariate Polynomial Ring in x0, x1, x2 over Rational Field sage: FreeModule(GF(7),3).category() Category of enumerated finite dimensional vector spaces with basis over (finite enumerated fields and subquotients of monoids and quotients of semigroups) sage: V = QQ^4; V.category() Category of finite dimensional vector spaces with basis over (number fields and quotient fields and metric spaces) sage: V = GF(5)**20; V.category() Category of enumerated finite dimensional vector spaces with basis over (finite enumerated fields and subquotients of monoids and quotients of semigroups) sage: FreeModule(ZZ,3).category() Category of finite dimensional modules with basis over (euclidean domains and infinite enumerated sets and metric spaces) sage: (QQ^0).category() Category of finite enumerated finite dimensional vector spaces with basis over (number fields and quotient fields and metric spaces)
- are_linearly_dependent(vecs)#
Return
True
if the vectorsvecs
are linearly dependent andFalse
otherwise.EXAMPLES:
sage: M = QQ^3 sage: vecs = [M([1,2,3]), M([4,5,6])] sage: M.are_linearly_dependent(vecs) False sage: vecs.append(M([3,3,3])) sage: M.are_linearly_dependent(vecs) True sage: R.<x> = QQ[] sage: M = FreeModule(R, 2) sage: vecs = [M([x^2+1, x+1]), M([x+2, 2*x+1])] sage: M.are_linearly_dependent(vecs) False sage: vecs.append(M([-2*x+1, -2*x^2+1])) sage: M.are_linearly_dependent(vecs) True
- base_field()#
Return the base field, which is the fraction field of the base ring of this module.
EXAMPLES:
sage: FreeModule(GF(3), 2).base_field() Finite Field of size 3 sage: FreeModule(ZZ, 2).base_field() Rational Field sage: FreeModule(PolynomialRing(GF(7),'x'), 2).base_field() Fraction Field of Univariate Polynomial Ring in x over Finite Field of size 7
- basis()#
Return the basis of this module.
EXAMPLES:
sage: FreeModule(Integers(12),3).basis() [ (1, 0, 0), (0, 1, 0), (0, 0, 1) ]
- basis_matrix(ring=None)#
Return the matrix whose rows are the basis for this free module.
INPUT:
ring
– (default:self.coordinate_ring()
) a ring over which the matrix is defined
EXAMPLES:
sage: FreeModule(Integers(12),3).basis_matrix() [1 0 0] [0 1 0] [0 0 1]
sage: M = FreeModule(GF(7),3).span([[2,3,4],[1,1,1]]); M Vector space of degree 3 and dimension 2 over Finite Field of size 7 Basis matrix: [1 0 6] [0 1 2] sage: M.basis_matrix() [1 0 6] [0 1 2]
sage: M = FreeModule(GF(7),3).span_of_basis([[2,3,4],[1,1,1]]) sage: M.basis_matrix() [2 3 4] [1 1 1]
sage: M = FreeModule(QQ,2).span_of_basis([[1,-1],[1,0]]); M Vector space of degree 2 and dimension 2 over Rational Field User basis matrix: [ 1 -1] [ 1 0] sage: M.basis_matrix() [ 1 -1] [ 1 0]
- cardinality()#
Return the cardinality of the free module.
OUTPUT:
Either an integer or
+Infinity
.EXAMPLES:
sage: k.<a> = FiniteField(9) sage: V = VectorSpace(k,3) sage: V.cardinality() 729 sage: W = V.span([[1,2,1],[0,1,1]]) sage: W.cardinality() 81 sage: R = IntegerModRing(12) sage: M = FreeModule(R,2) sage: M.cardinality() 144 sage: (QQ^3).cardinality() +Infinity
- codimension()#
Return the codimension of this free module, which is the dimension of the ambient space minus the dimension of this free module.
EXAMPLES:
sage: M = Matrix(3, 4, range(12)) sage: V = M.left_kernel(); V Free module of degree 3 and rank 1 over Integer Ring Echelon basis matrix: [ 1 -2 1] sage: V.dimension() 1 sage: V.codimension() 2
The codimension of an ambient space is always zero:
sage: (QQ^10).codimension() 0
- construction()#
The construction functor and base ring for self.
EXAMPLES:
sage: R = PolynomialRing(QQ,3,'x') sage: V = R^5 sage: V.construction() (VectorFunctor, Multivariate Polynomial Ring in x0, x1, x2 over Rational Field)
- coordinate_module(V)#
Suppose
V
is a submodule ofself
(or a module commensurable withself
), and thatself
is a free module over \(R\) of rank \(n\). Let \(\phi\) be the map fromself
to \(R^n\) that sends the basis vectors ofself
in order to the standard basis of \(R^n\). This function returns the image \(\phi(V)\).Warning
If there is no integer \(d\) such that \(dV\) is a submodule of
self
, then this function will give total nonsense.EXAMPLES:
We illustrate this function with some \(\ZZ\)-submodules of \(\QQ^3\):
sage: V = (ZZ^3).span([[1/2,3,5], [0,1,-3]]) sage: W = (ZZ^3).span([[1/2,4,2]]) sage: V.coordinate_module(W) Free module of degree 2 and rank 1 over Integer Ring User basis matrix: [1 4] sage: V.0 + 4*V.1 (1/2, 4, 2)
In this example, the coordinate module isn’t even in \(\ZZ^3\):
sage: W = (ZZ^3).span([[1/4,2,1]]) sage: V.coordinate_module(W) Free module of degree 2 and rank 1 over Integer Ring User basis matrix: [1/2 2]
The following more elaborate example illustrates using this function to write a submodule in terms of integral cuspidal modular symbols:
sage: M = ModularSymbols(54) sage: S = M.cuspidal_subspace() sage: K = S.integral_structure(); K Free module of degree 19 and rank 8 over Integer Ring Echelon basis matrix: [ 0 1 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0] ... sage: L = M[0].integral_structure(); L Free module of degree 19 and rank 2 over Integer Ring Echelon basis matrix: [ 0 1 1 0 -2 1 -1 1 -1 -2 2 0 0 0 0 0 0 0 0] [ 0 0 3 0 -3 2 -1 2 -1 -4 2 -1 -2 1 2 0 0 -1 1] sage: K.coordinate_module(L) Free module of degree 8 and rank 2 over Integer Ring User basis matrix: [ 1 1 1 -1 1 -1 0 0] [ 0 3 2 -1 2 -1 -1 -2] sage: K.coordinate_module(L).basis_matrix() * K.basis_matrix() [ 0 1 1 0 -2 1 -1 1 -1 -2 2 0 0 0 0 0 0 0 0] [ 0 0 3 0 -3 2 -1 2 -1 -4 2 -1 -2 1 2 0 0 -1 1]
- coordinate_ring()#
Return the ring over which the entries of the vectors are defined.
This is the same as
base_ring()
unless an explicit basis was given over the fraction field.EXAMPLES:
sage: M = ZZ^2 sage: M.coordinate_ring() Integer Ring
sage: M = (ZZ^2) * (1/2) sage: M.base_ring() Integer Ring sage: M.coordinate_ring() Rational Field
sage: R.<x> = QQ[] sage: L = R^2 sage: L.coordinate_ring() Univariate Polynomial Ring in x over Rational Field sage: L.span([(x,0), (1,x)]).coordinate_ring() Univariate Polynomial Ring in x over Rational Field sage: L.span([(x,0), (1,1/x)]).coordinate_ring() Fraction Field of Univariate Polynomial Ring in x over Rational Field sage: L.span([]).coordinate_ring() Univariate Polynomial Ring in x over Rational Field
- coordinate_vector(v, check=True)#
Return the vector whose coefficients give \(v\) as a linear combination of the basis for self.
INPUT:
v
– vectorcheck
– boolean (default:True
); ifTrue
, also verify that\(v\) is really in
self
.
OUTPUT: list
EXAMPLES:
sage: M = FreeModule(ZZ, 2); M0,M1=M.gens() sage: W = M.submodule([M0 + M1, M0 - 2*M1]) sage: W.coordinate_vector(2*M0 - M1) (2, -1)
- coordinates(v, check=True)#
Write \(v\) in terms of the basis for self.
INPUT:
v
– vectorcheck
– boolean (default:True
); ifTrue
, also verify that\(v\) is really in
self
.
OUTPUT: list
Returns a list \(c\) such that if \(B\) is the basis for self, then
\[\sum c_i B_i = v.\]If \(v\) is not in self, raise an
ArithmeticError
exception.EXAMPLES:
sage: M = FreeModule(ZZ, 2); M0,M1=M.gens() sage: W = M.submodule([M0 + M1, M0 - 2*M1]) sage: W.coordinates(2*M0-M1) [2, -1]
- dense_module()#
Return corresponding dense module.
EXAMPLES:
We first illustrate conversion with ambient spaces:
sage: M = FreeModule(QQ,3) sage: S = FreeModule(QQ,3, sparse=True) sage: M.sparse_module() Sparse vector space of dimension 3 over Rational Field sage: S.dense_module() Vector space of dimension 3 over Rational Field sage: M.sparse_module() == S True sage: S.dense_module() == M True sage: M.dense_module() == M True sage: S.sparse_module() == S True
Next we create a subspace:
sage: M = FreeModule(QQ,3, sparse=True) sage: V = M.span([ [1,2,3] ] ); V Sparse vector space of degree 3 and dimension 1 over Rational Field Basis matrix: [1 2 3] sage: V.sparse_module() Sparse vector space of degree 3 and dimension 1 over Rational Field Basis matrix: [1 2 3]
- dimension()#
Return the dimension of this free module.
EXAMPLES:
sage: M = FreeModule(FiniteField(19), 100) sage: W = M.submodule([M.gen(50)]) sage: W.dimension() 1
- direct_sum(other)#
Return the direct sum of
self
andother
as a free module.EXAMPLES:
sage: V = (ZZ^3).span([[1/2,3,5], [0,1,-3]]); V Free module of degree 3 and rank 2 over Integer Ring Echelon basis matrix: [1/2 0 14] [ 0 1 -3] sage: W = (ZZ^3).span([[1/2,4,2]]); W Free module of degree 3 and rank 1 over Integer Ring Echelon basis matrix: [1/2 4 2] sage: V.direct_sum(W) Free module of degree 6 and rank 3 over Integer Ring Echelon basis matrix: [1/2 0 14 0 0 0] [ 0 1 -3 0 0 0] [ 0 0 0 1/2 4 2]
- discriminant()#
Return the discriminant of this free module.
EXAMPLES:
sage: M = FreeModule(ZZ, 3) sage: M.discriminant() 1 sage: W = M.span([[1,2,3]]) sage: W.discriminant() 14 sage: W2 = M.span([[1,2,3], [1,1,1]]) sage: W2.discriminant() 6
- echelonized_basis_matrix()#
The echelonized basis matrix (not implemented for this module).
This example works because M is an ambient module. Submodule creation should exist for generic modules.
EXAMPLES:
sage: R = IntegerModRing(12) sage: S.<x,y> = R[] sage: M = FreeModule(S,3) sage: M.echelonized_basis_matrix() [1 0 0] [0 1 0] [0 0 1]
- free_module()#
Return this free module. (This is used by the
FreeModule
functor, and simply returns self.)EXAMPLES:
sage: M = FreeModule(ZZ, 3) sage: M.free_module() Ambient free module of rank 3 over the principal ideal domain Integer Ring
- gen(i=0)#
Return the \(i\)-th generator for
self
.Here \(i\) is between 0 and rank - 1, inclusive.
INPUT:
\(i\) – an integer (default 0)
OUTPUT: \(i\)-th basis vector for
self
.EXAMPLES:
sage: n = 5 sage: V = QQ^n sage: B = [V.gen(i) for i in range(n)] sage: B [(1, 0, 0, 0, 0), (0, 1, 0, 0, 0), (0, 0, 1, 0, 0), (0, 0, 0, 1, 0), (0, 0, 0, 0, 1)] sage: V.gens() == tuple(B) True
- gens()#
Return a tuple of basis elements of
self
.EXAMPLES:
sage: FreeModule(Integers(12),3).gens() ((1, 0, 0), (0, 1, 0), (0, 0, 1))
- gram_matrix()#
Return the gram matrix associated to this free module, defined to be \(G = B*A*B.transpose()\), where A is the inner product matrix (induced from the ambient space), and B the basis matrix.
EXAMPLES:
sage: V = VectorSpace(QQ,4) sage: u = V([1/2,1/2,1/2,1/2]) sage: v = V([0,1,1,0]) sage: w = V([0,0,1,1]) sage: M = span([u,v,w], ZZ) sage: M.inner_product_matrix() == V.inner_product_matrix() True sage: L = M.submodule_with_basis([u,v,w]) sage: L.inner_product_matrix() == M.inner_product_matrix() True sage: L.gram_matrix() [1 1 1] [1 2 1] [1 1 2]
- has_user_basis()#
Return
True
if the basis of this free module is specified by the user, as opposed to being the default echelon form.EXAMPLES:
sage: V = QQ^3 sage: W = V.subspace([[2,'1/2', 1]]) sage: W.has_user_basis() False sage: W = V.subspace_with_basis([[2,'1/2',1]]) sage: W.has_user_basis() True
- hom(im_gens, codomain=None, **kwds)#
Override the hom method to handle the case of morphisms given by left-multiplication of a matrix and the codomain is not given.
EXAMPLES:
sage: W = ZZ^2; W.hom(matrix(1, [1, 2]), side="right") Free module morphism defined as left-multiplication by the matrix [1 2] Domain: Ambient free module of rank 2 over the principal ideal domain Integer Ring Codomain: Ambient free module of rank 1 over the principal ideal domain Integer Ring sage: V = QQ^2; V.hom(identity_matrix(2), side="right") Vector space morphism represented as left-multiplication by the matrix: [1 0] [0 1] Domain: Vector space of dimension 2 over Rational Field Codomain: Vector space of dimension 2 over Rational Field
- inner_product_matrix()#
Return the default identity inner product matrix associated to this module.
By definition this is the inner product matrix of the ambient space, hence may be of degree greater than the rank of the module.
TODO: Differentiate the image ring of the inner product from the base ring of the module and/or ambient space. E.g. On an integral module over ZZ the inner product pairing could naturally take values in ZZ, QQ, RR, or CC.
EXAMPLES:
sage: M = FreeModule(ZZ, 3) sage: M.inner_product_matrix() [1 0 0] [0 1 0] [0 0 1]
- is_ambient()#
Returns False since this is not an ambient free module.
EXAMPLES:
sage: M = FreeModule(ZZ, 3).span([[1,2,3]]); M Free module of degree 3 and rank 1 over Integer Ring Echelon basis matrix: [1 2 3] sage: M.is_ambient() False sage: M = (ZZ^2).span([[1,0], [0,1]]) sage: M Free module of degree 2 and rank 2 over Integer Ring Echelon basis matrix: [1 0] [0 1] sage: M.is_ambient() False sage: M == M.ambient_module() True
- is_dense()#
Return
True
if the underlying representation of this module uses dense vectors, and False otherwise.EXAMPLES:
sage: FreeModule(ZZ, 2).is_dense() True sage: FreeModule(ZZ, 2, sparse=True).is_dense() False
- is_finite()#
Returns True if the underlying set of this free module is finite.
EXAMPLES:
sage: FreeModule(ZZ, 2).is_finite() False sage: FreeModule(Integers(8), 2).is_finite() True sage: FreeModule(ZZ, 0).is_finite() True
- is_full()#
Return
True
if the rank of this module equals its degree.EXAMPLES:
sage: FreeModule(ZZ, 2).is_full() True sage: M = FreeModule(ZZ, 2).span([[1,2]]) sage: M.is_full() False
- is_submodule(other)#
Return
True
ifself
is a submodule ofother
.EXAMPLES:
sage: M = FreeModule(ZZ,3) sage: V = M.ambient_vector_space() sage: X = V.span([[1/2,1/2,0],[1/2,0,1/2]], ZZ) sage: Y = V.span([[1,1,1]], ZZ) sage: N = X + Y sage: M.is_submodule(X) False sage: M.is_submodule(Y) False sage: Y.is_submodule(M) True sage: N.is_submodule(M) False sage: M.is_submodule(N) True sage: M = FreeModule(ZZ,2) sage: M.is_submodule(M) True sage: N = M.scale(2) sage: N.is_submodule(M) True sage: M.is_submodule(N) False sage: N = M.scale(1/2) sage: N.is_submodule(M) False sage: M.is_submodule(N) True
Since
basis()
is not implemented in general, submodule testing does not work for all PID’s. However, trivial cases are already used (and useful) for coercion, e.g.:sage: QQ(1/2) * vector(ZZ['x']['y'],[1,2,3,4]) (1/2, 1, 3/2, 2) sage: vector(ZZ['x']['y'],[1,2,3,4]) * QQ(1/2) (1/2, 1, 3/2, 2)
- matrix()#
Return the basis matrix of this module, which is the matrix whose rows are a basis for this module.
EXAMPLES:
sage: M = FreeModule(ZZ, 2) sage: M.matrix() [1 0] [0 1] sage: M.submodule([M.gen(0) + M.gen(1), M.gen(0) - 2*M.gen(1)]).matrix() [1 1] [0 3]
- ngens()#
Returns the number of basis elements of this free module.
EXAMPLES:
sage: FreeModule(ZZ, 2).ngens() 2 sage: FreeModule(ZZ, 0).ngens() 0 sage: FreeModule(ZZ, 2).span([[1,1]]).ngens() 1
- nonembedded_free_module()#
Returns an ambient free module that is isomorphic to this free module.
Thus if this free module is of rank \(n\) over a ring \(R\), then this function returns \(R^n\), as an ambient free module.
EXAMPLES:
sage: FreeModule(ZZ, 2).span([[1,1]]).nonembedded_free_module() Ambient free module of rank 1 over the principal ideal domain Integer Ring
- random_element(prob=1.0, *args, **kwds)#
Returns a random element of self.
INPUT:
- –
prob
- float. Each coefficient will be set to zero with probability \(1-prob\). Otherwise coefficients will be chosen randomly from base ring (and may be zero).
- –
*args, **kwds
- passed on torandom_element()
function of base ring.
EXAMPLES:
sage: M = FreeModule(ZZ, 2).span([[1, 1]]) sage: v = M.random_element() sage: v.parent() is M True sage: v in M True
Small entries are likely:
sage: for i in [-2, -1, 0, 1, 2]: ....: while vector([i, i]) != M.random_element(): ....: pass
Large entries appear as well:
sage: while abs(M.random_element()[0]) < 100: ....: pass
Passes extra positional or keyword arguments through:
sage: all(i in range(5, 10) for i in M.random_element(1.0, 5, 10)) True
- –
- rank()#
Return the rank of this free module.
EXAMPLES:
sage: FreeModule(Integers(6), 10000000).rank() 10000000 sage: FreeModule(ZZ, 2).span([[1,1], [2,2], [3,4]]).rank() 2
- relations()#
Return the module of relations of
self
.EXAMPLES:
sage: V = GF(2)^2 sage: V.relations() == V.zero_submodule() True sage: W = V.subspace([[1, 0]]) sage: W.relations() == V.zero_submodule() True sage: Q = V / W sage: Q.relations() == W True
- scale(other)#
Return the product of this module by the number other, which is the module spanned by other times each basis vector.
EXAMPLES:
sage: M = FreeModule(ZZ, 3) sage: M.scale(2) Free module of degree 3 and rank 3 over Integer Ring Echelon basis matrix: [2 0 0] [0 2 0] [0 0 2]
sage: a = QQ('1/3') sage: M.scale(a) Free module of degree 3 and rank 3 over Integer Ring Echelon basis matrix: [1/3 0 0] [ 0 1/3 0] [ 0 0 1/3]
- sparse_module()#
Return the corresponding sparse module with the same defining data.
EXAMPLES:
We first illustrate conversion with ambient spaces:
sage: M = FreeModule(Integers(8),3) sage: S = FreeModule(Integers(8),3, sparse=True) sage: M.sparse_module() Ambient sparse free module of rank 3 over Ring of integers modulo 8 sage: S.dense_module() Ambient free module of rank 3 over Ring of integers modulo 8 sage: M.sparse_module() is S True sage: S.dense_module() is M True sage: M.dense_module() is M True sage: S.sparse_module() is S True
Next we convert a subspace:
sage: M = FreeModule(QQ,3) sage: V = M.span([ [1,2,3] ] ); V Vector space of degree 3 and dimension 1 over Rational Field Basis matrix: [1 2 3] sage: V.sparse_module() Sparse vector space of degree 3 and dimension 1 over Rational Field Basis matrix: [1 2 3]
- uses_ambient_inner_product()#
Return
True
if the inner product on this module is the one induced by the ambient inner product.EXAMPLES:
sage: M = FreeModule(ZZ, 2) sage: W = M.submodule([[1,2]]) sage: W.uses_ambient_inner_product() True sage: W.inner_product_matrix() [1 0] [0 1]
sage: W.gram_matrix() [5]
- class sage.modules.free_module.FreeModule_generic_domain(base_ring, rank, degree, sparse=False, coordinate_ring=None, category=None)#
Bases:
FreeModule_generic
Base class for free modules over an integral domain.
- class sage.modules.free_module.FreeModule_generic_field(base_field, dimension, degree, sparse=False, category=None)#
Bases:
FreeModule_generic_pid
Base class for all free modules over fields.
- complement()#
Return the complement of
self
in theambient_vector_space()
.EXAMPLES:
sage: V = QQ^3 sage: V.complement() Vector space of degree 3 and dimension 0 over Rational Field Basis matrix: [] sage: V == V.complement().complement() True sage: W = V.span([[1, 0, 1]]) sage: X = W.complement(); X Vector space of degree 3 and dimension 2 over Rational Field Basis matrix: [ 1 0 -1] [ 0 1 0] sage: X.complement() == W True sage: X + W == V True
Even though we construct a subspace of a subspace, the orthogonal complement is still done in the ambient vector space \(\QQ^3\):
sage: V = QQ^3 sage: W = V.subspace_with_basis([[1,0,1],[-1,1,0]]) sage: X = W.subspace_with_basis([[1,0,1]]) sage: X.complement() Vector space of degree 3 and dimension 2 over Rational Field Basis matrix: [ 1 0 -1] [ 0 1 0]
All these complements are only done with respect to the inner product in the usual basis. Over finite fields, this means we can get complements which are only isomorphic to a vector space decomposition complement.
sage: F2 = GF(2,x) sage: V = F2^6 sage: W = V.span([[1,1,0,0,0,0]]) sage: W Vector space of degree 6 and dimension 1 over Finite Field of size 2 Basis matrix: [1 1 0 0 0 0] sage: W.complement() Vector space of degree 6 and dimension 5 over Finite Field of size 2 Basis matrix: [1 1 0 0 0 0] [0 0 1 0 0 0] [0 0 0 1 0 0] [0 0 0 0 1 0] [0 0 0 0 0 1] sage: W.intersection(W.complement()) Vector space of degree 6 and dimension 1 over Finite Field of size 2 Basis matrix: [1 1 0 0 0 0]
- echelonized_basis_matrix()#
Return basis matrix for
self
in row echelon form.EXAMPLES:
sage: V = FreeModule(QQ, 3).span_of_basis([[1,2,3],[4,5,6]]) sage: V.basis_matrix() [1 2 3] [4 5 6] sage: V.echelonized_basis_matrix() [ 1 0 -1] [ 0 1 2]
- intersection(other)#
Return the intersection of
self
and other, which must be R-submodules of a common ambient vector space.EXAMPLES:
sage: V = VectorSpace(QQ,3) sage: W1 = V.submodule([V.gen(0), V.gen(0) + V.gen(1)]) sage: W2 = V.submodule([V.gen(1), V.gen(2)]) sage: W1.intersection(W2) Vector space of degree 3 and dimension 1 over Rational Field Basis matrix: [0 1 0] sage: W2.intersection(W1) Vector space of degree 3 and dimension 1 over Rational Field Basis matrix: [0 1 0] sage: V.intersection(W1) Vector space of degree 3 and dimension 2 over Rational Field Basis matrix: [1 0 0] [0 1 0] sage: W1.intersection(V) Vector space of degree 3 and dimension 2 over Rational Field Basis matrix: [1 0 0] [0 1 0] sage: Z = V.submodule([]) sage: W1.intersection(Z) Vector space of degree 3 and dimension 0 over Rational Field Basis matrix: []
- is_subspace(other)#
True if this vector space is a subspace of other.
EXAMPLES:
sage: V = VectorSpace(QQ,3) sage: W = V.subspace([V.gen(0), V.gen(0) + V.gen(1)]) sage: W2 = V.subspace([V.gen(1)]) sage: W.is_subspace(V) True sage: W2.is_subspace(V) True sage: W.is_subspace(W2) False sage: W2.is_subspace(W) True
- linear_dependence(vectors, zeros='left', check=True)#
Returns a list of vectors giving relations of linear dependence for the input list of vectors. Can be used to check linear independence of a set of vectors.
INPUT:
vectors
– A list of vectors, all from the same vector space.zeros
– default:'left'
-'left'
or'right'
as a general preference for where zeros are located in the returned coefficientscheck
– default:True
- ifTrue
each item in the listvectors
is checked for membership inself
. Set toFalse
if you can be certain the vectors come from the vector space.
OUTPUT:
Returns a list of vectors. The scalar entries of each vector provide the coefficients for a linear combination of the input vectors that will equal the zero vector in
self
. Furthermore, the returned list is linearly independent in the vector space over the same base field with degree equal to the length of the listvectors
.The linear independence of
vectors
is equivalent to the returned list being empty, so this provides a test - see the examples below.The returned vectors are always independent, and with
zeros
set to'left'
they have 1’s in their first non-zero entries and a qualitative disposition to having zeros in the low-index entries. Withzeros
set to'right'
the situation is reversed with a qualitative disposition for zeros in the high-index entries.If the vectors in
vectors
are made the rows of a matrix \(V\) and the returned vectors are made the rows of a matrix \(R\), then the matrix product \(RV\) is a zero matrix of the proper size. And \(R\) is a matrix of full rank. This routine uses kernels of matrices to compute these relations of linear dependence, but handles all the conversions between sets of vectors and matrices. If speed is important, consider working with the appropriate matrices and kernels instead.EXAMPLES:
We begin with two linearly independent vectors, and add three non-trivial linear combinations to the set. We illustrate both types of output and check a selected relation of linear dependence.
sage: v1 = vector(QQ, [2, 1, -4, 3]) sage: v2 = vector(QQ, [1, 5, 2, -2]) sage: V = QQ^4 sage: V.linear_dependence([v1,v2]) [ ] sage: v3 = v1 + v2 sage: v4 = 3*v1 - 4*v2 sage: v5 = -v1 + 2*v2 sage: L = [v1, v2, v3, v4, v5] sage: relations = V.linear_dependence(L, zeros='left') sage: relations [ (1, 0, 0, -1, -2), (0, 1, 0, -1/2, -3/2), (0, 0, 1, -3/2, -7/2) ] sage: v2 + (-1/2)*v4 + (-3/2)*v5 (0, 0, 0, 0) sage: relations = V.linear_dependence(L, zeros='right') sage: relations [ (-1, -1, 1, 0, 0), (-3, 4, 0, 1, 0), (1, -2, 0, 0, 1) ] sage: z = sum([relations[2][i]*L[i] for i in range(len(L))]) sage: z == zero_vector(QQ, 4) True
A linearly independent set returns an empty list, a result that can be tested.
sage: v1 = vector(QQ, [0,1,-3]) sage: v2 = vector(QQ, [4,1,0]) sage: V = QQ^3 sage: relations = V.linear_dependence([v1, v2]); relations [ ] sage: relations == [] True
Exact results result from exact fields. We start with three linearly independent vectors and add in two linear combinations to make a linearly dependent set of five vectors.
sage: F = FiniteField(17) sage: v1 = vector(F, [1, 2, 3, 4, 5]) sage: v2 = vector(F, [2, 4, 8, 16, 15]) sage: v3 = vector(F, [1, 0, 0, 0, 1]) sage: (F^5).linear_dependence([v1, v2, v3]) == [] True sage: L = [v1, v2, v3, 2*v1+v2, 3*v2+6*v3] sage: (F^5).linear_dependence(L) [ (1, 0, 16, 8, 3), (0, 1, 2, 0, 11) ] sage: v1 + 16*v3 + 8*(2*v1+v2) + 3*(3*v2+6*v3) (0, 0, 0, 0, 0) sage: v2 + 2*v3 + 11*(3*v2+6*v3) (0, 0, 0, 0, 0) sage: (F^5).linear_dependence(L, zeros='right') [ (15, 16, 0, 1, 0), (0, 14, 11, 0, 1) ]
- quotient_abstract(sub, check=True, **kwds)#
Return an ambient free module isomorphic to the quotient space of
self
modulosub
, together with maps fromself
to the quotient, and a lifting map in the other direction.Use
self.quotient(sub)
to obtain the quotient module as an object equipped with natural maps in both directions, and a canonical coercion.INPUT:
sub
– a submodule ofself
or something that can be turned into one viaself.submodule(sub)
check
– (default:True
) whether or not to check that sub is a submodulefurther named arguments, that are currently ignored.
OUTPUT:
U
– the quotient as an abstract ambient free modulepi
– projection map to the quotientlift
– lifting map back from quotient
EXAMPLES:
sage: V = GF(19)^3 sage: W = V.span_of_basis([ [1,2,3], [1,0,1] ]) sage: U,pi,lift = V.quotient_abstract(W) sage: pi(V.2) (18) sage: pi(V.0) (1) sage: pi(V.0 + V.2) (0)
Another example involving a quotient of one subspace by another:
sage: A = matrix(QQ,4,4,[0,1,0,0, 0,0,1,0, 0,0,0,1, 0,0,0,0]) sage: V = (A^3).kernel() sage: W = A.kernel() sage: U, pi, lift = V.quotient_abstract(W) sage: [pi(v) == 0 for v in W.gens()] [True] sage: [pi(lift(b)) == b for b in U.basis()] [True, True]
- quotient_module(sub, check=True)#
Return the quotient of
self
by the given subspace sub.INPUT:
sub
– a submodule ofself
, or something that can be turned into one viaself.submodule(sub)
check
– (default:True
) whether or not to check thatsub
is a submodule
EXAMPLES:
sage: A = QQ^3; V = A.span([[1,2,3], [4,5,6]]) sage: Q = V.quotient( [V.0 + V.1] ); Q Vector space quotient V/W of dimension 1 over Rational Field where V: Vector space of degree 3 and dimension 2 over Rational Field Basis matrix: [ 1 0 -1] [ 0 1 2] W: Vector space of degree 3 and dimension 1 over Rational Field Basis matrix: [1 1 1] sage: Q(V.0 + V.1) (0)
We illustrate that the base rings must be the same:
sage: (QQ^2)/(ZZ^2) Traceback (most recent call last): ... ValueError: base rings must be the same
- scale(other)#
Return the product of
self
by the number other, which is the module spanned byother
times each basis vector. Sinceself
is a vector space this product equalsself
ifother
is nonzero, and is the zero vector space ifother
is 0.EXAMPLES:
sage: V = QQ^4 sage: V.scale(5) Vector space of dimension 4 over Rational Field sage: V.scale(0) Vector space of degree 4 and dimension 0 over Rational Field Basis matrix: []
sage: W = V.span([[1,1,1,1]]) sage: W.scale(2) Vector space of degree 4 and dimension 1 over Rational Field Basis matrix: [1 1 1 1] sage: W.scale(0) Vector space of degree 4 and dimension 0 over Rational Field Basis matrix: []
sage: V = QQ^4; V Vector space of dimension 4 over Rational Field sage: V.scale(3) Vector space of dimension 4 over Rational Field sage: V.scale(0) Vector space of degree 4 and dimension 0 over Rational Field Basis matrix: []
- span_of_basis(basis, base_ring=None, check=True, already_echelonized=False)#
Return the free K-module with the given basis, where K is the base field of
self
or user specified base_ring.Note that this span is a subspace of the ambient vector space, but need not be a subspace of self.
INPUT:
basis
- list of vectorscheck
- boolean (default:True
): whether or not to coerce entries of gens into base fieldalready_echelonized
- boolean (default:False
): set this if you know the gens are already in echelon form
EXAMPLES:
sage: V = VectorSpace(GF(7), 3) sage: W = V.subspace([[2,3,4]]); W Vector space of degree 3 and dimension 1 over Finite Field of size 7 Basis matrix: [1 5 2] sage: W.span_of_basis([[2,2,2], [3,3,0]]) Vector space of degree 3 and dimension 2 over Finite Field of size 7 User basis matrix: [2 2 2] [3 3 0]
The basis vectors must be linearly independent or a
ValueError
exception is raised:sage: W.span_of_basis([[2,2,2], [3,3,3]]) Traceback (most recent call last): ... ValueError: The given basis vectors must be linearly independent.
- subspace(gens, check=True, already_echelonized=False)#
Return the subspace of
self
spanned by the elements of gens.INPUT:
gens
- list of vectorscheck
- boolean (default:True
) verify that gens are all inself
.already_echelonized
- boolean (default:False
) set to True if you know the gens are in Echelon form.
EXAMPLES:
First we create a 1-dimensional vector subspace of an ambient \(3\)-dimensional space over the finite field of order \(7\):
sage: V = VectorSpace(GF(7), 3) sage: W = V.subspace([[2,3,4]]); W Vector space of degree 3 and dimension 1 over Finite Field of size 7 Basis matrix: [1 5 2]
Next we create an invalid subspace, but it’s allowed since
check=False
. This is just equivalent to computing the span of the element:sage: W.subspace([[1,1,0]], check=False) Vector space of degree 3 and dimension 1 over Finite Field of size 7 Basis matrix: [1 1 0]
With
check=True
(the default) the mistake is correctly detected and reported with anArithmeticError
exception:sage: W.subspace([[1,1,0]], check=True) Traceback (most recent call last): ... ArithmeticError: argument gens (= [[1, 1, 0]]) does not generate a submodule of self
- subspace_with_basis(gens, check=True, already_echelonized=False)#
Same as
self.submodule_with_basis(...)
.EXAMPLES:
We create a subspace with a user-defined basis.
sage: V = VectorSpace(GF(7), 3) sage: W = V.subspace_with_basis([[2,2,2], [1,2,3]]); W Vector space of degree 3 and dimension 2 over Finite Field of size 7 User basis matrix: [2 2 2] [1 2 3]
We then create a subspace of the subspace with user-defined basis.
sage: W1 = W.subspace_with_basis([[3,4,5]]); W1 Vector space of degree 3 and dimension 1 over Finite Field of size 7 User basis matrix: [3 4 5]
Notice how the basis for the same subspace is different if we merely use the
subspace
command.sage: W2 = W.subspace([[3,4,5]]); W2 Vector space of degree 3 and dimension 1 over Finite Field of size 7 Basis matrix: [1 6 4]
Nonetheless the two subspaces are equal (as mathematical objects):
sage: W1 == W2 True
- subspaces(dim)#
Iterate over all subspaces of dimension dim.
INPUT:
dim
- int, dimension of subspaces to be generated
EXAMPLES:
sage: V = VectorSpace(GF(3), 5) sage: len(list(V.subspaces(0))) 1 sage: len(list(V.subspaces(1))) 121 sage: len(list(V.subspaces(2))) 1210 sage: len(list(V.subspaces(3))) 1210 sage: len(list(V.subspaces(4))) 121 sage: len(list(V.subspaces(5))) 1
sage: V = VectorSpace(GF(3), 5) sage: V = V.subspace([V([1,1,0,0,0]),V([0,0,1,1,0])]) sage: list(V.subspaces(1)) [Vector space of degree 5 and dimension 1 over Finite Field of size 3 Basis matrix: [1 1 0 0 0], Vector space of degree 5 and dimension 1 over Finite Field of size 3 Basis matrix: [1 1 1 1 0], Vector space of degree 5 and dimension 1 over Finite Field of size 3 Basis matrix: [1 1 2 2 0], Vector space of degree 5 and dimension 1 over Finite Field of size 3 Basis matrix: [0 0 1 1 0]]
- vector_space(base_field=None)#
Return the vector space associated to self. Since
self
is a vector space this function simply returns self, unless the base field is different.EXAMPLES:
sage: V = span([[1,2,3]],QQ); V Vector space of degree 3 and dimension 1 over Rational Field Basis matrix: [1 2 3] sage: V.vector_space() Vector space of degree 3 and dimension 1 over Rational Field Basis matrix: [1 2 3]
- zero_submodule()#
Return the zero submodule of self.
EXAMPLES:
sage: (QQ^4).zero_submodule() Vector space of degree 4 and dimension 0 over Rational Field Basis matrix: []
- zero_subspace()#
Return the zero subspace of self.
EXAMPLES:
sage: (QQ^4).zero_subspace() Vector space of degree 4 and dimension 0 over Rational Field Basis matrix: []
- class sage.modules.free_module.FreeModule_generic_pid(base_ring, rank, degree, sparse=False, coordinate_ring=None, category=None)#
Bases:
FreeModule_generic_domain
Base class for all free modules over a PID.
- denominator()#
The denominator of the basis matrix of
self
(i.e. the LCM of the coordinate entries with respect to the basis of the ambient space).EXAMPLES:
sage: V = QQ^3 sage: L = V.span([[1,1/2,1/3], [-1/5,2/3,3]],ZZ) sage: L Free module of degree 3 and rank 2 over Integer Ring Echelon basis matrix: [ 1/5 19/6 37/3] [ 0 23/6 46/3] sage: L.denominator() 30
- index_in(other)#
Return the lattice index [other:self] of
self
in other, as an element of the base field. Whenself
is contained in other, the lattice index is the usual index. If the index is infinite, then this function returns infinity.EXAMPLES:
sage: L1 = span([[1,2]], ZZ) sage: L2 = span([[3,6]], ZZ) sage: L2.index_in(L1) 3
Note that the free modules being compared need not be integral.
sage: L1 = span([['1/2','1/3'], [4,5]], ZZ) sage: L2 = span([[1,2], [3,4]], ZZ) sage: L2.index_in(L1) 12/7 sage: L1.index_in(L2) 7/12 sage: L1.discriminant() / L2.discriminant() 49/144
The index of a lattice of infinite index is infinite.
sage: L1 = FreeModule(ZZ, 2) sage: L2 = span([[1,2]], ZZ) sage: L2.index_in(L1) +Infinity
- index_in_saturation()#
Return the index of this module in its saturation, i.e., its intersection with \(R^n\).
EXAMPLES:
sage: W = span([[2,4,6]], ZZ) sage: W.index_in_saturation() 2 sage: W = span([[1/2,1/3]], ZZ) sage: W.index_in_saturation() 1/6
- intersection(other)#
Return the intersection of
self
andother
.EXAMPLES:
We intersect two submodules one of which is clearly contained in the other:
sage: A = ZZ^2 sage: M1 = A.span([[1,1]]) sage: M2 = A.span([[3,3]]) sage: M1.intersection(M2) Free module of degree 2 and rank 1 over Integer Ring Echelon basis matrix: [3 3] sage: M1.intersection(M2) is M2 True
We intersection two submodules of \(\ZZ^3\) of rank \(2\), whose intersection has rank \(1\):
sage: A = ZZ^3 sage: M1 = A.span([[1,1,1], [1,2,3]]) sage: M2 = A.span([[2,2,2], [1,0,0]]) sage: M1.intersection(M2) Free module of degree 3 and rank 1 over Integer Ring Echelon basis matrix: [2 2 2]
We compute an intersection of two \(\ZZ\)-modules that are not submodules of \(\ZZ^2\):
sage: A = ZZ^2 sage: M1 = A.span([[1,2]]).scale(1/6) sage: M2 = A.span([[1,2]]).scale(1/15) sage: M1.intersection(M2) Free module of degree 2 and rank 1 over Integer Ring Echelon basis matrix: [1/3 2/3]
We intersect a \(\ZZ\)-module with a \(\QQ\)-vector space:
sage: A = ZZ^3 sage: L = ZZ^3 sage: V = QQ^3 sage: W = L.span([[1/2,0,1/2]]) sage: K = V.span([[1,0,1], [0,0,1]]) sage: W.intersection(K) Free module of degree 3 and rank 1 over Integer Ring Echelon basis matrix: [1/2 0 1/2] sage: K.intersection(W) Free module of degree 3 and rank 1 over Integer Ring Echelon basis matrix: [1/2 0 1/2]
We intersect two modules over the ring of integers of a number field:
sage: L.<w> = NumberField(x^2 - x + 2) sage: OL = L.ring_of_integers() sage: V = L**3 sage: W1 = V.span([[0,w/5,0], [1,0,-1/17]], OL) sage: W2 = V.span([[0,(1-w)/5,0]], OL) sage: W1.intersection(W2) Free module of degree 3 and rank 1 over Maximal Order in Number Field in w with defining polynomial x^2 - x + 2 Echelon basis matrix: [ 0 2/5 0]
- quotient_module(sub, check=True, **kwds)#
Return the quotient of
self
by the given submodule sub.INPUT:
sub
– a submodule ofself
, or something that can be turned into one viaself.submodule(sub)
check
– (default:True
) whether or not to check thatsub
is a submodulefurther named arguments, that are passed to the constructor of the quotient space
EXAMPLES:
sage: A = ZZ^3; V = A.span([[1,2,3], [4,5,6]]) sage: Q = V.quotient( [V.0 + V.1] ); Q Finitely generated module V/W over Integer Ring with invariants (0)
- saturation()#
Return the saturated submodule of \(R^n\) that spans the same vector space as self.
EXAMPLES:
We create a 1-dimensional lattice that is obviously not saturated and saturate it.
sage: L = span([[9,9,6]], ZZ); L Free module of degree 3 and rank 1 over Integer Ring Echelon basis matrix: [9 9 6] sage: L.saturation() Free module of degree 3 and rank 1 over Integer Ring Echelon basis matrix: [3 3 2]
We create a lattice spanned by two vectors, and saturate. Computation of discriminants shows that the index of lattice in its saturation is \(3\), which is a prime of congruence between the two generating vectors.
sage: L = span([[1,2,3], [4,5,6]], ZZ) sage: L.saturation() Free module of degree 3 and rank 2 over Integer Ring Echelon basis matrix: [ 1 0 -1] [ 0 1 2] sage: L.discriminant() 54 sage: L.saturation().discriminant() 6
Notice that the saturation of a non-integral lattice \(L\) is defined, but the result is integral hence does not contain \(L\):
sage: L = span([['1/2',1,3]], ZZ) sage: L.saturation() Free module of degree 3 and rank 1 over Integer Ring Echelon basis matrix: [1 2 6]
- span_of_basis(basis, base_ring=None, check=True, already_echelonized=False)#
Return the free R-module with the given basis, where R is the base ring of
self
or user specified base_ring.Note that this R-module need not be a submodule of self, nor even of the ambient space. It must, however, be contained in the ambient vector space, i.e., the ambient space tensored with the fraction field of R.
EXAMPLES:
sage: M = FreeModule(ZZ,3) sage: W = M.span_of_basis([M([1,2,3])])
Next we create two free \(\ZZ\)-modules, neither of which is a submodule of \(W\).
sage: W.span_of_basis([M([2,4,0])]) Free module of degree 3 and rank 1 over Integer Ring User basis matrix: [2 4 0]
The following module isn’t in the ambient module \(\ZZ^3\) but is contained in the ambient vector space \(\QQ^3\):
sage: V = M.ambient_vector_space() sage: W.span_of_basis([ V([1/5,2/5,0]), V([1/7,1/7,0]) ]) Free module of degree 3 and rank 2 over Integer Ring User basis matrix: [1/5 2/5 0] [1/7 1/7 0]
Of course the input basis vectors must be linearly independent:
sage: W.span_of_basis([ [1,2,0], [2,4,0] ]) Traceback (most recent call last): ... ValueError: The given basis vectors must be linearly independent.
- submodule_with_basis(basis, check=True, already_echelonized=False)#
Create the R-submodule of the ambient vector space with given basis, where R is the base ring of self.
INPUT:
basis
– a list of linearly independent vectorscheck
– whether or not to verify that each gen is inthe ambient vector space
OUTPUT:
FreeModule
– the \(R\)-submodule with given basis
EXAMPLES:
First we create a submodule of \(\\ZZ^3\):
sage: M = FreeModule(ZZ, 3) sage: B = M.basis() sage: N = M.submodule_with_basis([B[0]+B[1], 2*B[1]-B[2]]) sage: N Free module of degree 3 and rank 2 over Integer Ring User basis matrix: [ 1 1 0] [ 0 2 -1]
A list of vectors in the ambient vector space may fail to generate a submodule.
sage: V = M.ambient_vector_space() sage: X = M.submodule_with_basis([ V(B[0]+B[1])/2, V(B[1]-B[2])/2]) Traceback (most recent call last): ... ArithmeticError: The given basis does not generate a submodule of self.
However, we can still determine the R-span of vectors in the ambient space, or over-ride the submodule check by setting check to False.
sage: X = V.span([ V(B[0]+B[1])/2, V(B[1]-B[2])/2 ], ZZ) sage: X Free module of degree 3 and rank 2 over Integer Ring Echelon basis matrix: [ 1/2 0 1/2] [ 0 1/2 -1/2] sage: Y = M.submodule([ V(B[0]+B[1])/2, V(B[1]-B[2])/2 ], check=False) sage: X == Y True
Next we try to create a submodule of a free module over the principal ideal domain \(\QQ[x]\), using our general Hermite normal form implementation:
sage: R = PolynomialRing(QQ, 'x'); x = R.gen() sage: M = FreeModule(R, 3) sage: B = M.basis() sage: W = M.submodule_with_basis([x*B[0], 2*B[0]- x*B[2]]); W Free module of degree 3 and rank 2 over Univariate Polynomial Ring in x over Rational Field User basis matrix: [ x 0 0] [ 2 0 -x]
- vector_space_span(gens, check=True)#
Create the vector subspace of the ambient vector space with given generators.
INPUT:
gens
- a list of vector in selfcheck
- whether or not to verify that each gen is in the ambient vector space
OUTPUT: a vector subspace
EXAMPLES:
We create a \(2\)-dimensional subspace of \(\QQ^3\).
sage: V = VectorSpace(QQ, 3) sage: B = V.basis() sage: W = V.vector_space_span([B[0]+B[1], 2*B[1]-B[2]]) sage: W Vector space of degree 3 and dimension 2 over Rational Field Basis matrix: [ 1 0 1/2] [ 0 1 -1/2]
We create a subspace of a vector space over \(\QQ(i)\).
sage: R.<x> = QQ[] sage: K = NumberField(x^2 + 1, 'a'); a = K.gen() sage: V = VectorSpace(K, 3) sage: V.vector_space_span([2*V.gen(0) + 3*V.gen(2)]) Vector space of degree 3 and dimension 1 over Number Field in a with defining polynomial x^2 + 1 Basis matrix: [ 1 0 3/2]
We use the
vector_space_span
command to create a vector subspace of the ambient vector space of a submodule of \(\ZZ^3\).sage: M = FreeModule(ZZ,3) sage: W = M.submodule([M([1,2,3])]) sage: W.vector_space_span([M([2,3,4])]) Vector space of degree 3 and dimension 1 over Rational Field Basis matrix: [ 1 3/2 2]
- vector_space_span_of_basis(basis, check=True)#
Create the vector subspace of the ambient vector space with given basis.
INPUT:
basis
– a list of linearly independent vectorscheck
– whether or not to verify that each gen is inthe ambient vector space
OUTPUT: a vector subspace with user-specified basis
EXAMPLES:
sage: V = VectorSpace(QQ, 3) sage: B = V.basis() sage: W = V.vector_space_span_of_basis([B[0]+B[1], 2*B[1]-B[2]]) sage: W Vector space of degree 3 and dimension 2 over Rational Field User basis matrix: [ 1 1 0] [ 0 2 -1]
- zero_submodule()#
Return the zero submodule of this module.
EXAMPLES:
sage: V = FreeModule(ZZ,2) sage: V.zero_submodule() Free module of degree 2 and rank 0 over Integer Ring Echelon basis matrix: []
- class sage.modules.free_module.FreeModule_submodule_field(ambient, gens, check=True, already_echelonized=False, category=None)#
Bases:
FreeModule_submodule_with_basis_field
An embedded vector subspace with echelonized basis.
EXAMPLES:
Since this is an embedded vector subspace with echelonized basis, the echelon_coordinates() and user coordinates() agree:
sage: V = QQ^3 sage: W = V.span([[1,2,3],[4,5,6]]) sage: W Vector space of degree 3 and dimension 2 over Rational Field Basis matrix: [ 1 0 -1] [ 0 1 2]
sage: v = V([1,5,9]) sage: W.echelon_coordinates(v) [1, 5] sage: vector(QQ, W.echelon_coordinates(v)) * W.basis_matrix() (1, 5, 9) sage: v = V([1,5,9]) sage: W.coordinates(v) [1, 5] sage: vector(QQ, W.coordinates(v)) * W.basis_matrix() (1, 5, 9)
- coordinate_vector(v, check=True)#
Write \(v\) in terms of the user basis for
self
.INPUT:
v
– vectorcheck
– boolean (default:True
); ifTrue
, also verify that\(v\) is really in
self
OUTPUT: list
The output is a list \(c\) such that if \(B\) is the basis for
self
, then\[\sum c_i B_i = v.\]If \(v\) is not in
self
, raise anArithmeticError
exception.EXAMPLES:
sage: V = QQ^3 sage: W = V.span([[1,2,3],[4,5,6]]); W Vector space of degree 3 and dimension 2 over Rational Field Basis matrix: [ 1 0 -1] [ 0 1 2] sage: v = V([1,5,9]) sage: W.coordinate_vector(v) (1, 5) sage: W.coordinates(v) [1, 5] sage: vector(QQ, W.coordinates(v)) * W.basis_matrix() (1, 5, 9)
sage: V = VectorSpace(QQ,5, sparse=True) sage: W = V.subspace([[0,1,2,0,0], [0,-1,0,0,-1/2]]) sage: W.coordinate_vector([0,0,2,0,-1/2]) (0, 2)
- echelon_coordinates(v, check=True)#
Write \(v\) in terms of the echelonized basis of
self
.INPUT:
v
– vectorcheck
– boolean (default:True
); ifTrue
, also verify that \(v\) is really inself
OUTPUT: list
The output is a list \(c\) such that if \(B\) is the basis for
self
, then\[\sum c_i B_i = v.\]If \(v\) is not in
self
, raise anArithmeticError
exception.EXAMPLES:
sage: V = QQ^3 sage: W = V.span([[1,2,3],[4,5,6]]) sage: W Vector space of degree 3 and dimension 2 over Rational Field Basis matrix: [ 1 0 -1] [ 0 1 2]
sage: v = V([1,5,9]) sage: W.echelon_coordinates(v) [1, 5] sage: vector(QQ, W.echelon_coordinates(v)) * W.basis_matrix() (1, 5, 9)
- has_user_basis()#
Return
True
if the basis of this free module is specified by the user, as opposed to being the default echelon form.EXAMPLES:
sage: V = QQ^3 sage: W = V.subspace([[2,'1/2', 1]]) sage: W.has_user_basis() False sage: W = V.subspace_with_basis([[2,'1/2',1]]) sage: W.has_user_basis() True
- class sage.modules.free_module.FreeModule_submodule_pid(ambient, gens, check=True, already_echelonized=False, category=None)#
Bases:
FreeModule_submodule_with_basis_pid
An \(R\)-submodule of \(K^n\) where \(K\) is the fraction field of a principal ideal domain \(R\).
EXAMPLES:
sage: M = ZZ^3 sage: W = M.span_of_basis([[1,2,3],[4,5,19]]); W Free module of degree 3 and rank 2 over Integer Ring User basis matrix: [ 1 2 3] [ 4 5 19]
Generic tests, including saving and loading submodules and elements:
sage: TestSuite(W).run() sage: v = W.0 + W.1 sage: TestSuite(v).run()
- coordinate_vector(v, check=True)#
Write \(v\) in terms of the user basis for
self
.INPUT:
v
– vectorcheck
– boolean (default:True
); ifTrue
, also verify that\(v\) is really in
self
.
OUTPUT: list
The output is a list \(c\) such that if \(B\) is the basis for
self
, then\[\sum c_i B_i = v.\]If \(v\) is not in self, raise an
ArithmeticError
exception.EXAMPLES:
sage: V = ZZ^3 sage: W = V.span_of_basis([[1,2,3],[4,5,6]]) sage: W.coordinate_vector([1,5,9]) (5, -1)
- has_user_basis()#
Return
True
if the basis of this free module is specified by the user, as opposed to being the default echelon form.EXAMPLES:
sage: A = ZZ^3; A Ambient free module of rank 3 over the principal ideal domain Integer Ring sage: A.has_user_basis() False sage: W = A.span_of_basis([[2,'1/2',1]]) sage: W.has_user_basis() True sage: W = A.span([[2,'1/2',1]]) sage: W.has_user_basis() False
- class sage.modules.free_module.FreeModule_submodule_with_basis_field(ambient, basis, check=True, echelonize=False, echelonized_basis=None, already_echelonized=False, category=None)#
Bases:
FreeModule_generic_field
,FreeModule_submodule_with_basis_pid
An embedded vector subspace with a distinguished user basis.
EXAMPLES:
sage: M = QQ^3; W = M.submodule_with_basis([[1,2,3], [4,5,19]]); W Vector space of degree 3 and dimension 2 over Rational Field User basis matrix: [ 1 2 3] [ 4 5 19]
Since this is an embedded vector subspace with a distinguished user basis possibly different than the echelonized basis, the echelon_coordinates() and user coordinates() do not agree:
sage: V = QQ^3
sage: W = V.submodule_with_basis([[1,2,3], [4,5,6]]) sage: W Vector space of degree 3 and dimension 2 over Rational Field User basis matrix: [1 2 3] [4 5 6]
sage: v = V([1,5,9]) sage: W.echelon_coordinates(v) [1, 5] sage: vector(QQ, W.echelon_coordinates(v)) * W.echelonized_basis_matrix() (1, 5, 9)
sage: v = V([1,5,9]) sage: W.coordinates(v) [5, -1] sage: vector(QQ, W.coordinates(v)) * W.basis_matrix() (1, 5, 9)
Generic tests, including saving and loading submodules and elements:
sage: TestSuite(W).run() sage: K.<x> = FractionField(PolynomialRing(QQ,'x')) sage: M = K^3; W = M.span_of_basis([[1,1,x]]) sage: TestSuite(W).run()
- is_ambient()#
Return False since this is not an ambient module.
EXAMPLES:
sage: V = QQ^3 sage: V.is_ambient() True sage: W = V.span_of_basis([[1,2,3],[4,5,6]]) sage: W.is_ambient() False
- class sage.modules.free_module.FreeModule_submodule_with_basis_pid(ambient, basis, check=True, echelonize=False, echelonized_basis=None, already_echelonized=False, category=None)#
Bases:
FreeModule_generic_pid
Construct a submodule of a free module over PID with a distinguished basis.
INPUT:
ambient
– ambient free module over a principal ideal domain \(R\), i.e. \(R^n\);basis
– list of elements of \(K^n\), where \(K\) is the fraction field of \(R\). These elements must be linearly independent and will be used as the default basis of the constructed submodule;check
– (default:True
) ifFalse
, correctness of the input will not be checked and type conversion may be omitted, use with care;echelonize
– (default:False
) ifTrue
,basis
will be echelonized and the result will be used as the default basis of the constructed submodule;`` echelonized_basis`` – (default:
None
) if notNone
, must be the echelonized basis spanning the same submodule asbasis
;already_echelonized
– (default:False
) ifTrue
,basis
must be already given in the echelonized form.
OUTPUT:
\(R\)-submodule of \(K^n\) with the user-specified
basis
.
EXAMPLES:
sage: M = ZZ^3 sage: W = M.span_of_basis([[1,2,3],[4,5,6]]); W Free module of degree 3 and rank 2 over Integer Ring User basis matrix: [1 2 3] [4 5 6]
Now we create a submodule of the ambient vector space, rather than
M
itself:sage: W = M.span_of_basis([[1,2,3/2],[4,5,6]]); W Free module of degree 3 and rank 2 over Integer Ring User basis matrix: [ 1 2 3/2] [ 4 5 6]
- ambient()#
Return the ambient module or space for
self
.EXAMPLES:
sage: M = ZZ^3 sage: W = M.span_of_basis([[1,2,3],[4,5,6]]); W Free module of degree 3 and rank 2 over Integer Ring User basis matrix: [1 2 3] [4 5 6] sage: W.ambient() Ambient free module of rank 3 over the principal ideal domain Integer Ring
Now we create a submodule of the ambient vector space, rather than
M
itself:sage: W = M.span_of_basis([[1,2,3/2],[4,5,6]]); W Free module of degree 3 and rank 2 over Integer Ring User basis matrix: [ 1 2 3/2] [ 4 5 6] sage: W.ambient() Vector space of dimension 3 over Rational Field
A submodule of a submodule:
sage: M = ZZ^3 sage: W = M.span_of_basis([[1,2,3],[4,5,6]]); W Free module of degree 3 and rank 2 over Integer Ring User basis matrix: [1 2 3] [4 5 6] sage: U = W.span_of_basis([[5,7,9]]); U Free module of degree 3 and rank 1 over Integer Ring User basis matrix: [5 7 9] sage: U.ambient() Ambient free module of rank 3 over the principal ideal domain Integer Ring
- ambient_module()#
Return the ambient module related to the \(R\)-module self, which was used when creating this module, and is of the form \(R^n\). Note that
self
need not be contained in the ambient module, thoughself
will be contained in the ambient vector space.EXAMPLES:
sage: A = ZZ^3 sage: M = A.span_of_basis([[1,2,'3/7'],[4,5,6]]) sage: M Free module of degree 3 and rank 2 over Integer Ring User basis matrix: [ 1 2 3/7] [ 4 5 6] sage: M.ambient_module() Ambient free module of rank 3 over the principal ideal domain Integer Ring sage: M.is_submodule(M.ambient_module()) False
- ambient_vector_space()#
Return the ambient vector space in which this free module is embedded.
EXAMPLES:
sage: M = ZZ^3; M.ambient_vector_space() Vector space of dimension 3 over Rational Field
sage: N = M.span_of_basis([[1,2,'1/5']]) sage: N Free module of degree 3 and rank 1 over Integer Ring User basis matrix: [ 1 2 1/5] sage: M.ambient_vector_space() Vector space of dimension 3 over Rational Field sage: M.ambient_vector_space() is N.ambient_vector_space() True
If an inner product on the module is specified, then this is preserved on the ambient vector space.
sage: M = FreeModule(ZZ,4,inner_product_matrix=1) sage: V = M.ambient_vector_space() sage: V Ambient quadratic space of dimension 4 over Rational Field Inner product matrix: [1 0 0 0] [0 1 0 0] [0 0 1 0] [0 0 0 1] sage: N = M.submodule([[1,-1,0,0],[0,1,-1,0],[0,0,1,-1]]) sage: N.gram_matrix() [2 1 1] [1 2 1] [1 1 2] sage: V == N.ambient_vector_space() True
- basis()#
Return the user basis for this free module.
EXAMPLES:
sage: V = ZZ^3 sage: V.basis() [ (1, 0, 0), (0, 1, 0), (0, 0, 1) ] sage: M = V.span_of_basis([['1/8',2,1]]) sage: M.basis() [ (1/8, 2, 1) ]
- change_ring(R)#
Return the free module over \(R\) obtained by coercing each element of the basis of
self
into a vector over the fraction field of \(R\), then taking the resulting \(R\)-module.INPUT:
R
- a principal ideal domain
EXAMPLES:
sage: V = QQ^3 sage: W = V.subspace([[2, 1/2, 1]]) sage: W.change_ring(GF(7)) Vector space of degree 3 and dimension 1 over Finite Field of size 7 Basis matrix: [1 2 4]
sage: M = (ZZ^2) * (1/2) sage: N = M.change_ring(QQ) sage: N Vector space of degree 2 and dimension 2 over Rational Field Basis matrix: [1 0] [0 1] sage: N = M.change_ring(QQ['x']) sage: N Free module of degree 2 and rank 2 over Univariate Polynomial Ring in x over Rational Field Echelon basis matrix: [1/2 0] [ 0 1/2] sage: N.coordinate_ring() Univariate Polynomial Ring in x over Rational Field
The ring must be a principal ideal domain:
sage: M.change_ring(ZZ['x']) Traceback (most recent call last): ... TypeError: the new ring Univariate Polynomial Ring in x over Integer Ring should be a principal ideal domain
- construction()#
Returns the functorial construction of self, namely, the subspace of the ambient module spanned by the given basis.
EXAMPLES:
sage: M = ZZ^3 sage: W = M.span_of_basis([[1,2,3],[4,5,6]]); W Free module of degree 3 and rank 2 over Integer Ring User basis matrix: [1 2 3] [4 5 6] sage: c, V = W.construction() sage: c(V) == W True
- coordinate_vector(v, check=True)#
Write \(v\) in terms of the user basis for
self
.INPUT:
v
– vectorcheck
– boolean (default:True
); ifTrue
, also verify that\(v\) is really in
self
.
OUTPUT: list
The output is 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, raise an
ArithmeticError
exception.EXAMPLES:
sage: V = ZZ^3 sage: M = V.span_of_basis([['1/8',2,1]]) sage: M.coordinate_vector([1,16,8]) (8)
- echelon_coordinate_vector(v, check=True)#
Write \(v\) in terms of the echelonized basis for
self
.INPUT:
v
- vectorcheck
- boolean (default:True
); ifTrue
, also verify that \(v\) is really inself
.
Returns a list \(c\) such that if \(B\) is the echelonized basis for self, then
\[\sum c_i B_i = v.\]If \(v\) is not in self, raise an
ArithmeticError
exception.EXAMPLES:
sage: V = ZZ^3 sage: M = V.span_of_basis([['1/2',3,1], [0,'1/6',0]]) sage: B = M.echelonized_basis(); B [ (1/2, 0, 1), (0, 1/6, 0) ] sage: M.echelon_coordinate_vector(['1/2', 3, 1]) (1, 18)
- echelon_coordinates(v, check=True)#
Write \(v\) in terms of the echelonized basis for
self
.INPUT:
v
- vectorcheck
- boolean (default:True
); ifTrue
, also verify that \(v\) is really inself
.
OUTPUT: list
Returns a list \(c\) such that if \(B\) is the basis for self, then
\[\sum c_i B_i = v.\]If \(v\) is not in self, raise an
ArithmeticError
exception.EXAMPLES:
sage: A = ZZ^3 sage: M = A.span_of_basis([[1,2,'3/7'],[4,5,6]]) sage: M.coordinates([8,10,12]) [0, 2] sage: M.echelon_coordinates([8,10,12]) [8, -2] sage: B = M.echelonized_basis(); B [ (1, 2, 3/7), (0, 3, -30/7) ] sage: 8*B[0] - 2*B[1] (8, 10, 12)
We do an example with a sparse vector space:
sage: V = VectorSpace(QQ,5, sparse=True) sage: W = V.subspace_with_basis([[0,1,2,0,0], [0,-1,0,0,-1/2]]) sage: W.echelonized_basis() [ (0, 1, 0, 0, 1/2), (0, 0, 1, 0, -1/4) ] sage: W.echelon_coordinates([0,0,2,0,-1/2]) [0, 2]
- echelon_to_user_matrix()#
Return matrix that transforms the echelon basis to the user basis of self. This is a matrix \(A\) such that if \(v\) is a vector written with respect to the echelon basis for
self
then \(vA\) is that vector written with respect to the user basis of self.EXAMPLES:
sage: V = QQ^3 sage: W = V.span_of_basis([[1,2,3],[4,5,6]]) sage: W.echelonized_basis() [ (1, 0, -1), (0, 1, 2) ] sage: A = W.echelon_to_user_matrix(); A [-5/3 2/3] [ 4/3 -1/3]
The vector \((1,1,1)\) has coordinates \(v=(1,1)\) with respect to the echelonized basis for self. Multiplying \(vA\) we find the coordinates of this vector with respect to the user basis.
sage: v = vector(QQ, [1,1]); v (1, 1) sage: v * A (-1/3, 1/3) sage: u0, u1 = W.basis() sage: (-u0 + u1)/3 (1, 1, 1)
- echelonized_basis()#
Return the basis for
self
in echelon form.EXAMPLES:
sage: V = ZZ^3 sage: M = V.span_of_basis([['1/2',3,1], [0,'1/6',0]]) sage: M.basis() [ (1/2, 3, 1), (0, 1/6, 0) ] sage: B = M.echelonized_basis(); B [ (1/2, 0, 1), (0, 1/6, 0) ] sage: V.span(B) == M True
- echelonized_basis_matrix()#
Return basis matrix for
self
in row echelon form.EXAMPLES:
sage: V = FreeModule(ZZ, 3).span_of_basis([[1,2,3],[4,5,6]]) sage: V.basis_matrix() [1 2 3] [4 5 6] sage: V.echelonized_basis_matrix() [1 2 3] [0 3 6]
- has_user_basis()#
Return
True
if the basis of this free module is specified by the user, as opposed to being the default echelon form.EXAMPLES:
sage: V = ZZ^3; V.has_user_basis() False sage: M = V.span_of_basis([[1,3,1]]); M.has_user_basis() True sage: M = V.span([[1,3,1]]); M.has_user_basis() False
- lift()#
The lift (embedding) map from
self
to the ambient module or space.EXAMPLES:
sage: M = ZZ^3 sage: W = M.span_of_basis([[1,2,3],[4,5,6]]); W Free module of degree 3 and rank 2 over Integer Ring User basis matrix: [1 2 3] [4 5 6] sage: W.lift Generic morphism: From: Free module of degree 3 and rank 2 over Integer Ring User basis matrix: [1 2 3] [4 5 6] To: Ambient free module of rank 3 over the principal ideal domain Integer Ring sage: w = W([5,7,9]) sage: m = W.lift(w); m (5, 7, 9) sage: m.parent() Ambient free module of rank 3 over the principal ideal domain Integer Ring
- linear_combination_of_basis(v)#
Return the linear combination of the basis for
self
obtained from the coordinates of v.INPUT:
v
- list
EXAMPLES:
sage: V = span([[1,2,3], [4,5,6]], ZZ); V Free module of degree 3 and rank 2 over Integer Ring Echelon basis matrix: [1 2 3] [0 3 6] sage: V.linear_combination_of_basis([1,1]) (1, 5, 9)
This should raise an error if the resulting element is not in self:
sage: W = (QQ**2).span([[2, 0], [0, 8]], ZZ) sage: W.linear_combination_of_basis([1, -1/2]) Traceback (most recent call last): ... TypeError: element [2, -4] is not in free module
- relations()#
Return the submodule defining the relations of
self
as a subquotient (considering the ambient module as a quotient module).EXAMPLES:
sage: V = GF(2)^2 sage: W = V.subspace([[1, 0]]) sage: W.relations() == V.zero_submodule() True sage: Q = V / W sage: Q.relations() == W True sage: Q.zero_submodule().relations() == W True
- retract()#
The retract map from the ambient space.
This is a partial map, which gives an error for elements not in the subspace.
Calling this map on elements of the ambient space is the same as calling the element constructor of
self
.EXAMPLES:
sage: M = ZZ^3 sage: W = M.span_of_basis([[1,2,3],[4,5,6]]); W Free module of degree 3 and rank 2 over Integer Ring User basis matrix: [1 2 3] [4 5 6] sage: W.retract Generic morphism: From: Ambient free module of rank 3 over the principal ideal domain Integer Ring To: Free module of degree 3 and rank 2 over Integer Ring User basis matrix: [1 2 3] [4 5 6] sage: m = M([5, 7, 9]) sage: w = W.retract(m); w (5, 7, 9) sage: w.parent() Free module of degree 3 and rank 2 over Integer Ring User basis matrix: [1 2 3] [4 5 6]
- user_to_echelon_matrix()#
Return matrix that transforms a vector written with respect to the user basis of
self
to one written with respect to the echelon basis. The matrix acts from the right, as is usual in Sage.EXAMPLES:
sage: A = ZZ^3 sage: M = A.span_of_basis([[1,2,3],[4,5,6]]) sage: M.echelonized_basis() [ (1, 2, 3), (0, 3, 6) ] sage: M.user_to_echelon_matrix() [ 1 0] [ 4 -1]
The vector \(v=(5,7,9)\) in \(M\) is \((1,1)\) with respect to the user basis. Multiplying the above matrix on the right by this vector yields \((5,-1)\), which has components the coordinates of \(v\) with respect to the echelon basis.
sage: v0,v1 = M.basis(); v = v0+v1 sage: e0,e1 = M.echelonized_basis() sage: v (5, 7, 9) sage: 5*e0 + (-1)*e1 (5, 7, 9)
- vector_space(base_field=None)#
Return the vector space associated to this free module via tensor product with the fraction field of the base ring.
EXAMPLES:
sage: A = ZZ^3; A Ambient free module of rank 3 over the principal ideal domain Integer Ring sage: A.vector_space() Vector space of dimension 3 over Rational Field sage: M = A.span_of_basis([['1/3',2,'3/7'],[4,5,6]]); M Free module of degree 3 and rank 2 over Integer Ring User basis matrix: [1/3 2 3/7] [ 4 5 6] sage: M.vector_space() Vector space of degree 3 and dimension 2 over Rational Field User basis matrix: [1/3 2 3/7] [ 4 5 6]
- class sage.modules.free_module.Module_free_ambient(base_ring, degree, sparse=False, category=None)#
Bases:
Module
Base class for modules with elements represented by elements of a free module.
Modules whose elements are represented by elements of a free module (such as submodules, quotients, and subquotients of a free module) should be either a subclass of this class or
FreeModule_generic
, which itself is a subclass of this class. If the modules have bases and ranks, then useFreeModule_generic
. Otherwise, use this class.INPUT:
base_ring
– a commutative ringdegree
– a non-negative integer; degree of the ambient free modulesparse
– boolean (default:False
)category
– category (default:None
)
If
base_ring
is a field, then the default category is the category of finite-dimensional vector spaces over that field; otherwise it is the category of finite-dimensional free modules over that ring. In addition, the category is intersected with the category of finite enumerated sets if the ring is finite or the rank is 0.EXAMPLES:
sage: S.<x,y,z> = PolynomialRing(QQ) sage: M = S**2 sage: N = M.submodule([vector([x - y, z]), vector([y * z, x * z])]) sage: N.gens() [ (x - y, z), (y*z, x*z) ] sage: N.degree() 2
- coordinate_ring()#
Return the ring over which the entries of the vectors are defined.
EXAMPLES:
sage: S.<x,y,z> = PolynomialRing(QQ) sage: M = S**2 sage: N = M.submodule([vector([x - y, z]), vector([y * z, x * z])]) sage: N.coordinate_ring() Multivariate Polynomial Ring in x, y, z over Rational Field
- degree()#
Return the degree of this free module. This is the dimension of the ambient vector space in which it is embedded.
EXAMPLES:
sage: M = FreeModule(ZZ, 10) sage: W = M.submodule([M.gen(0), 2*M.gen(3) - M.gen(0), M.gen(0) + M.gen(3)]) sage: W.degree() 10 sage: W.rank() 2
- free_resolution(*args, **kwds)#
Return a free resolution of
self
.For input options, see
FreeResolution
.EXAMPLES:
sage: S.<x,y,z> = PolynomialRing(QQ) sage: M = S**2 sage: N = M.submodule([vector([x - y, z]), vector([y * z, x * z])]) sage: res = N.free_resolution() sage: res S^2 <-- S^2 <-- 0 sage: ascii_art(res.chain_complex()) [x - y y*z] [ z x*z] 0 <-- C_0 <-------------- C_1 <-- 0
- graded_free_resolution(*args, **kwds)#
Return a graded free resolution of
self
.For input options, see
GradedFiniteFreeResolution
.EXAMPLES:
sage: S.<x,y,z> = PolynomialRing(QQ) sage: M = S**2 sage: N = M.submodule([vector([x - y, z]), vector([y * z, x * z])]) sage: N.graded_free_resolution(shifts=[1, -1]) S(-1)⊕S(1) <-- S(-2)⊕S(-3) <-- 0 sage: N.graded_free_resolution(shifts=[2, 3]) S(-2)⊕S(-3) <-- S(-3)⊕S(-4) <-- 0 sage: N = M.submodule([vector([x^3 - y^6, z^2]), vector([y * z, x])]) sage: N.graded_free_resolution(degrees=[2, 1, 3], shifts=[2, 3]) S(-2)⊕S(-3) <-- S(-6)⊕S(-8) <-- 0
- is_sparse()#
Return
True
if the underlying representation of this module uses sparse vectors, andFalse
otherwise.EXAMPLES:
sage: FreeModule(ZZ, 2).is_sparse() False sage: FreeModule(ZZ, 2, sparse=True).is_sparse() True
- is_submodule(other)#
Return
True
ifself
is a submodule ofother
.EXAMPLES:
Submodule testing over general rings is not guaranteed to work in all cases. However, it will raise an error when it is unable to determine containment.
The zero module can always be tested:
sage: S.<x,y,z> = PolynomialRing(QQ) sage: M = S**2 sage: N = M.submodule([vector([x - y, z]), vector([y*z, x*z])]) sage: N.zero_submodule().is_submodule(M) True sage: N.zero_submodule().is_submodule(N) True sage: M.zero_submodule().is_submodule(N) True
It also respects which module it is constructed from:
sage: Q = M.quotient_module(N) sage: Q.zero_submodule().is_submodule(M) False sage: Q.zero_submodule().is_submodule(N) False sage: M.zero_submodule().is_submodule(Q) False sage: N.zero_submodule().is_submodule(Q) False
- quotient_module(sub, check=True)#
Return the quotient of
self
by the given subspacesub
.INPUT:
sub
– a submodule ofself
or something that can be turned into one viaself.submodule(sub)
check
– (default:True
) whether or not to check thatsub
is a submodule
EXAMPLES:
sage: S.<x,y,z> = PolynomialRing(QQ) sage: M = S**2 sage: N = M.submodule([vector([x - y, z]), vector([y * z, x * z])]) sage: M.quotient(N) Quotient module by Submodule of Ambient free module of rank 2 over the integral domain Multivariate Polynomial Ring in x, y, z over Rational Field Generated by the rows of the matrix: [x - y z] [ y*z x*z]
- relations_matrix()#
Return the matrix of relations of
self
.EXAMPLES:
sage: V = GF(2)^2 sage: V.relations_matrix() [] sage: W = V.subspace([[1, 0]]) sage: W.relations_matrix() [] sage: Q = V / W sage: Q.relations_matrix() [1 0] sage: S.<x,y,z> = PolynomialRing(QQ) sage: M = S**2 sage: M.relations_matrix() [] sage: N = M.submodule([vector([x - y, z]), vector([y*z, x*z])]) sage: Q = M.quotient_module(N) sage: Q.relations_matrix() [x - y z] [ y*z x*z]
- some_elements()#
Return some elements of this free module.
See
TestSuite
for a typical use case.OUTPUT:
An iterator.
EXAMPLES:
sage: F = FreeModule(ZZ, 2) sage: tuple(F.some_elements()) ((1, 0), (1, 1), (0, 1), (-1, 2), (-2, 3), ... (-49, 50)) sage: F = FreeModule(QQ, 3) sage: tuple(F.some_elements()) ((1, 0, 0), (1/2, 1/2, 1/2), (1/2, -1/2, 2), (-2, 0, 1), (-1, 42, 2/3), (-2/3, 3/2, -3/2), (4/5, -4/5, 5/4), ... (46/103823, -46/103823, 103823/46)) sage: F = FreeModule(SR, 2) sage: tuple(F.some_elements()) ((1, 0), (some_variable, some_variable))
- span(gens, base_ring=None, check=True, already_echelonized=False)#
Return the \(R\)-span of
gens
, where \(R\) is thebase_ring
.The default \(R\) is the base ring of
self
. Note that this span need not be a submodule ofself
, nor even of the ambient space. It must, however, be contained in the ambient vector space, i.e., the ambient space tensored with the fraction field of \(R\).INPUT:
gens
– a list of vectorsbase_ring
– (optional) a ringcheck
– boolean (default:True
): whether or not to coerce entries of gens into base fieldalready_echelonized
– boolean (default:False
); set this if you know the gens are already in echelon form
EXAMPLES:
sage: V = VectorSpace(GF(7), 3) sage: W = V.subspace([[2,3,4]]); W Vector space of degree 3 and dimension 1 over Finite Field of size 7 Basis matrix: [1 5 2] sage: W.span([[1,1,1]]) Vector space of degree 3 and dimension 1 over Finite Field of size 7 Basis matrix: [1 1 1]
Over a general ring:
sage: S.<x,y,z> = PolynomialRing(QQ) sage: M = S**2 sage: M.span([vector([x - y, z]), vector([y*z, x*z])]) Submodule of Ambient free module of rank 2 over the integral domain Multivariate Polynomial Ring in x, y, z over Rational Field Generated by the rows of the matrix: [x - y z] [ y*z x*z]
Over a PID:
sage: V = FreeModule(ZZ,3) sage: W = V.submodule([V.gen(0)]) sage: W.span([V.gen(1)]) Free module of degree 3 and rank 1 over Integer Ring Echelon basis matrix: [0 1 0] sage: W.submodule([V.gen(1)]) Traceback (most recent call last): ... ArithmeticError: argument gens (= [(0, 1, 0)]) does not generate a submodule of self sage: V.span([[1,0,0],[1/5,4,0],[6,3/4,0]]) Free module of degree 3 and rank 2 over Integer Ring Echelon basis matrix: [1/5 0 0] [ 0 1/4 0]
It also works with other things than integers:
sage: R.<x>=QQ[] sage: L=R^1 sage: a=L.span([(1/x,)]) sage: a Free module of degree 1 and rank 1 over Univariate Polynomial Ring in x over Rational Field Echelon basis matrix: [1/x] sage: b=L.span([(1/x,)]) sage: a(b.gens()[0]) (1/x) sage: L2 = R^2 sage: L2.span([[(x^2+x)/(x^2-3*x+2),1/5],[(x^2+2*x)/(x^2-4*x+3),x]]) Free module of degree 2 and rank 2 over Univariate Polynomial Ring in x over Rational Field Echelon basis matrix: [x/(x^3 - 6*x^2 + 11*x - 6) 2/15*x^2 - 17/75*x - 1/75] [ 0 x^3 - 11/5*x^2 - 3*x + 4/5]
Note that the
base_ring
can make a huge difference. We repeat the previous example over the fraction field of R and get a simpler vector space.sage: L2.span([[(x^2+x)/(x^2-3*x+2),1/5],[(x^2+2*x)/(x^2-4*x+3),x]],base_ring=R.fraction_field()) Vector space of degree 2 and dimension 2 over Fraction Field of Univariate Polynomial Ring in x over Rational Field Basis matrix: [1 0] [0 1]
- submodule(gens, check=True, already_echelonized=False)#
Create the \(R\)-submodule of the ambient module with given generators, where \(R\) is the base ring of
self
.INPUT:
gens
– a list of free module elements or a free modulecheck
– (default:True
) whether or not to verify that the gens are inself
OUTPUT:
The submodule spanned by the vectors in the list
gens
. The basis for the subspace is always put in reduced row echelon form (if possible).EXAMPLES:
We create a submodule of \(\ZZ^3\):
sage: M = FreeModule(ZZ, 3) sage: B = M.basis() sage: W = M.submodule([B[0]+B[1], 2*B[1]-B[2]]) sage: W Free module of degree 3 and rank 2 over Integer Ring Echelon basis matrix: [ 1 1 0] [ 0 2 -1]
We create a submodule of a submodule:
sage: W.submodule([3*B[0] + 3*B[1]]) Free module of degree 3 and rank 1 over Integer Ring Echelon basis matrix: [3 3 0]
We try to create a submodule that isn’t really a submodule, which results in an
ArithmeticError
exception:sage: W.submodule([B[0] - B[1]]) Traceback (most recent call last): ... ArithmeticError: argument gens (= [(1, -1, 0)]) does not generate a submodule of self
Next we create a submodule of a free module over the principal ideal domain \(\QQ[x]\), which uses the general Hermite normal form functionality:
sage: R = PolynomialRing(QQ, 'x'); x = R.gen() sage: M = FreeModule(R, 3) sage: B = M.basis() sage: W = M.submodule([x*B[0], 2*B[1]- x*B[2]]); W Free module of degree 3 and rank 2 over Univariate Polynomial Ring in x over Rational Field Echelon basis matrix: [ x 0 0] [ 0 2 -x] sage: W.ambient_module() Ambient free module of rank 3 over the principal ideal domain Univariate Polynomial Ring in x over Rational Field
Over a generic ring:
sage: S.<x,y,z> = PolynomialRing(QQ) sage: A = S**2 sage: A.submodule([vector([x - y,z]), vector([y*z, x*z])]) Submodule of Ambient free module of rank 2 over the integral domain Multivariate Polynomial Ring in x, y, z over Rational Field Generated by the rows of the matrix: [x - y z] [ y*z x*z]
- zero()#
Return the zero vector in this module.
EXAMPLES:
sage: M = FreeModule(ZZ, 2) sage: M.zero() (0, 0) sage: M.span([[1,1]]).zero() (0, 0) sage: M.zero_submodule().zero() (0, 0) sage: M.zero_submodule().zero().is_mutable() False
- zero_submodule()#
Return the zero submodule of this module.
EXAMPLES:
sage: S.<x,y,z> = PolynomialRing(QQ) sage: M = S**2 sage: M.zero_submodule() Submodule of Ambient free module of rank 2 over the integral domain Multivariate Polynomial Ring in x, y, z over Rational Field Generated by the rows of the matrix: []
- zero_vector()#
Return the zero vector in this module.
EXAMPLES:
sage: M = FreeModule(ZZ, 2) sage: M.zero_vector() (0, 0) sage: M(0) (0, 0) sage: M.span([[1,1]]).zero_vector() (0, 0) sage: M.zero_submodule().zero_vector() (0, 0)
- class sage.modules.free_module.RealDoubleVectorSpace_class(n)#
Bases:
FreeModule_ambient_field
- coordinates(v)#
- sage.modules.free_module.VectorSpace(K, dimension_or_basis_keys, sparse, inner_product_matrix, with_basis=None, dimension=False, basis_keys=None, **args)#
EXAMPLES:
The base can be complicated, as long as it is a field.
sage: V = VectorSpace(FractionField(PolynomialRing(ZZ,'x')),3) sage: V Vector space of dimension 3 over Fraction Field of Univariate Polynomial Ring in x over Integer Ring sage: V.basis() [ (1, 0, 0), (0, 1, 0), (0, 0, 1) ]
The base must be a field or a
TypeError
is raised.sage: VectorSpace(ZZ,5) Traceback (most recent call last): ... TypeError: Argument K (= Integer Ring) must be a field.
- sage.modules.free_module.basis_seq(V, vecs)#
This converts a list vecs of vectors in V to an Sequence of immutable vectors.
Should it? I.e. in most
other
parts of the system the return type of basis or generators is a tuple.EXAMPLES:
sage: V = VectorSpace(QQ,2) sage: B = V.gens() sage: B ((1, 0), (0, 1)) sage: v = B[0] sage: v[0] = 0 # immutable Traceback (most recent call last): ... ValueError: vector is immutable; please change a copy instead (use copy()) sage: sage.modules.free_module.basis_seq(V, V.gens()) [ (1, 0), (0, 1) ]
- sage.modules.free_module.element_class(R, is_sparse)#
The class of the vectors (elements of a free module) with base ring R and boolean is_sparse.
EXAMPLES:
sage: FF = FiniteField(2) sage: P = PolynomialRing(FF,'x') sage: sage.modules.free_module.element_class(QQ, is_sparse=True) <class 'sage.modules.free_module_element.FreeModuleElement_generic_sparse'> sage: sage.modules.free_module.element_class(QQ, is_sparse=False) <class 'sage.modules.vector_rational_dense.Vector_rational_dense'> sage: sage.modules.free_module.element_class(ZZ, is_sparse=True) <class 'sage.modules.free_module_element.FreeModuleElement_generic_sparse'> sage: sage.modules.free_module.element_class(ZZ, is_sparse=False) <class 'sage.modules.vector_integer_dense.Vector_integer_dense'> sage: sage.modules.free_module.element_class(FF, is_sparse=True) <class 'sage.modules.free_module_element.FreeModuleElement_generic_sparse'> sage: sage.modules.free_module.element_class(FF, is_sparse=False) <class 'sage.modules.vector_mod2_dense.Vector_mod2_dense'> sage: sage.modules.free_module.element_class(GF(7), is_sparse=False) <class 'sage.modules.vector_modn_dense.Vector_modn_dense'> sage: sage.modules.free_module.element_class(P, is_sparse=True) <class 'sage.modules.free_module_element.FreeModuleElement_generic_sparse'> sage: sage.modules.free_module.element_class(P, is_sparse=False) <class 'sage.modules.free_module_element.FreeModuleElement_generic_dense'>
- sage.modules.free_module.is_FreeModule(M)#
Return True if M inherits from FreeModule_generic.
EXAMPLES:
sage: from sage.modules.free_module import is_FreeModule sage: V = ZZ^3 sage: is_FreeModule(V) True sage: W = V.span([ V.random_element() for i in range(2) ]) sage: is_FreeModule(W) True
- sage.modules.free_module.span(gens, base_ring=None, check=True, already_echelonized=False)#
Return the span of the vectors in
gens
using scalars frombase_ring
.INPUT:
gens
- a list of either vectors or lists of ring elements used to generate the spanbase_ring
- default:None
- a principal ideal domain for the ring of scalarscheck
- default:True
- passed to thespan()
method of the ambient modulealready_echelonized
- default:False
- set toTrue
if the vectors form the rows of a matrix in echelon form, in order to skip the computation of an echelonized basis for the span.
OUTPUT:
A module (or vector space) that is all the linear combinations of the free module elements (or vectors) with scalars from the ring (or field) given by
base_ring
. See the examples below describing behavior when the base ring is not specified and/or the module elements are given as lists that do not carry explicit base ring information.EXAMPLES:
The vectors in the list of generators can be given as lists, provided a base ring is specified and the elements of the list are in the ring (or the fraction field of the ring). If the base ring is a field, the span is a vector space.
sage: V = span([[1,2,5], [2,2,2]], QQ); V Vector space of degree 3 and dimension 2 over Rational Field Basis matrix: [ 1 0 -3] [ 0 1 4] sage: span([V.gen(0)], QuadraticField(-7,'a')) Vector space of degree 3 and dimension 1 over Number Field in a with defining polynomial x^2 + 7 with a = 2.645751311064591?*I Basis matrix: [ 1 0 -3] sage: span([[1,2,3], [2,2,2], [1,2,5]], GF(2)) Vector space of degree 3 and dimension 1 over Finite Field of size 2 Basis matrix: [1 0 1]
If the base ring is not a field, then a module is created. The entries of the vectors can lie outside the ring, if they are in the fraction field of the ring.
sage: span([[1,2,5], [2,2,2]], ZZ) Free module of degree 3 and rank 2 over Integer Ring Echelon basis matrix: [ 1 0 -3] [ 0 2 8] sage: span([[1,1,1], [1,1/2,1]], ZZ) Free module of degree 3 and rank 2 over Integer Ring Echelon basis matrix: [ 1 0 1] [ 0 1/2 0] sage: R.<x> = QQ[] sage: M= span( [[x, x^2+1], [1/x, x^3]], R); M Free module of degree 2 and rank 2 over Univariate Polynomial Ring in x over Rational Field Echelon basis matrix: [ 1/x x^3] [ 0 x^5 - x^2 - 1] sage: M.basis()[0][0].parent() Fraction Field of Univariate Polynomial Ring in x over Rational Field
A base ring can be inferred if the generators are given as a list of vectors.
sage: span([vector(QQ, [1,2,3]), vector(QQ, [4,5,6])]) Vector space of degree 3 and dimension 2 over Rational Field Basis matrix: [ 1 0 -1] [ 0 1 2] sage: span([vector(QQ, [1,2,3]), vector(ZZ, [4,5,6])]) Vector space of degree 3 and dimension 2 over Rational Field Basis matrix: [ 1 0 -1] [ 0 1 2] sage: span([vector(ZZ, [1,2,3]), vector(ZZ, [4,5,6])]) Free module of degree 3 and rank 2 over Integer Ring Echelon basis matrix: [1 2 3] [0 3 6]