Vector Spaces¶
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class
sage.categories.vector_spaces.
VectorSpaces
(K)¶ Bases:
sage.categories.category_types.Category_module
The category of (abstract) vector spaces over a given field
??? with an embedding in an ambient vector space ???
EXAMPLES:
sage: VectorSpaces(QQ) Category of vector spaces over Rational Field sage: VectorSpaces(QQ).super_categories() [Category of modules over Rational Field]
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class
CartesianProducts
(category, *args)¶ Bases:
sage.categories.cartesian_product.CartesianProductsCategory
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extra_super_categories
()¶ The category of vector spaces is closed under Cartesian products:
sage: C = VectorSpaces(QQ) sage: C.CartesianProducts() Category of Cartesian products of vector spaces over Rational Field sage: C in C.CartesianProducts().super_categories() True
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class
DualObjects
(category, *args)¶ Bases:
sage.categories.dual.DualObjectsCategory
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extra_super_categories
()¶ Returns the dual category
EXAMPLES:
The category of algebras over the Rational Field is dual to the category of coalgebras over the same field:
sage: C = VectorSpaces(QQ) sage: C.dual() Category of duals of vector spaces over Rational Field sage: C.dual().super_categories() # indirect doctest [Category of vector spaces over Rational Field]
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class
ElementMethods
¶ Bases:
object
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class
Filtered
(base_category)¶ Bases:
sage.categories.filtered_modules.FilteredModulesCategory
Category of filtered vector spaces.
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class
Graded
(base_category)¶ Bases:
sage.categories.graded_modules.GradedModulesCategory
Category of graded vector spaces.
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class
ParentMethods
¶ Bases:
object
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dimension
()¶ Return the dimension of this vector space.
EXAMPLES:
sage: M = FreeModule(FiniteField(19), 100) sage: W = M.submodule([M.gen(50)]) sage: W.dimension() 1 sage: M = FiniteRankFreeModule(QQ, 3) sage: M.dimension() 3 sage: M.tensor_module(1,2).dimension() 27
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class
TensorProducts
(category, *args)¶ Bases:
sage.categories.tensor.TensorProductsCategory
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extra_super_categories
()¶ The category of vector spaces is closed under tensor products:
sage: C = VectorSpaces(QQ) sage: C.TensorProducts() Category of tensor products of vector spaces over Rational Field sage: C in C.TensorProducts().super_categories() True
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class
WithBasis
(base_category)¶ Bases:
sage.categories.category_with_axiom.CategoryWithAxiom_over_base_ring
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class
CartesianProducts
(category, *args)¶ Bases:
sage.categories.cartesian_product.CartesianProductsCategory
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extra_super_categories
()¶ The category of vector spaces with basis is closed under Cartesian products:
sage: C = VectorSpaces(QQ).WithBasis() sage: C.CartesianProducts() Category of Cartesian products of vector spaces with basis over Rational Field sage: C in C.CartesianProducts().super_categories() True
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class
Filtered
(base_category)¶ Bases:
sage.categories.filtered_modules.FilteredModulesCategory
Category of filtered vector spaces with basis.
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example
(base_ring=None)¶ Return an example of a graded vector space with basis, as per
Category.example()
.EXAMPLES:
sage: Modules(QQ).WithBasis().Graded().example() An example of a graded module with basis: the free module on partitions over Rational Field
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class
Graded
(base_category)¶ Bases:
sage.categories.graded_modules.GradedModulesCategory
Category of graded vector spaces with basis.
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example
(base_ring=None)¶ Return an example of a graded vector space with basis, as per
Category.example()
.EXAMPLES:
sage: Modules(QQ).WithBasis().Graded().example() An example of a graded module with basis: the free module on partitions over Rational Field
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class
TensorProducts
(category, *args)¶ Bases:
sage.categories.tensor.TensorProductsCategory
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extra_super_categories
()¶ The category of vector spaces with basis is closed under tensor products:
sage: C = VectorSpaces(QQ).WithBasis() sage: C.TensorProducts() Category of tensor products of vector spaces with basis over Rational Field sage: C in C.TensorProducts().super_categories() True
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is_abelian
()¶ Return whether this category is abelian.
This is always
True
since the base ring is a field.EXAMPLES:
sage: VectorSpaces(QQ).WithBasis().is_abelian() True
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class
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additional_structure
()¶ Return
None
.Indeed, the category of vector spaces defines no additional structure: a bimodule morphism between two vector spaces is a vector space morphism.
See also
Todo
Should this category be a
CategoryWithAxiom
?EXAMPLES:
sage: VectorSpaces(QQ).additional_structure()
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base_field
()¶ Returns the base field over which the vector spaces of this category are all defined.
EXAMPLES:
sage: VectorSpaces(QQ).base_field() Rational Field
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super_categories
()¶ EXAMPLES:
sage: VectorSpaces(QQ).super_categories() [Category of modules over Rational Field]
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class