Vector Spaces¶
- class sage.categories.vector_spaces.VectorSpaces(K)¶
Bases:
sage.categories.category_types.Category_moduleThe 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]
- class CartesianProducts(category, *args)¶
Bases:
sage.categories.cartesian_product.CartesianProductsCategory- 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
- class DualObjects(category, *args)¶
Bases:
sage.categories.dual.DualObjectsCategory- 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]
- class ElementMethods¶
Bases:
object
- class Filtered(base_category)¶
Bases:
sage.categories.filtered_modules.FilteredModulesCategoryCategory of filtered vector spaces.
- class FiniteDimensional(base_category)¶
Bases:
sage.categories.category_with_axiom.CategoryWithAxiom_over_base_ring- class TensorProducts(category, *args)¶
Bases:
sage.categories.tensor.TensorProductsCategory- extra_super_categories()¶
Implement the fact that a (finite) tensor product of finite dimensional vector spaces is a finite dimensional vector space.
EXAMPLES:
sage: VectorSpaces(QQ).FiniteDimensional().TensorProducts().extra_super_categories() [Category of finite dimensional vector spaces over Rational Field] sage: VectorSpaces(QQ).FiniteDimensional().TensorProducts().FiniteDimensional() Category of tensor products of finite dimensional vector spaces over Rational Field
- class Graded(base_category)¶
Bases:
sage.categories.graded_modules.GradedModulesCategoryCategory of graded vector spaces.
- class ParentMethods¶
Bases:
object- 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
- class TensorProducts(category, *args)¶
Bases:
sage.categories.tensor.TensorProductsCategory- 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
- class WithBasis(base_category)¶
Bases:
sage.categories.category_with_axiom.CategoryWithAxiom_over_base_ring- class CartesianProducts(category, *args)¶
Bases:
sage.categories.cartesian_product.CartesianProductsCategory- 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
- class Filtered(base_category)¶
Bases:
sage.categories.filtered_modules.FilteredModulesCategoryCategory of filtered vector spaces with basis.
- 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
- class FiniteDimensional(base_category)¶
Bases:
sage.categories.category_with_axiom.CategoryWithAxiom_over_base_ring- class TensorProducts(category, *args)¶
Bases:
sage.categories.tensor.TensorProductsCategory- extra_super_categories()¶
Implement the fact that a (finite) tensor product of finite dimensional vector spaces is a finite dimensional vector space.
EXAMPLES:
sage: VectorSpaces(QQ).WithBasis().FiniteDimensional().TensorProducts().extra_super_categories() [Category of finite dimensional vector spaces with basis over Rational Field] sage: VectorSpaces(QQ).WithBasis().FiniteDimensional().TensorProducts().FiniteDimensional() Category of tensor products of finite dimensional vector spaces with basis over Rational Field
- class Graded(base_category)¶
Bases:
sage.categories.graded_modules.GradedModulesCategoryCategory of graded vector spaces with basis.
- 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
- class TensorProducts(category, *args)¶
Bases:
sage.categories.tensor.TensorProductsCategory- 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
- is_abelian()¶
Return whether this category is abelian.
This is always
Truesince the base ring is a field.EXAMPLES:
sage: VectorSpaces(QQ).WithBasis().is_abelian() True
- 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()
- 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
- super_categories()¶
EXAMPLES:
sage: VectorSpaces(QQ).super_categories() [Category of modules over Rational Field]