Vector Spaces¶
- class sage.categories.vector_spaces.VectorSpaces(K)[source]¶
- Bases: - 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] - >>> from sage.all import * >>> VectorSpaces(QQ) Category of vector spaces over Rational Field >>> VectorSpaces(QQ).super_categories() [Category of modules over Rational Field] - class CartesianProducts(category, *args)[source]¶
- Bases: - CartesianProductsCategory- extra_super_categories()[source]¶
- 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 - >>> from sage.all import * >>> C = VectorSpaces(QQ) >>> C.CartesianProducts() Category of Cartesian products of vector spaces over Rational Field >>> C in C.CartesianProducts().super_categories() True 
 
 - class DualObjects(category, *args)[source]¶
- Bases: - DualObjectsCategory- extra_super_categories()[source]¶
- Return 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] - >>> from sage.all import * >>> C = VectorSpaces(QQ) >>> C.dual() Category of duals of vector spaces over Rational Field >>> C.dual().super_categories() # indirect doctest [Category of vector spaces over Rational Field] 
 
 - class Filtered(base_category)[source]¶
- Bases: - FilteredModulesCategory- Category of filtered vector spaces. 
 - class FiniteDimensional(base_category)[source]¶
- Bases: - CategoryWithAxiom_over_base_ring- class TensorProducts(category, *args)[source]¶
- Bases: - TensorProductsCategory- extra_super_categories()[source]¶
- 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 - >>> from sage.all import * >>> VectorSpaces(QQ).FiniteDimensional().TensorProducts().extra_super_categories() [Category of finite dimensional vector spaces over Rational Field] >>> VectorSpaces(QQ).FiniteDimensional().TensorProducts().FiniteDimensional() Category of tensor products of finite dimensional vector spaces over Rational Field 
 
 
 - class Graded(base_category)[source]¶
- Bases: - GradedModulesCategory- Category of graded vector spaces. 
 - class ParentMethods[source]¶
- Bases: - object- dimension()[source]¶
- Return the dimension of this vector space. - EXAMPLES: - sage: M = FreeModule(FiniteField(19), 100) # needs sage.modules sage: W = M.submodule([M.gen(50)]) # needs sage.modules sage: W.dimension() # needs sage.modules 1 sage: M = FiniteRankFreeModule(QQ, 3) # needs sage.modules sage: M.dimension() # needs sage.modules 3 sage: M.tensor_module(1, 2).dimension() # needs sage.modules 27 - >>> from sage.all import * >>> M = FreeModule(FiniteField(Integer(19)), Integer(100)) # needs sage.modules >>> W = M.submodule([M.gen(Integer(50))]) # needs sage.modules >>> W.dimension() # needs sage.modules 1 >>> M = FiniteRankFreeModule(QQ, Integer(3)) # needs sage.modules >>> M.dimension() # needs sage.modules 3 >>> M.tensor_module(Integer(1), Integer(2)).dimension() # needs sage.modules 27 
 
 - class TensorProducts(category, *args)[source]¶
- Bases: - TensorProductsCategory- extra_super_categories()[source]¶
- 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 - >>> from sage.all import * >>> C = VectorSpaces(QQ) >>> C.TensorProducts() Category of tensor products of vector spaces over Rational Field >>> C in C.TensorProducts().super_categories() True 
 
 - class WithBasis(base_category)[source]¶
- Bases: - CategoryWithAxiom_over_base_ring- class CartesianProducts(category, *args)[source]¶
- Bases: - CartesianProductsCategory- extra_super_categories()[source]¶
- 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 - >>> from sage.all import * >>> C = VectorSpaces(QQ).WithBasis() >>> C.CartesianProducts() Category of Cartesian products of vector spaces with basis over Rational Field >>> C in C.CartesianProducts().super_categories() True 
 
 - class Filtered(base_category)[source]¶
- Bases: - FilteredModulesCategory- Category of filtered vector spaces with basis. - example(base_ring=None)[source]¶
- Return an example of a graded vector space with basis, as per - Category.example().- EXAMPLES: - sage: Modules(QQ).WithBasis().Graded().example() # needs sage.combinat sage.modules An example of a graded module with basis: the free module on partitions over Rational Field - >>> from sage.all import * >>> Modules(QQ).WithBasis().Graded().example() # needs sage.combinat sage.modules An example of a graded module with basis: the free module on partitions over Rational Field 
 
 - class FiniteDimensional(base_category)[source]¶
- Bases: - CategoryWithAxiom_over_base_ring- class TensorProducts(category, *args)[source]¶
- Bases: - TensorProductsCategory- extra_super_categories()[source]¶
- 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 - >>> from sage.all import * >>> VectorSpaces(QQ).WithBasis().FiniteDimensional().TensorProducts().extra_super_categories() [Category of finite dimensional vector spaces with basis over Rational Field] >>> VectorSpaces(QQ).WithBasis().FiniteDimensional().TensorProducts().FiniteDimensional() Category of tensor products of finite dimensional vector spaces with basis over Rational Field 
 
 
 - class Graded(base_category)[source]¶
- Bases: - GradedModulesCategory- Category of graded vector spaces with basis. - example(base_ring=None)[source]¶
- Return an example of a graded vector space with basis, as per - Category.example().- EXAMPLES: - sage: Modules(QQ).WithBasis().Graded().example() # needs sage.combinat sage.modules An example of a graded module with basis: the free module on partitions over Rational Field - >>> from sage.all import * >>> Modules(QQ).WithBasis().Graded().example() # needs sage.combinat sage.modules An example of a graded module with basis: the free module on partitions over Rational Field 
 
 - class TensorProducts(category, *args)[source]¶
- Bases: - TensorProductsCategory- extra_super_categories()[source]¶
- 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 - >>> from sage.all import * >>> C = VectorSpaces(QQ).WithBasis() >>> C.TensorProducts() Category of tensor products of vector spaces with basis over Rational Field >>> C in C.TensorProducts().super_categories() True 
 
 
 - additional_structure()[source]¶
- 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() - >>> from sage.all import * >>> VectorSpaces(QQ).additional_structure()