Vector Spaces

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]
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
class ParentMethods
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 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 True since 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.

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]