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An abelian group with scalar multiplication by a field.
wp-vectornlab-vectorstacks-algebra-textEach relation below ends at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
For a field extension , scalar multiplication by makes a vector space over ; its dimension is the extension degree.
Pass canonically from a stronger object to structure it determines, or forget part of the data while retaining a valid weaker structure. The carrier may change under a canonical induced construction.
milne-fields-galois-theoryThis is an authored directed relation from the source endpoint to the target endpoint.
Add compatible field action.
Feed several structures into a construction junction and impose compatibility between them.
wp-vectornlab-vectorstacks-algebra-textEach relation below starts at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
Forgetting the distinguished zero of a vector space while retaining its translation action yields an affine space; abstractly, an affine space is a torsor for a vector space.
Pass canonically from a stronger object to structure it determines, or forget part of the data while retaining a valid weaker structure. The carrier may change under a canonical induced construction.
wp-vectornlab-vectorstacks-algebra-textwp-affine_spacenlab-affine_spacetong-relativityThis is an authored directed relation from the source endpoint to the target endpoint.
Choose a bilinear map ; do not yet require associativity or a unit.
Equip an existing carrier or structured object with additional chosen data, when such compatible data exists. Use a construction junction when several independently meaningful inputs must coexist on the same carrier or interact compatibly.
wp-algebra-fieldstacks-algebra-textThis is an authored directed relation from the source endpoint to the target endpoint.
A vector space together with a quadratic form canonically determines a new algebra, the quotient of its tensor algebra by the relations . This changes the carrier rather than merely adding data to the vector space.
Apply a standard functorial or canonical construction whose output is not merely a reduct of the input and is not generally an equivalent presentation of the same object.
wp-clifford_algebranlab-clifford_algebraThis is an authored directed relation from the source endpoint to the target endpoint.
Add coassociative comultiplication and a counit.
Equip an existing carrier or structured object with additional chosen data, when such compatible data exists. Use a construction junction when several independently meaningful inputs must coexist on the same carrier or interact compatibly.
wp-coalgebranlab-coalgebrastacks-algebra-textThis is an authored directed relation from the source endpoint to the target endpoint.
A complex vector space is a vector space over .
A more specific theory, model, entity class, or regime is obtained by restricting or extending the scope of a broader framework.
wp-complex_vector_spacenlab-complex_vector_spaceThis is an authored directed relation from the source endpoint to the target endpoint.
An indexed family of vector spaces has a direct sum characterized by finitely supported components and the coproduct universal property.
Apply a standard functorial or canonical construction whose output is not merely a reduct of the input and is not generally an equivalent presentation of the same object.
wp-direct-sum-vector-spacenlab-direct-sum-vector-spaceThis is an authored directed relation from the source endpoint to the target endpoint.
The algebraic dual is canonically constructed as the vector space of linear functionals.
Apply a standard functorial or canonical construction whose output is not merely a reduct of the input and is not generally an equivalent presentation of the same object.
wp-dual_vector_spacenlab-dual_vector_spaceThis is an authored directed relation from the source endpoint to the target endpoint.
The exterior algebra is the quotient of the tensor algebra by the two-sided ideal generated by all .
Apply a standard functorial or canonical construction whose output is not merely a reduct of the input and is not generally an equivalent presentation of the same object.
wp-exterior_algebranlab-exterior_algebraThis is an authored directed relation from the source endpoint to the target endpoint.
For a real or complex vector space, choose a positive-definite symmetric bilinear form (real) or Hermitian sesquilinear form (complex).
Equip an existing carrier or structured object with additional chosen data, when such compatible data exists. Use a construction junction when several independently meaningful inputs must coexist on the same carrier or interact compatibly.
wp-inner-productnlab-inner-product-spaceprinceton-functional-analysisThis is an authored directed relation from the source endpoint to the target endpoint.
Supply the Vector space structure as one jointly required input.
Feed several structures into a construction junction and impose compatibility between them.
wp-group-representationnlab-representationetingof-representation-theoryThis is an authored directed relation from the source endpoint to the target endpoint.
Supply a -vector space over the same field as the Lie-algebra input.
Feed several structures into a construction junction and impose compatibility between them.
mathlib-lie-moduleThis is an authored directed relation from the source endpoint to the target endpoint.
Supply the vector-space operations.
Feed several structures into a construction junction and impose compatibility between them.
wp-top-vectorprinceton-functional-analysisThis is an authored directed relation from the source endpoint to the target endpoint.
Add an alternating bilinear bracket satisfying Jacobi.
Equip an existing carrier or structured object with additional chosen data, when such compatible data exists. Use a construction junction when several independently meaningful inputs must coexist on the same carrier or interact compatibly.
wp-lie-algebrastacks-algebra-textThis is an authored directed relation from the source endpoint to the target endpoint.
A linear operator is additional data: a chosen linear map from a vector space to itself, or more generally between specified vector spaces.
Equip an existing carrier or structured object with additional chosen data, when such compatible data exists. Use a construction junction when several independently meaningful inputs must coexist on the same carrier or interact compatibly.
wp-linear_operatornlab-linear_operatorThis is an authored directed relation from the source endpoint to the target endpoint.
Equip the scalar field with a real-valued absolute value and choose with and .
Equip an existing carrier or structured object with additional chosen data, when such compatible data exists. Use a construction junction when several independently meaningful inputs must coexist on the same carrier or interact compatibly.
mathlib-seminormmathlib-normed-moduleThis is an authored directed relation from the source endpoint to the target endpoint.
A symplectic vector space adds a nondegenerate alternating bilinear form.
Equip an existing carrier or structured object with additional chosen data, when such compatible data exists. Use a construction junction when several independently meaningful inputs must coexist on the same carrier or interact compatibly.
wp-symplectic_vector_spacenlab-symplectic_vector_spaceThis is an authored directed relation from the source endpoint to the target endpoint.
The tensor product is characterized by its universal property for bilinear maps.
Apply a standard functorial or canonical construction whose output is not merely a reduct of the input and is not generally an equivalent presentation of the same object.
wp-tensor_product_vector_spacenlab-tensor_product_vector_space