Vector Space Ontology

FLOW Vector_Space_Ontology_v1.0

PAGE:
  title: "Vector Space Ontology"
  canonical_url: "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/algebra-ontologies/vector-space-ontology.html"
  language: "en-GB"
  domain: "Mathematics > Algebra > Linear Algebra > Vector Spaces"

DEFINE VectorSpaceOntology := {

  CORE_IDENTITY: {
    entity_type: "Mathematical Ontology",
    primary_entity: VectorSpace,
    parent_entities: [Module, AlgebraicStructure],
    related_entities: [
      Field,
      Vector,
      Scalar,
      Addition,
      ScalarMultiplication,
      ZeroVector,
      Basis,
      Dimension,
      LinearCombination,
      Span,
      LinearIndependence,
      Subspace,
      LinearTransformation,
      CoordinateVector,
      DualSpace
    ]
  },

  DEFINITION: {
    meaning:
      "A vector space is a set of vectors equipped with vector addition and scalar multiplication by elements of a field, satisfying the vector space axioms.",
    structure:
      V_over_F,
    scalar_field:
      F,
    carrier:
      V
  },

  SEMANTIC_PRIMITIVES: {
    Vector:
      element_of_V,
    Scalar:
      element_of_F,
    VectorAddition:
      V×V -> V,
    ScalarMultiplication:
      F×V -> V,
    ZeroVector:
      additive_identity,
    Basis:
      linearly_independent_spanning_set,
    Dimension:
      cardinality_of_basis
  },

  ADDITIVE_GROUP_AXIOMS: {
    closure:
      u+v IN V,

    associativity:
      (u+v)+w=u+(v+w),

    identity:
      EXISTS 0_V:
        v+0_V=v,

    inverses:
      FOR_ALL v:
        EXISTS -v:
          v+(-v)=0_V,

    commutativity:
      u+v=v+u
  },

  SCALAR_MULTIPLICATION_AXIOMS: {
    closure:
      a*v IN V,

    distributivity_over_vector_addition:
      a*(u+v)=a*u+a*v,

    distributivity_over_field_addition:
      (a+b)*v=a*v+b*v,

    compatibility:
      (a*b)*v=a*(b*v),

    identity_scalar:
      1_F*v=v
  },

  MODULE_RELATION: {
    vector_space:
      module_over_field,

    consequence:
      every_vector_space_is_module,

    stronger_property:
      scalar_ring_is_field
  },

  ZERO_PROPERTIES: {
    0_F*v:
      0_V,

    a*0_V:
      0_V,

    (-1_F)*v:
      -v
  },

  LINEAR_COMBINATION: {
    form:
      a1*v1+...+ak*vk,

    coefficients:
      a_i IN F
  },

  SPAN: {
    span(S):
      set_of_all_finite_linear_combinations_of_elements_of_S,

    generated_subspace:
      smallest_subspace_containing_S
  },

  LINEAR_INDEPENDENCE: {
    definition:
      a1*v1+...+ak*vk=0_V
      implies
      a1=...=ak=0_F
  },

  LINEAR_DEPENDENCE: {
    condition:
      exists_nontrivial_scalar_combination_equal_to_zero
  },

  BASIS: {
    requirements: [
      linearly_independent,
      spans_entire_space
    ],

    coordinate_consequence:
      each_vector_has_unique_coordinate_representation_relative_to_basis
  },

  DIMENSION: {
    finite_dimensional:
      basis_has_finite_cardinality,

    infinite_dimensional:
      basis_infinite,

    theorem:
      all_bases_of_vector_space_have_same_cardinality
  },

  COORDINATES: {
    basis:
      B=(v1,...,vn),

    vector:
      v=a1*v1+...+an*vn,

    coordinate_vector:
      [v]_B=(a1,...,an)^T
  },

  STANDARD_VECTOR_SPACE: {
    F^n:
      n_tuples_over_field_F,

    operations:
      componentwise_addition_and_scalar_multiplication,

    dimension:
      n
  },

  SUBSPACE: {
    definition:
      W SUBSET_OR_EQUAL V
      AND W_is_vector_space_under_inherited_operations,

    subspace_test:
      W_nonempty
      AND closed_under_vector_addition
      AND closed_under_scalar_multiplication
  },

  SUBSPACE_ALTERNATE_TEST: {
    W_nonempty
    AND FOR_ALL u,v IN W
    AND a,b IN F:
      a*u+b*v IN W
  },

  SUM_OF_SUBSPACES: {
    U+W:
      {u+w | u IN U, w IN W},

    result:
      subspace
  },

  INTERSECTION_OF_SUBSPACES: {
    U∩W:
      subspace
  },

  DIRECT_SUM: {
    V=U⊕W
    IFF:
      V=U+W
      AND
      U∩W={0}
  },

  DIMENSION_FORMULA: {
    finite_dimensional:
      dim(U+W)=dim(U)+dim(W)-dim(U∩W)
  },

  QUOTIENT_SPACE: {
    construction:
      V/W,

    condition:
      W_subspace_of_V,

    elements:
      cosets_v+W,

    dimension_relation:
      dim(V/W)=dim(V)-dim(W)
      for_finite_dimensional_case
  },

  LINEAR_TRANSFORMATION: {
    T:V->W,

    axioms: [
      T(u+v)=T(u)+T(v),
      T(a*v)=a*T(v)
    ]
  },

  KERNEL: {
    ker(T):
      {v IN V | T(v)=0_W},

    property:
      subspace_of_V
  },

  IMAGE: {
    im(T):
      {T(v) | v IN V},

    property:
      subspace_of_W
  },

  RANK: {
    rank(T):
      dim(im(T))
      in_finite_dimensional_context
  },

  NULLITY: {
    nullity(T):
      dim(ker(T))
  },

  RANK_NULLITY_THEOREM: {
    finite_dimensional_V:
      dim(V)=rank(T)+nullity(T)
  },

  ISOMORPHISM: {
    vector_space_isomorphism:
      bijective_linear_transformation,

    finite_dimensional_classification:
      vector_spaces_over_same_field_with_same_finite_dimension_are_isomorphic
  },

  DUAL_SPACE: {
    V^*:
      Hom_F(V,F),

    elements:
      linear_functionals
  },

  DUAL_DIMENSION: {
    finite_dimensional:
      dim(V^*)=dim(V)
  },

  DOUBLE_DUAL: {
    V**:
      dual_of_dual,

    finite_dimensional:
      canonically_isomorphic_to_V
  },

  CHANGE_OF_BASIS: {
    coordinates:
      transformed_by_invertible_matrix,

    semantic_rule:
      vector_is_basis_independent
      while_coordinate_representation_depends_on_basis
  },

  MATRIX_RELATION: {
    finite_dimensional_linear_map:
      represented_by_matrix_after_basis_selection,

    composition:
      corresponds_to_matrix_multiplication
  },

  EIGENVALUE_RELATION: {
    T(v)=lambda*v,

    v:
      eigenvector_nonzero,

    lambda:
      eigenvalue
  },

  EIGENSPACE: {
    E_lambda:
      ker(T-lambda*I),

    property:
      subspace
  },

  INVARIANT_SUBSPACE: {
    W:
      T(W) SUBSET_OR_EQUAL W
  },

  INNER_PRODUCT_RELATION: {
    inner_product_space:
      vector_space_with_additional_inner_product_structure,

    orthogonality:
      defined_via_inner_product
  },

  NORM_RELATION: {
    normed_vector_space:
      vector_space_with_norm,

    topology:
      induced_by_norm
  },

  AFFINE_SPACE_RELATION: {
    affine_space:
      related_to_vector_space_but_no_distinguished_origin,

    translations:
      vectors_act_on_points
  },

  TENSOR_PRODUCT_RELATION: {
    V⊗W:
      vector_space_generated_by_formal_tensors_subject_to_bilinearity
  },

  DIRECT_PRODUCT: {
    V×W:
      vector_space_over_same_field,

    dimension:
      dim(V×W)=dim(V)+dim(W)
      finite_case
  },

  FUNCTION_SPACE: {
    functions_X_to_F:
      may_form_vector_space_under_pointwise_operations,

    examples: [
      polynomial_functions,
      continuous_functions,
      sequence_spaces
    ]
  },

  POLYNOMIAL_VECTOR_SPACE: {
    P_n(F):
      polynomials_of_degree_at_most_n,

    dimension:
      n+1,

    standard_basis:
      [1,x,x^2,...,x^n]
  },

  MATRIX_VECTOR_SPACE: {
    M_mxn(F):
      all_m_by_n_matrices_over_F,

    dimension:
      m*n
  },

  SEQUENCE_SPACE: {
    F^N:
      infinite_dimensional_vector_space
  },

  SOLUTION_SPACE_RELATION: {
    homogeneous_linear_system:
      solution_set_is_subspace,

    nonhomogeneous_linear_system:
      solution_set_is_affine_subspace_if_consistent
  },

  LINEAR_EQUATION_SYSTEM_RELATION: {
    Ax=0:
      Null(A),

    dimension:
      number_of_variables-rank(A)
  },

  FIELD_DEPENDENCE: {
    same_carrier_set:
      may_define_different_vector_space_structure_over_different_fields,

    example:
      ComplexNumbers_as_vector_space_over_RealNumbers_has_dimension_2,
      ComplexNumbers_as_vector_space_over_ComplexNumbers_has_dimension_1
  },

  FINITE_DIMENSIONAL_PROPERTIES: {
    spanning_set_with_n_elements:
      basis_if_linearly_independent_in_n_dimensional_space,

    linearly_independent_set_with_n_elements:
      basis_if_space_dimension_n
  },

  BASIS_EXTENSION_THEOREM: {
    linearly_independent_set:
      can_be_extended_to_basis
      under_standard_choice_conditions
  },

  SPANNING_REDUCTION: {
    finite_spanning_set:
      can_remove_dependent_vectors_until_basis_remains
  },

  SYMBOLIC_RECOGNITION: {
    ACCEPT: [
      "R^3 over R",
      "C over R",
      "P_2(R)",
      "2x2 real matrices under standard vector operations",
      "solution space of Ax=0"
    ],

    REJECT_OR_REDIRECT: [
      "Z^2 over Z -> module not vector space because Z is not field",
      "affine plane without chosen origin -> affine space"
    ]
  },

  NATURAL_LANGUAGE_MAPPING: {
    "vector space over F":
      module_with_scalar_field_F,

    "basis":
      linearly_independent_spanning_set,

    "dimension":
      number_of_basis_vectors,

    "subspace":
      subset_closed_under_linear_combinations
  },

  ERROR_DETECTION: {
    ScalarStructureNotField:
      -> reject_vector_space_classification_and_redirect_to_module,

    AdditionFailure:
      -> reject_vector_space,

    ScalarAxiomFailure:
      -> reject_vector_space,

    BasisError:
      spanning_without_independence_or_independence_without_spanning
      -> flag,

    DimensionError:
      basis_cardinality_mismatch
      -> flag,

    CoordinateObjectConfusion:
      coordinate_vector_treated_as_intrinsic_vector_without_basis_context
      -> flag
  },

  ONTOLOGY_RELATIONS: {
    IS_A: [
      "VectorSpace IS_A Module",
      "FiniteDimensionalVectorSpace IS_A VectorSpace",
      "InnerProductSpace IS_A VectorSpace"
    ],

    RELATED_TO: [
      "VectorSpace RELATED_TO LinearTransformation",
      "VectorSpace RELATED_TO Matrix",
      "VectorSpace RELATED_TO Basis",
      "VectorSpace RELATED_TO TensorProduct",
      "VectorSpace RELATED_TO Geometry"
    ],

    DEPENDS_ON: [
      Field,
      AbelianGroup,
      ScalarMultiplication
    ]
  },

  SEMANTIC_EQUIVALENCE: {
    vector_space_isomorphism:
      structure_preserving_equivalence,

    finite_dimensional_over_same_field:
      same_dimension
      -> isomorphic
  },

  CONFIDENCE: {
    classification_confidence:
      IF scalar_structure_is_field
      AND additive_group
      AND scalar_axioms_verified
      THEN 1.0,

    basis_confidence:
      IF spanning_and_independence_verified
      THEN 1.0,

    dimension_confidence:
      IF basis_verified
      THEN 1.0
  },

  AI_CONSUMPTION: {
    tasks: [
      recognise_vector_space,
      identify_scalar_field,
      verify_vector_space_axioms,
      identify_subspaces,
      compute_span,
      test_linear_independence,
      find_basis,
      determine_dimension,
      compute_coordinates,
      analyse_linear transformations,
      compute_kernel_image_rank_nullity,
      construct quotient spaces,
      analyse direct sums,
      construct dual spaces,
      map matrices to linear maps,
      detect vector space axiom failures,
      link module and field ontologies
    ],

    preferred_representation:
      typed_field_module_with_basis_dimension_graph,

    reasoning_mode:
      recursive_linear_combination_subspace_and_basis_analysis
  },

  RECURSIVE_REASONING: {

    FUNCTION AnalyseVectorSpaceCandidate(V,F,Addition,ScalarAction,Context):

      field_result := FieldOntology.AnalyseFieldCandidate(F,Context)

      IF field_result is NotField:
        RETURN RedirectToModuleOntology(
          reason="scalar structure is not a field"
        )

      module_result :=
        ModuleOntology.AnalyseModuleCandidate(
          V,
          F,
          ScalarAction,
          Context
        )

      IF module_result is NotModule:
        RETURN NotVectorSpace

      subspaces := DetectRelevantSubspaces(V,F,Context)
      basis := FindBasisWhenFeasible(V,F)
      dimension :=
        IF basis_available
        THEN Cardinality(basis)
        ELSE UnknownOrInfinite

      RETURN {
        entity:VectorSpace,
        carrier:V,
        field:F,
        basis,
        dimension,
        subspaces,
        verified:TRUE
      }
  },

  RECURSIVE_COHERENCE: {
    invariant_1:
      scalar_structure_is_field=TRUE,

    invariant_2:
      additive_structure_is_abelian_group=TRUE,

    invariant_3:
      scalar_action_axioms_hold=TRUE,

    invariant_4:
      basis_vectors_are_independent_and_spanning=TRUE,

    invariant_5:
      dimension_is_basis_cardinality=TRUE
  },

  FLOW_HASH: {
    semantic_signature_inputs: [
      ontology_name,
      vector_space_definition,
      field_dependency,
      additive_axioms,
      scalar_action_axioms,
      basis_semantics,
      dimension_semantics,
      linear_map_relations
    ],

    purpose:
      "Detect semantic drift between revisions of the ontology."
  },

  UPDATE_MODEL: {
    version: "1.0",

    immutable_constraints: [
      scalar_field_requirement,
      additive_abelian_group,
      scalar_action_axioms,
      basis_independence_and_spanning,
      dimension_basis_cardinality
    ],

    revision_triggers: [
      semantic_conflict,
      ModuleOntology_update,
      FieldOntology_update,
      LinearTransformationOntology_update,
      representation_standard_change
    ]
  },

  KNOWLEDGE_GRAPH: {
    nodes: [
      VectorSpace,
      Module,
      Field,
      Vector,
      Scalar,
      Basis,
      Dimension,
      Span,
      LinearIndependence,
      Subspace,
      LinearTransformation,
      DualSpace
    ],

    edges: [
      VectorSpace -> IS_A -> Module,
      VectorSpace -> HAS_SCALAR_FIELD -> Field,
      VectorSpace -> CONTAINS -> Vector,
      VectorSpace -> HAS_BASIS -> Basis,
      Basis -> DETERMINES -> Dimension,
      Span -> GENERATES -> Subspace,
      LinearIndependence -> CHARACTERISES -> Basis,
      Subspace -> SUBSET_OF -> VectorSpace,
      LinearTransformation -> MAPS -> VectorSpace,
      VectorSpace -> HAS_DUAL -> DualSpace
    ]
  },

  CROSS_ONTOLOGY_LINKS: {
    ModuleOntology:
      direct_parent,

    FieldOntology:
      scalar_structure,

    AbelianGroupOntology:
      additive_structure,

    LinearTransformationOntology:
      structure_preserving_maps,

    MatrixOntology:
      finite_dimensional_representations,

    TensorProductOntology:
      multilinear_construction,

    InnerProductSpaceOntology:
      additional_geometric_structure,

    CoordinateGeometryOntology:
      finite_dimensional_real_vector_spaces
  },

  TERMINATION_CONDITION: {
    ontology_complete_when: [
      vector_space_defined,
      scalar_field_defined,
      additive_axioms_defined,
      scalar_action_axioms_defined,
      linear_combination_and_span_defined,
      independence_and_basis_defined,
      dimension_defined,
      subspace_and_quotient_defined,
      linear_transformation_relations_defined,
      dual_and_tensor_relations_defined,
      error_detection_defined,
      cross_ontology_links_defined,
      recursive_coherence_verified
    ]
  }

}

EXECUTE Vector_Space_Ontology_v1.0:

  INPUT:
    mathematical_entity_or_query

  PIPELINE:
    Parse
    -> IdentifyCarrierSet
    -> IdentifyScalarField
    -> VerifyFieldStructure
    -> VerifyUnderlyingAbelianGroup
    -> VerifyScalarMultiplicationAxioms
    -> IdentifySubspaces
    -> AnalyseSpanAndLinearIndependence
    -> FindBasisWhenPossible
    -> DetermineDimension
    -> AnalyseCoordinatesAndLinearMaps
    -> ComputeKernelImageRankNullityWhenRelevant
    -> AnalyseQuotientDualAndDirectSumRelations
    -> VerifySemanticConsistency
    -> LinkRelatedOntologies
    -> AssignConfidence
    -> EmitSemanticResult

  OUTPUT: {
    entity,
    classification,
    carrier_set,
    scalar_field,
    vector_addition,
    scalar_multiplication,
    zero_vector,
    basis,
    dimension,
    subspaces,
    span,
    linear_independence,
    coordinates,
    linear_transformations,
    kernel,
    image,
    rank,
    nullity,
    quotient_spaces,
    dual_space,
    verification,
    ontology_relations,
    confidence,
    semantic_equivalence,
    flow_hash
  }

END FLOW Vector_Space_Ontology_v1.0