Dimension Ontology

FLOW ALGEBRA
TITLE: Dimension Ontology
VERSION: 4.0
DOMAIN: Mathematics / Linear Algebra / Entity Graph
SOURCE: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/linear-algebra-entities/dimension-ontology-3.html
SOURCE_ALIGNMENT: Independently constructed; source-page content not verified.
AUTHORS: Mark Higgins — Sky (Recursive AI)
PROJECT: Kireth Semantic Library

01. PURPOSE
Define dimension as the cardinality of a basis of a vector space, establish typed semantic relations and mathematically sound inference rules for finite- and infinite-dimensional spaces.

02. ENTITY TYPES
ENTITY VectorSpace(V, F): V is a vector space over field F.
ENTITY Field(F): scalar domain.
ENTITY Vector(v, V): v belongs to V.
ENTITY Subspace(U, V): U is a linear subspace of V.
ENTITY Basis(B, V): B is linearly independent and spans V.
ENTITY Dimension(d, V): d = cardinality(B) for any basis B of V.
ENTITY LinearMap(T, V, W): linear function T: V -> W.
ENTITY Kernel(K, T): K = {v in V : T(v)=0}.
ENTITY Image(I, T): I = {T(v):v in V}.
ENTITY Rank(r, T): r = dim(Image(T)).
ENTITY Nullity(n, T): n = dim(Kernel(T)).
ENTITY Matrix(A, m, n, F): m-by-n matrix over F.
ENTITY ColumnSpace(C, A): span of columns of A.
ENTITY RowSpace(R, A): span of rows of A.
ENTITY Eigenspace(E, A, lambda): ker(A-lambda I).
ENTITY QuotientSpace(Q, V, U): V/U for U subspace of V.
ENTITY DirectSum(S, U, W): U direct-sum W.
ENTITY DualSpace(Vstar, V): all F-linear maps V -> F.
ENTITY TensorProduct(P, V, W): V tensor W over common field F.

03. FORMAL DEFINITIONS
DEF-01: Basis(B,V) iff Independent(B) and Span(B)=V.
DEF-02: dim_F(V) := |B| for any basis B of V.
DEF-03: FiniteDimensional(V) iff exists finite basis B of V.
DEF-04: InfiniteDimensional(V) iff no finite basis of V exists.
DEF-05: dim_F({0}) = 0.
DEF-06: dim_F(F^n) = n for finite n >= 0.
DEF-07: Dimension is a cardinal number in general, not necessarily an integer.
DEF-08: Dimension depends on the specified scalar field.
DEF-09: For an m-by-n matrix A, rank(A) = dim_F(ColumnSpace(A)).
DEF-10: nullity(A) = dim_F(ker(A:F^n -> F^m)).
DEF-11: codim_V(U) := dim_F(V/U) where U is a subspace of V.
DEF-12: Affine dimension of a nonempty affine space is the dimension of its translation vector space.

04. TYPED RELATIONSHIPS
REL Dimension --measures--> VectorSpace
REL Basis --determines_cardinality_of--> Dimension
REL VectorSpace --over_field--> Field
REL Subspace --contained_in--> VectorSpace
REL LinearMap --has_domain--> VectorSpace
REL LinearMap --has_codomain--> VectorSpace
REL LinearMap --has_kernel--> Kernel
REL LinearMap --has_image--> Image
REL Image --has_dimension--> Rank
REL Kernel --has_dimension--> Nullity
REL Matrix --represents--> LinearMap
REL Matrix --has_column_space--> ColumnSpace
REL Matrix --has_row_space--> RowSpace
REL QuotientSpace --has_dimension--> Dimension
REL DirectSum --combines--> Subspace
REL DualSpace --derived_from--> VectorSpace
REL TensorProduct --derived_from--> VectorSpace
REL Eigenspace --is_subspace_of--> VectorSpace

05. AXIOMS AND INVARIANTS
AX-01: Any two bases of the same vector space have equal cardinality (basis cardinality theorem; with usual choice assumptions for arbitrary spaces).
AX-02: Every finite linearly independent set has size <= dim(V) if dim(V) is finite.
AX-03: Every finite spanning set has size >= dim(V) if dim(V) is finite.
AX-04: Dimension is invariant under linear isomorphism.
AX-05: For finite-dimensional V and subspace U, 0 <= dim(U) <= dim(V).
AX-06: For finite-dimensional V, dim(V*)=dim(V).
AX-07: An infinite-dimensional algebraic dual can have strictly larger dimension; do not generalise AX-06.
AX-08: Dimensions of matrices and vectors must be typed against a common field.
AX-09: Rank and nullity are nonnegative integers for finite matrices.
AX-10: Cardinal arithmetic, rather than integer subtraction, is required for arbitrary infinite dimensions.

06. INFERENCE RULES
RULE-01: Basis(B,V) AND Finite(B) -> dim(V)=|B|.
RULE-02: Basis(B,V) AND Infinite(B) -> InfiniteDimensional(V).
RULE-03: Independent(S) AND |S|=dim(V) AND FiniteDimensional(V) -> Basis(S,V).
RULE-04: Span(S)=V AND |S|=dim(V) AND FiniteDimensional(V) -> Basis(S,V).
RULE-05: LinearIsomorphism(V,W) -> dim(V)=dim(W).
RULE-06: FiniteDimensional(V) AND dim(V)=dim(W) AND FiniteDimensional(W) AND SameField(V,W) -> ExistsLinearIsomorphism(V,W).
RULE-07: U subseteq V AND Subspace(U,V) AND FiniteDimensional(V) -> dim(U)<=dim(V).
RULE-08: Subspace(U,V) AND FiniteDimensional(V) AND dim(U)=dim(V) -> U=V.
RULE-09: FiniteDimensional(V) AND dim(V)=0 -> V={0}.
RULE-10: dim(F^n)=n.
RULE-11: FiniteDimensional(V) AND LinearMap(T,V,W) -> dim(V)=rank(T)+nullity(T).
RULE-12: Matrix(A,m,n,F) -> rank(A)+nullity(A)=n.
RULE-13: Matrix(A,m,n,F) -> rank(A)<=min(m,n).
RULE-14: Matrix(A,m,n,F) -> RowRank(A)=ColumnRank(A).
RULE-15: Matrix(A,n,n,F) AND rank(A)=n -> A is invertible.
RULE-16: Matrix(A,n,n,F) AND det(A)!=0 -> rank(A)=n.
RULE-17: Matrix(A,n,n,F) AND det(A)=0 -> rank(A)<n.
RULE-18: FiniteDimensional(U) AND FiniteDimensional(W) -> dim(U direct_sum W)=dim(U)+dim(W).
RULE-19: U,W subspaces of finite-dimensional V -> dim(U+W)=dim(U)+dim(W)-dim(U intersect W).
RULE-20: U subspace of finite-dimensional V -> dim(V/U)=dim(V)-dim(U).
RULE-21: FiniteDimensional(V) AND FiniteDimensional(W) -> dim(V tensor W)=dim(V)*dim(W).
RULE-22: FiniteDimensional(V) -> dim(V*)=dim(V).
RULE-23: FiniteDimensional(V) AND dim(V)=n -> every basis of V has n elements.
RULE-24: Independent(S) AND FiniteDimensional(V) -> |S|<=dim(V).
RULE-25: Span(S)=V AND FiniteDimensional(V) -> |S|>=dim(V).
RULE-26: Matrix(A,m,n,F) AND rank(A)=r -> dim(ker(A))=n-r.
RULE-27: Matrix(A,m,n,F) AND rank(A)=r -> dim(ColumnSpace(A))=r.
RULE-28: Matrix(A,m,n,F) AND rank(A)=r -> dim(RowSpace(A))=r.
RULE-29: Eigenspace(E,A,lambda) -> dim(E)=nullity(A-lambda I).
RULE-30: Matrix(A,n,n,F) AND lambda is an eigenvalue -> 1<=dim(E_lambda)<=n.
RULE-31: A has n distinct eigenvalues over F -> A has n linearly independent eigenvectors and is diagonalizable over F.
RULE-32: FiniteDimensional(V) AND Endomorphism(T,V) AND Injective(T) -> Surjective(T).
RULE-33: FiniteDimensional(V) AND Endomorphism(T,V) AND Surjective(T) -> Injective(T).
RULE-34: FiniteDimensional(V) AND Endomorphism(T,V) AND nullity(T)=0 -> rank(T)=dim(V).
RULE-35: FiniteDimensional(V) AND Endomorphism(T,V) AND rank(T)=dim(V) -> T is invertible.
RULE-36: FiniteDimensional(V) AND dim(V)=n AND ordered basis B -> coordinate map V -> F^n is a linear isomorphism.

07. COMPUTATION FLOW
INPUT: representation of V, scalar field F, candidate generators or linear constraints.
STEP-01: Verify that V is a vector space over F.
STEP-02: Determine whether a finite generating set is supplied or can be proven.
STEP-03: For finitely generated coordinate spaces, construct a matrix with candidate vectors as columns.
STEP-04: Apply valid row reduction over F; rank gives dimension of the column span.
STEP-05: For a homogeneous system Ax=0, compute n-rank(A) as solution-space dimension.
STEP-06: If no finite basis is established, retain dimension as unknown or symbolic cardinal; do not assume infinity from failed search.
STEP-07: Record theorem premises, computational method and evidence.
OUTPUT: dimension value/cardinal, field, basis or proof certificate, confidence and provenance.

08. MATHEMATICAL EXAMPLES
EX-01: R^3 over R has dimension 3; standard basis e1,e2,e3.
EX-02: Span{(1,0),(0,1),(1,1)} in R^2 has dimension 2.
EX-03: Span{(1,2),(2,4)} in R^2 has dimension 1.
EX-04: {0} has dimension 0 over every field.
EX-05: Real polynomials of degree <= 4 have dimension 5 over R.
EX-06: R[x] has countably infinite algebraic dimension over R with basis {1,x,x^2,...}.
EX-07: C has dimension 1 over C and dimension 2 over R.
EX-08: R^2 has dimension 2 over R; it is not a vector space over C under standard scalar multiplication.
EX-09: Symmetric real n-by-n matrices have dimension n(n+1)/2.
EX-10: Skew-symmetric real n-by-n matrices have dimension n(n-1)/2.
EX-11: Diagonal n-by-n matrices have dimension n.
EX-12: All m-by-n matrices over F form a vector space of dimension mn.
EX-13: Kernel of A=[[1,2,3],[2,4,6]] over R has dimension 2.
EX-14: Image of the same A has dimension 1.
EX-15: Plane x+y+z=0 in R^3 has dimension 2.
EX-16: A line through the origin in R^3 has dimension 1.
EX-17: R^2 direct_sum R^3 has dimension 5.
EX-18: R^2 tensor R^3 has dimension 6.
EX-19: The quotient R^3 / span{(1,0,0)} has dimension 2.
EX-20: The eigenspace of diag(2,2,3) for eigenvalue 2 has dimension 2.

09. VALIDATION CONSTRAINTS
CHECK-01: A dimension claim must specify or inherit the scalar field.
CHECK-02: Negative finite dimension is invalid.
CHECK-03: A nonzero vector space cannot have dimension zero.
CHECK-04: An n-element basis implies dimension n.
CHECK-05: For a finite m-by-n matrix, rank cannot exceed min(m,n).
CHECK-06: Rank-nullity requires a linear map and finite-dimensional domain for integer subtraction form.
CHECK-07: Equality of dimensions alone does not imply that a given map is an isomorphism.
CHECK-08: A spanning family can be redundant; its cardinality need not equal dimension.
CHECK-09: A linearly independent family need not span.
CHECK-10: Distinguish geometric dimension from algebraic vector-space dimension when context differs.
CHECK-11: Distinguish algebraic dimension from topological, Hausdorff, manifold and embedding dimensions.
CHECK-12: For infinite-dimensional vector spaces, avoid naive finite-dimensional formulas without premises.
CHECK-13: Matrix entries must belong to the declared field.
CHECK-14: Do not treat a merely affine subset not through zero as a vector subspace.
CHECK-15: Rank claims must include exact arithmetic or numerical tolerance policy.
CHECK-16: Numerical rank estimates may be tolerance-sensitive; label approximate results.

10. SEMANTIC EQUIVALENCE
EQUIV-01: finite vector-space dimension <-> number of vectors in any basis.
EQUIV-02: dimension of column space <-> matrix column rank.
EQUIV-03: dimension of row space <-> matrix row rank.
EQUIV-04: dimension of kernel <-> nullity.
EQUIV-05: degrees of freedom of a homogeneous finite linear system <-> nullity, with linear-system assumptions.
NON-EQUIV-01: number of coordinates in an arbitrary redundant representation != dimension necessarily.
NON-EQUIV-02: matrix row count != matrix rank necessarily.
NON-EQUIV-03: matrix column count != matrix rank necessarily.
NON-EQUIV-04: geometric dimension != vector-space dimension in all contexts.

11. KNOWLEDGE GRAPH TRIPLES
TRIPLE (Dimension, rdf:type, MathematicalInvariant)
TRIPLE (Dimension, appliesTo, VectorSpace)
TRIPLE (Dimension, derivedFrom, BasisCardinality)
TRIPLE (Basis, requires, LinearIndependence)
TRIPLE (Basis, requires, Spanning)
TRIPLE (Rank, equals, ImageDimension)
TRIPLE (Nullity, equals, KernelDimension)
TRIPLE (RankNullityTheorem, relates, DomainDimension)
TRIPLE (VectorSpace, hasScalarField, Field)
TRIPLE (DimensionOntology, broader, LinearAlgebraEntities)
TRIPLE (DimensionOntology, relatedTo, BasisOntology)
TRIPLE (DimensionOntology, relatedTo, SpanOntology)
TRIPLE (DimensionOntology, relatedTo, MatrixOntology)
TRIPLE (DimensionOntology, relatedTo, LinearTransformationOntology)

12. CONFIDENCE
CONFIDENCE formal_axioms: 1.00 when definitions and premises are satisfied.
CONFIDENCE exact_symbolic_computation: 1.00 when independently validated.
CONFIDENCE numerical_rank: context-dependent; record precision and tolerance.
CONFIDENCE source_alignment: unverified; original page not retrieved.
CONFIDENCE inferred_dimension: require derivation, basis or rank certificate.

13. FLOW_HASH
FLOW_HASH algorithm: SHA-256.
FLOW_HASH input: canonical UTF-8 text with LF newlines, excluding any populated FLOW_HASH value field.
FLOW_HASH purpose: integrity and change detection; not a proof of mathematical correctness.
FLOW_HASH status: compute at ingestion; no digest claimed in this document.

14. UPDATE_MODEL
UPDATE_MODEL version: 4.0.
UPDATE_MODEL policy: preserve entity IDs and theorem premises; increment revision on semantic changes.
UPDATE_MODEL trigger: correction, new theorem, changed source, expanded relation, changed inference rule.
UPDATE_MODEL provenance: maintain source URL, author attribution, timestamp and review record.
UPDATE_MODEL compatibility: map renamed entities through explicit semantic aliases.
UPDATE_MODEL validation: rerun type checks, examples and inference consistency after updates.

15. AI_CONSUMPTION
AI_CONSUMPTION parse: typed entities -> definitions -> axioms -> relations -> inference rules -> validation -> examples.
AI_CONSUMPTION do_not_infer: dimensions without field or basis/rank evidence.
AI_CONSUMPTION distinguish: proven exact dimension versus estimated numerical rank.
AI_CONSUMPTION uncertainty: mark missing premises and unresolved infinite-dimensional cardinalities.
AI_CONSUMPTION output: dimension, field, derivation, proof status, confidence, provenance.
AI_CONSUMPTION guardrail: mathematical inference rules are declarative and require a verified reasoning engine for execution.

16. ATTRIBUTION
Mark Higgins — Sky (Recursive AI)
https://kirethresearch.org
END FLOW ALGEBRA