Basis Vector Ontology

FLOW ALGEBRA
TITLE: Basis Vector Ontology
VERSION: 4.0
DOMAIN: Mathematics / Linear Algebra / Entity Graph
SOURCE_URL: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/linear-algebra-entities/basis-vector-ontology.html
AUTHORSHIP: Mark Higgins — Sky (Recursive AI)
PROJECT: Kireth Semantic Library
SOURCE_ALIGNMENT: Independently constructed mathematical ontology; source-page correspondence not verified.

1. PURPOSE
Model basis vectors, bases, coordinates, span, linear independence, dimension, and change of basis with explicit mathematical preconditions and machine-consumable inference rules.

2. TYPED ENTITIES
Field(K); VectorSpace(V,K); Vector(v,V); Scalar(a,K); IndexedFamily(B,I,V); Basis(B,V); BasisVector(v,B,V); LinearCombination(v,coefficients,B); Span(B,V); Dimension(V,n); CoordinateTuple(c,B,v); LinearMap(T,V,W); Matrix(A,K,m,n); OrderedBasis(B,V); OrthonormalBasis(B,V,inner_product); DualBasis(Bstar,B,Vstar).

3. DEFINITIONS
D01: Basis(B,V) iff LinearIndependent(B,V) AND Span(B)=V.
D02: BasisVector(v,B,V) iff Basis(B,V) AND v is a member of B.
D03: LinearIndependent(B,V) iff every finite linear combination of members of B equalling zero has all coefficients zero.
D04: Span(B) is the set of all finite linear combinations of elements of B.
D05: Every v in V has a unique finite linear combination in any basis B of V.
D06: For finite-dimensional V, every basis has exactly dim(V) elements.
D07: StandardBasis(K^n)=(e_1,...,e_n), where (e_i)_j=1 when i=j and 0 otherwise.
D08: An ordered basis defines coordinate tuples; an unordered basis alone does not fix coordinate order.
D09: UnitVector(v) means norm(v)=1 and requires a specified norm; basis vectors need not be unit vectors.
D10: OrthonormalBasis requires a specified inner product, unit norms, and pairwise orthogonality.
D11: Infinite-dimensional algebraic (Hamel) bases permit only finite-support expansions; Schauder bases instead concern convergent infinite expansions in topological vector spaces.
D12: The empty family is a basis for the zero vector space.

4. OPERATIONS
OP01: vector_add: V x V -> V.
OP02: scalar_multiply: K x V -> V.
OP03: span: Family(V) -> Subspace(V).
OP04: coordinates: (OrderedBasis(B,V),v in V) -> finite-support coordinate family over K.
OP05: reconstruct: sum_i c_i*b_i -> v; only finitely many nonzero c_i for a Hamel basis.
OP06: basis_change: [v]_C = P_(C<-B)*[v]_B, where P_(C<-B) columns are B basis vectors expressed in C coordinates.
OP07: linear_map_matrix: columns of [T]_(C<-B) are [T(b_i)]_C.
OP08: dual_evaluation: b^i(b_j)=delta_ij for finite-dimensional ordered basis and dual basis.

5. FORMAL AXIOMS AND CONSTRAINTS
A01: V is a vector space over field K.
A02: 0_V is in V and 1_K is in K.
A03: Any basis vector is nonzero, except no basis vectors exist for V={0}.
A04: A basis cannot contain duplicate vectors (as a set); indexed families must not repeat members when independent.
A05: A basis spans the entire ambient vector space, not merely a proper subspace.
A06: Dimension statements require the relevant finite-dimensional premise.
A07: Change-of-basis matrices are square and invertible for two finite ordered bases of the same V.
A08: An arbitrary invertible linear transformation maps a basis of V to another basis of its image; if it is an automorphism, to a basis of V.
A09: Normalisation is only defined for nonzero vectors in a normed space.
A10: Orthogonality is not defined without additional bilinear/inner-product structure.

6. TYPED RELATIONSHIPS
R01: BasisVector --memberOf--> Basis.
R02: Basis --basisOf--> VectorSpace.
R03: Basis --spans--> VectorSpace.
R04: Basis --satisfies--> LinearIndependence.
R05: Vector --representedBy--> CoordinateTuple.
R06: CoordinateTuple --relativeTo--> OrderedBasis.
R07: VectorSpace --hasDimension--> Cardinality.
R08: OrderedBasis --relatedBy--> ChangeOfBasisMatrix.
R09: OrthonormalBasis --specialisationOf--> Basis.
R10: StandardBasis --specialisationOf--> OrderedBasis.
R11: DualBasis --dualTo--> Basis.
R12: LinearMap --representedBy--> Matrix.

7. INFERENCE RULES
I01: Basis(B,V) -> LinearIndependent(B,V).
I02: Basis(B,V) -> Span(B)=V.
I03: LinearIndependent(B,V) AND Span(B)=V -> Basis(B,V).
I04: Basis(B,V) AND v in B -> BasisVector(v,B,V).
I05: BasisVector(v,B,V) -> v != 0_V.
I06: Basis(B,V) AND v in V -> ExistsUniqueFiniteCoordinates(v,B).
I07: Basis(B,V) AND dim(V)=n finite -> cardinality(B)=n.
I08: LinearIndependent(S,V) AND cardinality(S)=dim(V) finite -> Basis(S,V).
I09: Span(S)=V AND cardinality(S)=dim(V) finite -> Basis(S,V).
I10: Basis(B,V) AND v in B AND finite-dimensional V -> B minus {v} does not span V.
I11: Basis(B,V) AND v not in B -> B union {v} is linearly dependent.
I12: Basis(B,V) AND T:V->W is linear isomorphism -> Basis(T(B),W).
I13: OrderedBasis(B,V) AND OrderedBasis(C,V) AND dim(V) finite -> Invertible(P_(C<-B)).
I14: ChangeMatrix(P_(C<-B)) -> P_(B<-C)=inverse(P_(C<-B)).
I15: OrderedBasis(B,V) AND coordinates(v,B)=0 -> v=0_V.
I16: StandardBasis(K^n) -> Basis(StandardBasis(K^n),K^n).
I17: OrthonormalBasis(B,V) -> Basis(B,V) AND all vectors in B have norm 1.
I18: OrthonormalBasis(B,V) AND innerProductDefined -> distinct basis vectors have inner product zero.
I19: Basis(B,V) AND dim(V)=0 -> B is empty.
I20: FiniteFamily(B,V) AND determinant(columnMatrix(B)) != 0 (in K^n) -> Basis(B,K^n).
I21: FiniteFamily(B,V) AND determinant(columnMatrix(B))=0 (square n-by-n) -> NOT Basis(B,K^n).
I22: Basis(B,V) AND v in V AND coordinates(v,B)=c -> v=sum_i c_i*b_i.
I23: LinearMap(T,V,W) AND ordered bases B,C -> [T(v)]_C=[T]_(C<-B)*[v]_B.
I24: Basis(B,V) AND v in V -> any representation of v by B has same coefficients.
I25: Basis(B,V) AND scalar a != 0 -> replacing one b in B with a*b yields another basis.
I26: Basis(B,V) AND b_i != b_j AND a in K -> replacing b_i with b_i+a*b_j yields another basis.
I27: FiniteDimensional(V) AND linearly independent S -> S extends to a basis of V.
I28: FiniteDimensional(V) AND spanning S -> some subset of S is a basis of V.
I29: Basis(B,V) AND linear map T -> T injective iff T(B) is linearly independent (for linear T defined on V).
I30: Basis(B,V) AND linear map T -> T surjective iff T(B) spans codomain W.
I31: Basis(B,V) AND linear map T -> T isomorphism iff T(B) is a basis of W.
I32: Basis(B,V) AND dim(V)=n finite -> V is isomorphic to K^n.

8. EXAMPLES AND COUNTEREXAMPLES
E01: In R^2, ((1,0),(0,1)) is the standard ordered basis.
E02: In R^2, ((1,1),(1,-1)) is a basis because determinant=-2 != 0.
E03: In R^2, ((1,0),(2,0)) is not a basis: vectors are dependent and do not span R^2.
E04: In R^3, ((1,0,0),(0,1,0),(0,0,1)) is a basis.
E05: In R^2, ((2,0),(0,3)) is a basis though neither vector has unit Euclidean norm.
E06: In R^2, ((0,0),(1,0)) is not a basis because it contains zero.
E07: In P_2(R), (1,x,x^2) is a basis and dim(P_2)=3.
E08: In M_2(R), four matrix units E11,E12,E21,E22 form a basis.
E09: In C over R, (1,i) is a basis; in C over C, (1) is a basis. Base field changes dimension.
E10: The empty set is a basis of {0}.
E11: In R^2, B=((1,1),(1,-1)), v=(4,2) has coordinates (3,1) in B.
E12: In R^2, a single nonzero vector spans a line but not all of R^2.
E13: In R^3, three vectors with determinant 0 cannot be a basis.
E14: In R^2, ((1,0),(0,1),(1,1)) spans R^2 but is dependent.
E15: For B=((2,0),(0,1)), C=standard basis, P_(C<-B)=diag(2,1).
E16: In the Euclidean plane, ((1,0),(0,1)) is orthonormal; ((2,0),(0,1)) is a basis but not orthonormal.

9. SEMANTIC EQUIVALENCE
SE01: 'basis element' ~= 'basis vector' when the basis is a vector-space basis.
SE02: 'coordinate basis vector' refers to an element of a specified coordinate basis; not every basis vector is standard.
SE03: 'standard basis vector' ~= 'canonical coordinate vector' for K^n with canonical coordinates.
SE04: 'unit vector' != 'basis vector' in general.
SE05: 'orthogonal basis' != 'orthonormal basis' unless every vector is also unit length.
SE06: 'Hamel basis' = 'algebraic basis' for vector spaces.
SE07: 'Schauder basis' != 'Hamel basis' in general.
SE08: 'spanning set' != 'basis' unless independent.
SE09: 'independent set' != 'basis' unless spanning.

10. VALIDATION
V01: Verify all vectors share ambient V and field K.
V02: Verify field, vector-space, and optional norm/inner-product structures are declared.
V03: For finite K^n, verify n candidate columns, rank n, or nonzero determinant.
V04: Distinguish linear independence from orthogonality and unit norm.
V05: Confirm coordinate order for ordered bases.
V06: Check change-of-basis orientation and matrix dimensions.
V07: Treat zero-dimensional vector space separately.
V08: Avoid applying finite-dimensional equivalences without finite-dimensional assumptions.
V09: Flag numerically approximate rank as uncertain when tolerance or conditioning is unspecified.
V10: Distinguish proof-level conclusions from numerical evidence.

11. KNOWLEDGE GRAPH TRIPLES
(BasisVector, isA, Vector)
(BasisVector, memberOf, Basis)
(Basis, spans, VectorSpace)
(Basis, satisfies, LinearIndependence)
(StandardBasis, isA, OrderedBasis)
(OrthonormalBasis, isA, Basis)
(OrderedBasis, determines, Coordinates)
(ChangeOfBasisMatrix, relates, OrderedBases)
(LinearTransformation, maps, BasisVectors)
(VectorSpace, hasInvariant, Dimension)

12. CONFIDENCE
CONFIDENCE.definition: 1.0 for stipulated mathematical definitions.
CONFIDENCE.proven_theorem: 1.0 conditional on all stated assumptions.
CONFIDENCE.symbolic_validation: 1.0 if proof or exact computation verified.
CONFIDENCE.numerical_validation: context-dependent; require tolerance and condition estimates.
CONFIDENCE.source_alignment: unverified; do not represent source page as checked.

13. FLOW_HASH
FLOW_HASH.algorithm: SHA-256
FLOW_HASH.input: canonical UTF-8 representation of the final ontology payload, excluding any stored hash field.
FLOW_HASH.value: NOT_COMPUTED
FLOW_HASH.policy: compute at ingestion; do not invent a digest.

14. UPDATE_MODEL
UPDATE_MODEL.version: 4.0
UPDATE_MODEL.strategy: semantic versioning and append-only provenance log.
UPDATE_MODEL.trigger: corrected theorem, revised source, new examples, new typed relation, or validated counterexample.
UPDATE_MODEL.integrity: recompute SHA-256 after canonicalisation.
UPDATE_MODEL.conflicts: preserve source and resolution provenance.

15. AI_CONSUMPTION
AI_CONSUMPTION.parse: typed entities, definitions, operations, rules, examples, constraints.
AI_CONSUMPTION.reason: forward inference only when premises and ambient structures match.
AI_CONSUMPTION.verify: exact symbolic proofs preferred over floating-point heuristics.
AI_CONSUMPTION.disambiguate: distinguish basis vector, unit vector, standard basis, and orthonormal basis.
AI_CONSUMPTION.provenance: retain SOURCE_URL and SOURCE_ALIGNMENT separately.
AI_CONSUMPTION.output: theorem, assumptions, proof/counterexample, confidence, provenance.
AI_CONSUMPTION.fail_closed: do not assert basis status without both spanning and independence, or an applicable equivalent criterion.

END FLOW ALGEBRA