Vector Ontology

FLOW ALGEBRA
TITLE: Vector Ontology
VERSION: 4.0
LIBRARY: Kireth Semantic Library
SOURCE_URL: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/linear-algebra-entities/vector-ontology-2.html
AUTHORSHIP: Mark Higgins — Sky (Recursive AI)
SOURCE_ALIGNMENT: Independently constructed mathematical ontology; source page not verified.

01. PURPOSE
Represent vectors as typed mathematical entities, with their ambient spaces, fields, operations, representations, properties, and inference rules. Do not conflate abstract vectors with coordinate tuples or physical directed quantities.

02. TYPES
TYPE Field
TYPE VectorSpace {scalar_field: Field, addition: V×V→V, scalar_multiplication: F×V→V}
TYPE Vector {parent_space: VectorSpace}
TYPE Scalar {field: Field}
TYPE Basis {space: VectorSpace, elements: indexed_set<Vector>}
TYPE CoordinateTuple {basis: Basis, entries: indexed_family<Scalar>}
TYPE LinearMap {domain: VectorSpace, codomain: VectorSpace}
TYPE Subspace {parent_space: VectorSpace, elements: set<Vector>}
TYPE NormedSpace {vector_space: VectorSpace, norm: V→R_nonnegative}
TYPE InnerProductSpace {vector_space: VectorSpace, inner_product: V×V→F}
TYPE ZeroVector <: Vector
TYPE UnitVector <: Vector [requires normed space]
TYPE Eigenvector <: Vector [requires endomorphism and nonzero vector]

03. FOUNDATIONAL SEMANTICS
ENTITY Vector(v) REQUIRES parent_space(v)=V
ENTITY Scalar(a) REQUIRES field(a)=field(V)
RELATION belongs_to(v,V)
RELATION has_scalar_field(V,F)
RELATION has_representation(v,tuple,basis)
RELATION spans(S,W)
RELATION linearly_independent(S)
RELATION orthogonal_to(u,v) [inner product required]
RELATION image_of(v,T,u) [T(u)=v]
RELATION eigenvector_of(v,T,lambda)
RELATION is_zero_vector(v,V)
RELATION is_unit_vector(v,norm)

04. VECTOR SPACE AXIOMS
For all u,v,w in V and a,b in F:
A01: u+v = v+u.
A02: (u+v)+w = u+(v+w).
A03: exists unique 0_V: v+0_V=v.
A04: for each v exists unique -v: v+(-v)=0_V.
A05: a(u+v)=au+av.
A06: (a+b)v=av+bv.
A07: a(bv)=(ab)v.
A08: 1_F v=v.
Derived: 0_F v=0_V; a0_V=0_V; (-1_F)v=-v.

05. OPERATIONS AND SIGNATURES
OP vector_add: V×V→V
OP vector_negate: V→V
OP scalar_multiply: F×V→V
OP linear_combination: finite_list<F×V>→V
OP coordinate_map: V×Basis(V)→F^I [finite-dimensional basis: F^n]
OP linear_map_apply: LinearMap(V,W)×V→W
OP dot_product: R^n×R^n→R [standard Euclidean structure]
OP inner_product: V×V→F [additional structure; conjugate symmetry over C]
OP norm: V→R_nonnegative [additional structure]
OP cross_product: R^3×R^3→R^3 [orientation and Euclidean structure required]
OP tensor_product: V×W→V⊗W [bilinear canonical map]
OP outer_product: F^m×F^n→F^(m×n) [chosen coordinate bases]

06. MATHEMATICAL DEFINITIONS
D01: A vector is an element of a vector space.
D02: A zero vector is the additive identity in a specified vector space.
D03: A nonzero vector satisfies v≠0_V.
D04: A linear combination of v_1,...,v_n is sum_i a_i v_i.
D05: span(S) is the set of all finite linear combinations of elements of S.
D06: S is linearly independent iff every finite relation sum_i a_i v_i=0 has all coefficients zero.
D07: A basis is a linearly independent spanning set.
D08: Coordinates depend on a chosen ordered basis, whereas the vector itself does not.
D09: In a normed space, unit(v) iff ||v||=1.
D10: In an inner-product space, u⊥v iff <u,v>=0.
D11: For linear T:V→V, v is an eigenvector iff v≠0 and T(v)=lambda v.
D12: A vector's dimension refers to its ambient space or coordinate representation; vectors do not independently possess a canonical number of coordinates.

07. INFERENCE RULES
R01: vector(v,V) AND vector(w,V) -> vector(v+w,V).
R02: vector(v,V) AND scalar(a,F(V)) -> vector(av,V).
R03: vector(v,V) -> v+0_V=v.
R04: vector(v,V) -> v+(-v)=0_V.
R05: vector(v,V) -> 0_F v=0_V.
R06: vector(v,V) -> 1_F v=v.
R07: vector(v,V) -> (-1_F)v=-v.
R08: finite S subset V -> span(S) is subspace of V.
R09: S linearly independent AND spans(S,V) -> basis(S,V).
R10: basis(B,V) AND finite_dim(V,n) -> |B|=n.
R11: basis(B,V) -> every v in V has unique finite basis expansion.
R12: v in span(S) -> exists finite coefficients and vectors from S giving v.
R13: T linear AND T(v+w) defined -> T(v+w)=T(v)+T(w).
R14: T linear AND scalar(a) -> T(av)=aT(v).
R15: T linear -> T(0_V)=0_W.
R16: T linear -> ker(T) subspace of domain(T).
R17: T linear -> im(T) subspace of codomain(T).
R18: finite_dim(V) AND T:V→W linear -> dim(V)=dim(ker T)+dim(im T).
R19: norm(v)=0 -> v=0_V.
R20: v≠0 AND norm_defined(v) -> norm(v/norm(v))=1 [real or complex normed space].
R21: inner_product(u,v)=0 -> orthogonal(u,v).
R22: inner_product(v,v)=0 -> v=0_V.
R23: v eigenvector of T with eigenvalue lambda -> v≠0_V.
R24: T(v)=lambda v AND v≠0 -> eigenvector(v,T,lambda).
R25: vectors v,w in V -> v-w=v+(-w).
R26: finite S contains 0_V -> S linearly dependent.
R27: finite S contains duplicate vectors -> S linearly dependent.
R28: S linearly independent -> every subset of S linearly independent.
R29: S spans V -> any superset within V spans V.
R30: finite_dim(V,n) AND S has n independent vectors -> S is basis.
R31: finite_dim(V,n) AND S has n spanning vectors -> S is basis.
R32: v and w in same V AND v=w -> their coordinates agree relative to the same basis.
R33: coordinates equal relative to same basis -> vectors equal.
R34: B,C bases of finite-dimensional V -> coordinate_change_matrix(B,C) invertible.
R35: scalar a≠0 AND av=0_V -> v=0_V [field scalars].
R36: u+v=u+w -> v=w.

08. VALIDATION CONSTRAINTS
C01: Vector addition operands must belong to the same ambient vector space or have an explicit embedding/coercion.
C02: Scalar multiplication requires a scalar from the space's underlying field.
C03: Basis and coordinate data must reference the same vector space.
C04: Unit vector requires an explicitly declared norm.
C05: Orthogonality requires an explicitly declared inner product or equivalent structure.
C06: Dot and cross products must not be assumed on arbitrary abstract vector spaces.
C07: Eigenvector claims require a nonzero vector and a linear endomorphism.
C08: Distinguish vector equality from equality of coordinates in different bases.
C09: Rank-nullity requires finite-dimensional domain for the numerical dimension equation.
C10: Never infer a norm from vector-space axioms alone.
C11: For complex inner products, record the chosen linearity convention.
C12: Infinite-dimensional coordinate expansions may require finite support for a Hamel basis; infinite convergent expansions need topology and a suitable basis concept.

09. EXAMPLES
E01: V=R^2, v=(2,3), w=(1,-1), v+w=(3,2).
E02: 2(2,3)=(4,6).
E03: 0_(R^2)=(0,0).
E04: e1=(1,0), e2=(0,1) form the standard basis of R^2.
E05: (2,3)=2e1+3e2.
E06: span{(1,0)} is the x-axis in R^2.
E07: {(1,0),(2,0)} is linearly dependent.
E08: ||(3,4)||_2=5.
E09: (3/5,4/5) is a Euclidean unit vector.
E10: (1,0)·(0,1)=0.
E11: T(x,y)=(2x,3y) is linear.
E12: (1,0) is eigenvector of T with eigenvalue 2.
E13: ker(T)= {(0,0)} for T(x,y)=(2x,3y).
E14: A change of basis alters coordinates, not the underlying vector.
E15: Polynomials of degree at most 2 form a vector space over R.
E16: Continuous real-valued functions on [0,1] form a vector space over R.

10. SEMANTIC EQUIVALENCE
EQ01: vector = element of vector space [mathematical abstract sense].
EQ02: null vector = zero vector [when same ambient space].
EQ03: linear span = span = generated linear subspace.
EQ04: position vector = coordinate/geometric representation [not synonymous with every vector].
EQ05: unit vector = normalised vector [only when norm is one; zero cannot be normalised].
EQ06: coordinate vector = representation relative to a basis [not identical to basis-independent vector entity].

11. KNOWLEDGE GRAPH TRIPLES
(Vector, rdf:type, MathematicalEntity)
(Vector, belongsTo, VectorSpace)
(VectorSpace, hasScalarField, Field)
(VectorSpace, supportsOperation, VectorAddition)
(VectorSpace, supportsOperation, ScalarMultiplication)
(ZeroVector, subclassOf, Vector)
(UnitVector, subclassOf, Vector)
(BasisVector, subclassOf, Vector)
(Eigenvector, subclassOf, Vector)
(LinearCombination, produces, Vector)
(Basis, spans, VectorSpace)
(Basis, satisfies, LinearIndependence)
(LinearTransformation, maps, Vector)
(Norm, measures, VectorMagnitude)

12. CONFIDENCE
CONFIDENCE mathematical_definitions: 0.99
CONFIDENCE formal_axioms: 0.99
CONFIDENCE inference_rules: 0.98
CONFIDENCE examples: 0.99
CONFIDENCE source_page_alignment: unverified
CONFIDENCE_POLICY: Scores describe editorial confidence, not calibrated probabilities or proof certificates.

13. FLOW_HASH
FLOW_HASH.algorithm: SHA-256
FLOW_HASH.canonicalisation: UTF-8; LF newlines; NFC; omit FLOW_HASH.value field before hashing.
FLOW_HASH.value: TO_BE_COMPUTED_ON_PUBLICATION
FLOW_HASH.policy: Recalculate whenever canonical content changes; do not claim an uncomputed hash.

14. UPDATE_MODEL
UPDATE_MODEL.version: 4.0
UPDATE_MODEL.entity_id: kireth:linear-algebra:vector
UPDATE_MODEL.revision_policy: increment revision and recompute content hash on semantic changes
UPDATE_MODEL.dependencies: VectorSpace, Scalar, Basis, LinearTransformation, Subspace, Norm, InnerProduct
UPDATE_MODEL.conflict_policy: prefer verified formal mathematical definitions; preserve alternative conventions with provenance
UPDATE_MODEL.last_review: 2026-10-13

15. AI_CONSUMPTION
AI_CONSUMPTION.mode: typed_semantic_reasoning
AI_CONSUMPTION.require_parent_space: true
AI_CONSUMPTION.enforce_field_compatibility: true
AI_CONSUMPTION.enforce_operation_preconditions: true
AI_CONSUMPTION.distinguish_abstract_and_coordinate_vectors: true
AI_CONSUMPTION.proof_obligations: explicit
AI_CONSUMPTION.unknown_claims: do_not_infer
AI_CONSUMPTION.source_provenance: preserve
AI_CONSUMPTION.validation: axioms, typing, dimensional consistency, counterexamples
AI_CONSUMPTION.output: typed triples, rules, examples, assumptions, confidence, provenance

END FLOW ALGEBRA