Vector Space Ontology

FLOW ALGEBRA
TITLE: Vector Space Ontology
VERSION: 4.0
DOMAIN: Mathematical Entities / Algebraic Entities / Vector Space
SOURCE_URL: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/algebraic-entities/vector-space-ontology-2.html
SOURCE_ALIGNMENT: Independently constructed; source-page wording not verified.
AUTHORS: Mark Higgins — Sky (Recursive AI)
SITE: https://kirethresearch.org

01. PURPOSE
Define typed mathematical objects, axioms, relationships, validation and inference for vector spaces over fields, with explicit assumptions and proof obligations.

02. ENTITY TYPES
Field(K); Scalar(a:K); VectorSpace(V,K); Vector(v:V); ZeroVector(0_V); VectorAddition(add:V×V→V); ScalarMultiplication(scale:K×V→V); Subspace(W,V); Span(S,V); LinearCombination(v,S); LinearIndependence(S); Basis(B,V); Dimension(V,cardinal); LinearMap(T,V,W); Kernel(T); Image(T); QuotientSpace(V,W); DirectSum(V,W); CoordinateVector(v,B); DualSpace(V*); LinearFunctional(f,V,K); Isomorphism(T); Eigenvector(T,v); Eigenvalue(T,lambda).

03. CORE SIGNATURE
field(K)
vector_space(V, K, add, scale, zero)
add: V × V -> V
scale: K × V -> V
zero: V
neg: V -> V
subspace(W, V)
linear_map(T, V, W)

04. VECTOR SPACE AXIOMS
For all u,v,w in V and a,b in K:
VS01 add(u,v) in V.
VS02 add(u,v) = add(v,u).
VS03 add(add(u,v),w) = add(u,add(v,w)).
VS04 add(v,0_V) = v.
VS05 add(v,neg(v)) = 0_V.
VS06 scale(a,v) in V.
VS07 scale(a,add(u,v)) = add(scale(a,u),scale(a,v)).
VS08 scale(a+b,v) = add(scale(a,v),scale(b,v)).
VS09 scale(a*b,v) = scale(a,scale(b,v)).
VS10 scale(1_K,v) = v.
Assume K is a field and V is a set with total operations of stated types.

05. DERIVED IDENTITIES
D01 scale(0_K,v) = 0_V.
D02 scale(a,0_V) = 0_V.
D03 scale(-1_K,v) = neg(v).
D04 scale(-a,v) = neg(scale(a,v)).
D05 add(v,w) = 0_V implies w = neg(v).
D06 scale(a,v) = 0_V implies (a = 0_K or v = 0_V).
D07 cancellation: add(u,w)=add(v,w) implies u=v.

06. SUBSPACES
SUB01 W subseteq V and W nonempty.
SUB02 u,v in W implies u+v in W.
SUB03 a in K and u in W implies a*u in W.
SUB04 Equivalent test: W nonempty and closed under a*u+b*v for a,b in K.
SUB05 {0_V} and V are subspaces.
SUB06 Intersection of any family of subspaces is a subspace.
SUB07 Union of two subspaces is a subspace iff one contains the other.
SUB08 W1+W2 = {w1+w2 | w1 in W1, w2 in W2} is a subspace.

07. SPAN AND LINEAR COMBINATIONS
SPAN01 span(S) = {sum(i=1..n) a_i*v_i | n finite, a_i in K, v_i in S}.
SPAN02 span(empty_set) = {0_V}.
SPAN03 S subseteq span(S).
SPAN04 span(S) is the smallest subspace containing S.
SPAN05 S subseteq T implies span(S) subseteq span(T).
SPAN06 span(span(S)) = span(S).
SPAN07 A generating set S satisfies span(S)=V.

08. INDEPENDENCE, BASIS, DIMENSION
BAS01 S linearly independent iff every finite linear combination equal to 0_V has all coefficients zero, for distinct vectors from S.
BAS02 S linearly dependent iff some finite nontrivial linear combination is zero.
BAS03 A basis B is independent and spans V.
BAS04 Every vector has a unique finite linear combination representation in a basis.
BAS05 Every vector space has a basis assuming the axiom of choice; finite-dimensional spaces need no such general appeal.
BAS06 Any two bases of a vector space have equal cardinality (under standard choice assumptions in the infinite case).
BAS07 dim(V) = cardinality of any basis.
BAS08 For finite-dimensional V, independent sets have at most dim(V) members.
BAS09 For finite-dimensional V, spanning sets have at least dim(V) members.
BAS10 In an n-dimensional space, any n independent vectors form a basis.
BAS11 In an n-dimensional space, any n spanning vectors form a basis.

09. LINEAR MAPS
MAP01 T:V->W is linear iff T(u+v)=T(u)+T(v) and T(a*v)=a*T(v).
MAP02 T(0_V)=0_W.
MAP03 T(-v)=-T(v).
MAP04 ker(T)={v in V | T(v)=0_W} is a subspace of V.
MAP05 im(T)={T(v) | v in V} is a subspace of W.
MAP06 T injective iff ker(T)={0_V}.
MAP07 T surjective iff im(T)=W.
MAP08 T isomorphism iff linear and bijective.
MAP09 If V finite-dimensional, dim(V)=dim(ker T)+dim(im T) (rank-nullity).
MAP10 Composition of compatible linear maps is linear.
MAP11 Inverse of a bijective linear map is linear.

10. QUOTIENTS, SUMS, DUALS
Q01 For W subspace V, V/W consists of additive cosets v+W.
Q02 (u+W)+(v+W)=(u+v)+W.
Q03 a*(v+W)=(a*v)+W.
Q04 For finite-dimensional V, dim(V/W)=dim(V)-dim(W).
Q05 V=U direct_sum W iff V=U+W and U intersection W={0_V}.
Q06 For finite-dimensional U,W, dim(U+W)=dim(U)+dim(W)-dim(U intersection W).
Q07 V*=Hom_K(V,K) is the algebraic dual.
Q08 For finite-dimensional V, dim(V*)=dim(V).
Q09 Canonical evaluation V->V** is injective; it is an isomorphism for finite-dimensional V.

11. MATRIX AND COORDINATE REPRESENTATIONS
CO01 Fix ordered bases B of V and C of W to represent T by a matrix [T]_(C<-B).
CO02 [T(v)]_C = [T]_(C<-B) * [v]_B.
CO03 Matrix composition corresponds to linear-map composition with compatible bases.
CO04 A change-of-basis matrix is invertible.
CO05 Matrix rank equals dimension of the image of its associated linear map.
CO06 Eigenvectors/eigenvalues require an endomorphism T:V->V and nonzero v satisfying T(v)=lambda*v.

12. TYPED RELATIONSHIPS
VectorSpace --OVER_FIELD--> Field
VectorSpace --HAS_VECTOR--> Vector
VectorSpace --HAS_OPERATION--> VectorAddition
VectorSpace --HAS_OPERATION--> ScalarMultiplication
Subspace --SUBSPACE_OF--> VectorSpace
Basis --BASIS_OF--> VectorSpace
Basis --SPANS--> VectorSpace
Basis --IS_LINEarly_INDEPENDENT_IN--> VectorSpace
LinearMap --DOMAIN--> VectorSpace
LinearMap --CODOMAIN--> VectorSpace
LinearMap --HAS_KERNEL--> Subspace
LinearMap --HAS_IMAGE--> Subspace
QuotientSpace --QUOTIENT_OF--> VectorSpace
DualSpace --DUAL_OF--> VectorSpace
CoordinateVector --RELATIVE_TO--> Basis

13. FORMAL INFERENCE RULES
R01 field(K) and valid_axioms(V,K) -> vector_space(V,K).
R02 vector_space(V,K) -> additive_group(V) and abelian(V,+).
R03 vector_space(V,K) and W nonempty and closed_linear_combinations(W) -> subspace(W,V).
R04 subspace(W,V) -> vector_space(W,K) with restricted operations.
R05 subspace(U,V) and subspace(W,V) -> subspace(U intersection W,V).
R06 subset(S,V) -> subspace(span(S),V).
R07 independent(B) and span(B)=V -> basis(B,V).
R08 basis(B,V) -> unique_coordinates_for_all_vectors(V,B).
R09 basis(B,V) and finite(B) -> finite_dimensional(V).
R10 linear(T,V,W) -> subspace(kernel(T),V).
R11 linear(T,V,W) -> subspace(image(T),W).
R12 linear(T,V,W) and kernel(T)={0_V} -> injective(T).
R13 linear(T,V,W) and image(T)=W -> surjective(T).
R14 linear(T,V,W) and injective(T) and surjective(T) -> isomorphism(T).
R15 finite_dimensional(V) and linear(T,V,W) -> dim(V)=nullity(T)+rank(T).
R16 subspace(W,V) -> vector_space(V/W,K).
R17 subspace(U,V) and subspace(W,V) and U intersection W={0_V} and U+W=V -> direct_sum(V,U,W).
R18 finite_dimensional(V) and basis(B,V) -> dim(V)=size(B).
R19 vector_space(V,K) and vector_space(W,K) -> vector_space(Hom_K(V,W),K).
R20 linear(T,V,W) and linear(S,W,X) -> linear(S∘T,V,X).
R21 finite_dimensional(V) and dim(V)=dim(W) and injective(T) and linear(T,V,W) -> bijective(T).
R22 finite_dimensional(V) and dim(V)=dim(W) and surjective(T) and linear(T,V,W) -> bijective(T).
R23 vector_space(V,K) and scalar(a,K) and vector(v,V) -> vector(a*v,V).
R24 linear(T,V,W) and v in kernel(T) -> T(v)=0_W.
R25 basis(B,V) and v in V -> exists_unique_finite_coordinate_representation(v,B).
R26 subspace(U,V) and subspace(W,V) -> subspace(U+W,V).
R27 finite_dimensional(V) and subspace(W,V) -> dim(W)<=dim(V).
R28 finite_dimensional(V) and subspace(W,V) and dim(W)=dim(V) -> W=V.
R29 isomorphism(T,V,W) -> dim(V)=dim(W).
R30 vector_space(V,K) and v in V -> 0_K*v=0_V.
R31 field(K) and a != 0_K and a*v=0_V -> v=0_V.
R32 finite_dimensional(V) and endomorphism(T,V) and eigenpair(T,lambda,v) -> v!=0_V and T(v)=lambda*v.

14. VALIDATION CONSTRAINTS
VAL01 Reject scalar multiplication where scalars are not in the specified field.
VAL02 Reject structures lacking an additive identity or additive inverses.
VAL03 Reject nonassociative addition or failure of either distributive law.
VAL04 Reject claims of subspace without zero vector and closure checks.
VAL05 Reject empty set as a subspace (for standard vector spaces).
VAL06 Reject a purported basis that is dependent or does not span.
VAL07 Reject dimension equality conclusions without necessary finiteness or cardinal assumptions.
VAL08 Reject rank-nullity when finite-dimensional domain is not established.
VAL09 Reject unqualified eigenvector zero vector.
VAL10 Record ordered bases for matrix/coordinate claims.
VAL11 Do not infer orthogonality, norm or inner product from vector-space axioms alone.
VAL12 Do not infer topological completeness or convergence without additional structure.
VAL13 Distinguish finite linear combinations from infinite sums.
VAL14 State field characteristic when relevant.
VAL15 Do not conflate a module over a ring with a vector space over a field.
VAL16 Do not conflate abstract vector spaces with Euclidean coordinate spaces.

15. WORKED EXAMPLES
EX01 K^n over field K with componentwise addition and scalar multiplication: vector space, dimension n.
EX02 R^2 with standard basis ((1,0),(0,1)): basis and dimension 2.
EX03 Polynomial ring K[x] regarded as a K-vector space: infinite-dimensional with basis {1,x,x^2,...}.
EX04 Polynomials of degree at most n: dimension n+1.
EX05 Continuous real-valued functions on [0,1]: real vector space under pointwise operations.
EX06 Solutions to homogeneous linear system Ax=0: subspace, namely kernel(A).
EX07 Solutions to inhomogeneous Ax=b with b != 0: generally affine, not a subspace (if nonempty).
EX08 Span{(1,0),(2,0)} in R^2: x-axis, dimension 1; generators dependent.
EX09 T:R^3->R^2, T(x,y,z)=(x,y): rank 2, nullity 1.
EX10 R^3 / span{(0,0,1)}: quotient of dimension 2.
EX11 R^2 = span{(1,0)} direct_sum span{(0,1)}.
EX12 Complex numbers C as a real vector space: dimension 2; as a complex vector space: dimension 1.
EX13 Finite field F_p^n over F_p: dimension n and p^n vectors.
EX14 R with operation x⊕y=x+y+1 and usual scalar multiplication: not a vector space under those mixed operations.
EX15 Any nonzero linear functional on finite-dimensional V has kernel of codimension 1.
EX16 Zero vector space {0} has empty basis and dimension 0.

16. SEMANTIC EQUIVALENCE
SEM01 vector space == linear space (standard mathematical usage).
SEM02 linear span == span.
SEM03 linearly independent == independent (when linear context explicit).
SEM04 linear transformation == linear map (common contexts).
SEM05 null space == kernel (for linear operators/matrices).
SEM06 column space == image (for a matrix-associated map with standard coordinate conventions).
SEM07 rank == dimension(image) (for linear maps where dimension defined).
SEM08 nullity == dimension(kernel) (for finite-dimensional settings).
SEM09 vector space != affine space.
SEM10 vector space != inner-product space (additional structure).
SEM11 vector space != normed space (additional structure).
SEM12 vector space != module over arbitrary ring (field requirement distinguishes).

17. KNOWLEDGE GRAPH TRIPLES
(VectorSpace, isA, AlgebraicStructure)
(VectorSpace, definedOver, Field)
(VectorSpace, hasUnderlyingAdditiveStructure, AbelianGroup)
(Subspace, isA, VectorSpace)
(Subspace, subsetOf, VectorSpace)
(Basis, spans, VectorSpace)
(Basis, satisfies, LinearIndependence)
(Dimension, measuredBy, BasisCardinality)
(LinearMap, preserves, VectorAddition)
(LinearMap, preserves, ScalarMultiplication)
(Kernel, isA, Subspace)
(Image, isA, Subspace)
(QuotientSpace, constructedFrom, VectorSpace)
(DualSpace, consistsOf, LinearFunctional)

18. CONFIDENCE
Mathematical axioms and explicitly stated standard theorems: confidence 1.00 conditional on formal assumptions.
Examples: confidence 1.00 where independently algebraically verifiable.
Source-page fidelity: unverified; do not assign an invented source confidence.
Machine inference confidence must not replace a proof obligation.

19. FLOW_HASH
canonicalise: UTF-8; LF newlines; normalise stable field ordering; preserve mathematical symbols and source URL.
hash_algorithm: SHA-256
flow_hash: COMPUTE_ON_INGEST (no fabricated digest)

20. UPDATE_MODEL
version: 4.0
change_detection: compare canonical content hashes; log changes to axioms, definitions, assumptions and inference rules.
revision_policy: increment version for semantic changes; preserve provenance and previous revision identifiers.
validation_on_update: rerun typing checks, axiom checks, example verification and rule consistency.

21. AI_CONSUMPTION
Parse entities as typed nodes and relations as directed edges.
Treat mathematical implications as conditional rules, not empirical predictions.
Check field and dimension assumptions before executing derived rules.
Distinguish definitions, theorems, examples, aliases and counterexamples.
Maintain provenance to SOURCE_URL while marking exact source-page alignment unverified.
Use proof obligations for every nontrivial derived claim.
Do not silently assume norms, metrics, inner products or topologies.

END FLOW ALGEBRA