Linear Transformation Ontology

FLOW ALGEBRA
TITLE: Linear Transformation Ontology
VERSION: 4.0
SOURCE_URL: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/linear-algebra-entities/linear-transformation-ontology-2.html
SOURCE_ALIGNMENT: Independently constructed mathematical model; page-specific alignment not verified.
AUTHORS: Mark Higgins — Sky (Recursive AI)
PROJECT: Kireth Semantic Library
DATE: 2026-10-13

01 | PURPOSE
Represent linear maps between vector spaces, their defining axioms, composition, matrices, kernels, images, rank, nullity, invertibility, and semantic inference.

02 | TYPED ENTITIES
ENTITY VectorSpace<V, F> : additive abelian group with scalar action by field F.
ENTITY LinearTransformation<T, V, W, F> : function T: V -> W preserving addition and scalar multiplication.
ENTITY Vector<v, V> : element of V.
ENTITY Scalar<a, F> : element of F.
ENTITY Basis<B, V> : linearly independent spanning family of V.
ENTITY Matrix<A, m, n, F> : m-by-n array over F.
ENTITY Kernel<K, T> : {v in V | T(v)=0_W}.
ENTITY Image<I, T> : {T(v) | v in V}.
ENTITY Rank<r, T> : dimension of Image(T), when finite.
ENTITY Nullity<q, T> : dimension of Kernel(T), when finite.
ENTITY Isomorphism<T> : bijective linear transformation.
ENTITY Endomorphism<T> : linear map V -> V.
ENTITY Automorphism<T> : invertible endomorphism.
ENTITY Projection<P> : endomorphism with P∘P=P.
ENTITY IdempotentMatrix<A> : square matrix with A²=A.
ENTITY Eigenpair<lambda,v,T> : v != 0 and T(v)=lambda*v, for T:V->V.

03 | FORMAL AXIOMS
AXIOM LT01: forall u,v in V: T(u+v)=T(u)+T(v).
AXIOM LT02: forall a in F, v in V: T(a*v)=a*T(v).
AXIOM LT03: forall a,b in F, u,v in V: T(a*u+b*v)=a*T(u)+b*T(v).
AXIOM LT04: T(0_V)=0_W.
AXIOM LT05: T(-v)=-T(v).
AXIOM LT06: Domain and codomain share the same scalar field F (unless a separate semilinear structure is specified).
AXIOM LT07: Linear maps are determined by their values on a basis of V.

04 | CORE OPERATIONS
OP APPLY(T,v) -> T(v) in W.
OP ADD(T,S) -> (T+S)(v)=T(v)+S(v), when codomains agree.
OP SCALE(a,T) -> (aT)(v)=a*T(v).
OP COMPOSE(S,T) -> (S∘T)(v)=S(T(v)), when types match.
OP IDENTITY(V) -> id_V(v)=v.
OP RESTRICT(T,U) -> T|_U: U -> W, for subspace U subseteq V.
OP MATRIX(T,B,C) -> [T]_(C<-B), with bases B of V and C of W.
OP KERNEL(T) -> ker(T).
OP IMAGE(T) -> im(T).
OP PREIMAGE(T,Y) -> {v | T(v) in Y}.

05 | MATHEMATICAL RELATIONSHIPS
REL LinearTransformation IS_A Function.
REL LinearTransformation HAS_DOMAIN VectorSpace.
REL LinearTransformation HAS_CODOMAIN VectorSpace.
REL LinearTransformation PRESERVES VectorAddition.
REL LinearTransformation PRESERVES ScalarMultiplication.
REL LinearTransformation HAS_KERNEL Subspace.
REL LinearTransformation HAS_IMAGE Subspace.
REL LinearTransformation REPRESENTED_BY Matrix GIVEN OrderedBases.
REL LinearTransformation COMPOSES_WITH LinearTransformation WHEN CodomainDomainCompatible.
REL Isomorphism IS_A LinearTransformation.
REL Automorphism IS_A Endomorphism.
REL Eigenvector ASSOCIATED_WITH Eigenvalue AND Endomorphism.
REL Rank MEASURES DimensionOfImage.
REL Nullity MEASURES DimensionOfKernel.

06 | MATRIX REPRESENTATION
For ordered bases B=(v_1,...,v_n) and C=(w_1,...,w_m), column j of A=[T]_(C<-B) is [T(v_j)]_C.
COORDINATE_RULE: [T(v)]_C = A [v]_B.
DIMENSION_RULE: A has m rows and n columns when dim(W)=m and dim(V)=n.
COMPOSITION_RULE: [S∘T]_(D<-B)=[S]_(D<-C)[T]_(C<-B).
BASIS_CHANGE_RULE: A_new=Q^(-1) A_old P, where P converts new domain coordinates to old and Q converts new codomain coordinates to old.
ENDOMORPHISM_SIMILARITY: A_new=P^(-1) A_old P when the same basis change is used in domain and codomain.

07 | INFERENCE RULES
RULE LT-R01: LINEAR(T) => T(0)=0.
RULE LT-R02: LINEAR(T) => T(-v)=-T(v).
RULE LT-R03: LINEAR(T) => T(sum_i a_i*v_i)=sum_i a_i*T(v_i), for finite sums.
RULE LT-R04: LINEAR(T) AND LINEAR(S) AND SAME_DOMAIN_CODOMAIN => LINEAR(T+S).
RULE LT-R05: LINEAR(T) AND a in F => LINEAR(aT).
RULE LT-R06: LINEAR(T:V->W) AND LINEAR(S:W->X) => LINEAR(S∘T).
RULE LT-R07: LINEAR(T) => KERNEL(T) IS_SUBSPACE_OF V.
RULE LT-R08: LINEAR(T) => IMAGE(T) IS_SUBSPACE_OF W.
RULE LT-R09: LINEAR(T) AND ker(T)={0} => INJECTIVE(T).
RULE LT-R10: LINEAR(T) AND INJECTIVE(T) => ker(T)={0}.
RULE LT-R11: LINEAR(T) AND im(T)=W => SURJECTIVE(T).
RULE LT-R12: LINEAR(T) AND INJECTIVE(T) AND SURJECTIVE(T) => ISOMORPHISM(T).
RULE LT-R13: ISOMORPHISM(T) => LINEAR(T^(-1)).
RULE LT-R14: FINITE_DIM(V) AND LINEAR(T) => dim(V)=rank(T)+nullity(T).
RULE LT-R15: dim(V)=dim(W)<infinity AND LINEAR(T:V->W) => (INJECTIVE(T) IFF SURJECTIVE(T)).
RULE LT-R16: LINEAR(T) AND BASIS(B,V) => T DETERMINED_BY {T(b): b in B}.
RULE LT-R17: LINEAR(T) AND SPANS(S,V) => SPANS(T(S),im(T)).
RULE LT-R18: LINEAR(T) AND INDEPENDENT(T(B)) AND BASIS(B,V) => INJECTIVE(T).
RULE LT-R19: LINEAR(T) AND T(B) BASIS_OF W AND B BASIS_OF V => ISOMORPHISM(T).
RULE LT-R20: LINEAR(T) AND FINITE_DIM(V) => rank(T)<=min(dim(V),dim(W)) when W finite-dimensional.
RULE LT-R21: LINEAR(T) AND T∘T=T => PROJECTION(T).
RULE LT-R22: PROJECTION(T) => V=im(T) direct_sum ker(T).
RULE LT-R23: LINEAR(T) AND T∘T=0 => im(T) subseteq ker(T).
RULE LT-R24: LINEAR(T) AND T(v)=lambda*v AND v!=0 => EIGENVECTOR(v,T,lambda).
RULE LT-R25: LINEAR(T) AND T(v)=0 AND v!=0 => ZERO_EIGENVALUE(T), for endomorphisms.
RULE LT-R26: LINEAR(T) AND FINITE_DIM(V) AND T:V->V => det([T])!=0 IFF INVERTIBLE(T).
RULE LT-R27: LINEAR(T) AND FINITE_DIM(V) AND T:V->V => rank(T)=dim(V) IFF INVERTIBLE(T).
RULE LT-R28: LINEAR(T) AND LINEAR(S) AND MATCHING_TYPES => rank(S∘T)<=min(rank(S),rank(T)), for finite ranks.
RULE LT-R29: LINEAR(T) AND U SUBSPACE_OF V => LINEAR(T|_U).
RULE LT-R30: LINEAR(T) AND Y SUBSPACE_OF W => PREIMAGE(T,Y) IS_SUBSPACE_OF V.
RULE LT-R31: LINEAR(T) AND dim(V)=n<infinity => dim(im(T))<=n.
RULE LT-R32: LINEAR(T) AND KERNEL(T)=V => ZERO_TRANSFORMATION(T).
RULE LT-R33: LINEAR(T) AND im(T)={0} => ZERO_TRANSFORMATION(T).
RULE LT-R34: LINEAR(T) AND LINEAR(S) AND S∘T=id_V => INJECTIVE(T).
RULE LT-R35: LINEAR(T) AND LINEAR(S) AND T∘S=id_W => SURJECTIVE(T).
RULE LT-R36: LINEAR(T) AND LINEAR(S) AND S∘T=id_V AND T∘S=id_W => INVERSE_PAIR(T,S).

08 | VALIDATION CONSTRAINTS
CHECK VC01: Domain and codomain must be vector spaces over a compatible field.
CHECK VC02: Addition preservation must hold for all pairs of domain vectors.
CHECK VC03: Homogeneity must hold for all domain vectors and scalars.
CHECK VC04: Matrix row count equals codomain dimension; column count equals domain dimension.
CHECK VC05: Kernel membership requires domain typing and zero codomain image.
CHECK VC06: Image membership requires a domain preimage.
CHECK VC07: Rank-nullity only when domain dimension is finite.
CHECK VC08: Determinant and eigenvalue rules require finite-dimensional endomorphisms.
CHECK VC09: Invertibility requires a two-sided inverse, not merely a one-sided map without further hypotheses.
CHECK VC10: Never infer linearity solely from T(0)=0.
CHECK VC11: Affine maps T(v)=Av+b are linear only when b=0.
CHECK VC12: Coordinate formulas must declare their ordered bases.
CHECK VC13: Matrix multiplication order in composition must be preserved.
CHECK VC14: Distinguish infinite-dimensional algebraic rank from numerical matrix rank.
CHECK VC15: For complex spaces, conjugate-linear maps are not complex-linear unless separately justified.
CHECK VC16: Eigenvectors must be nonzero.

09 | WORKED EXAMPLES
EXAMPLE E01: T:R^2->R^2, T(x,y)=(2x,3y); matrix diag(2,3), rank 2, nullity 0.
EXAMPLE E02: T:R^3->R^2, T(x,y,z)=(x+y,z); kernel={(t,-t,0):t in R}, rank 2, nullity 1.
EXAMPLE E03: P:R^2->R^2, P(x,y)=(x,0); P^2=P, im(P)=span((1,0)), ker(P)=span((0,1)).
EXAMPLE E04: D:P_3(R)->P_2(R), D(p)=p'; rank 3, nullity 1.
EXAMPLE E05: Evaluation E:P_2(R)->R, E(p)=p(1); linear, rank 1, nullity 2.
EXAMPLE E06: T:R^2->R^2, T(x,y)=(x+1,y) is not linear since T(0,0)!=(0,0).
EXAMPLE E07: Rotation by theta on R^2 is linear with matrix [[cos(theta),-sin(theta)],[sin(theta),cos(theta)]].
EXAMPLE E08: Trace tr:M_n(F)->F is linear.
EXAMPLE E09: Transpose M_(m,n)(F)->M_(n,m)(F) is linear.
EXAMPLE E10: T:R^2->R, T(x,y)=x-y; surjective, not injective.
EXAMPLE E11: Inclusion i:U->V for subspace U is injective linear.
EXAMPLE E12: Zero map Z:V->W has rank 0; nullity dim(V) when finite.
EXAMPLE E13: T:R^2->R^2 with A=[[1,1],[0,1]] is invertible, det(A)=1.
EXAMPLE E14: Projection onto a line is linear when the projection is through the origin.
EXAMPLE E15: Complex conjugation C->C is real-linear but not complex-linear.
EXAMPLE E16: A change-of-basis matrix represents identity between two coordinate descriptions, not a change to the underlying vector.

10 | SEMANTIC_EQUIVALENCE
ALIAS linear map == linear transformation.
ALIAS vector space homomorphism == linear transformation.
ALIAS linear operator == endomorphism (common convention; usage may vary).
ALIAS null space == kernel (for a represented matrix or map).
ALIAS range == image (in linear algebra; disambiguate from codomain).
ALIAS one-to-one == injective.
ALIAS onto == surjective.
ALIAS nonsingular linear operator == invertible endomorphism (finite-dimensional context).
NON_EQUIVALENCE: image != codomain in general.
NON_EQUIVALENCE: affine transformation != linear transformation in general.
NON_EQUIVALENCE: matrix != basis-independent linear transformation.

11 | KNOWLEDGE GRAPH TRIPLES
(LinearTransformation, preserves, VectorAddition)
(LinearTransformation, preserves, ScalarMultiplication)
(LinearTransformation, hasDomain, VectorSpace)
(LinearTransformation, hasCodomain, VectorSpace)
(Kernel, isSubspaceOf, Domain)
(Image, isSubspaceOf, Codomain)
(MatrixRepresentation, dependsOn, OrderedBases)
(RankNullityTheorem, relates, Rank)
(RankNullityTheorem, relates, Nullity)
(Isomorphism, isA, BijectiveLinearTransformation)
(Projection, satisfies, Idempotence)
(LinearTransformation, composesWith, LinearTransformation)

12 | CONFIDENCE
CONFIDENCE formal_axioms: 1.00
CONFIDENCE standard_finite_dimensional_theorems: 1.00
CONFIDENCE examples_symbolic: 1.00
CONFIDENCE semantic_aliases: 0.96
CONFIDENCE source_page_alignment: unverified
POLICY: Confidence scores describe model assertions, not empirically calibrated probabilities.

13 | FLOW_HASH
FLOW_HASH.algorithm: SHA-256
FLOW_HASH.scope: canonical UTF-8 content excluding FLOW_HASH.value itself
FLOW_HASH.value: GENERATED_ON_INGEST
FLOW_HASH.policy: Recalculate after canonicalisation and every material update; never invent a digest.

14 | UPDATE_MODEL
UPDATE_MODEL.version: 4.0
UPDATE_MODEL.change_policy: append change log; retain source provenance; revalidate dependent theorems.
UPDATE_MODEL.trigger: corrected definition, new mathematical proof, ontology alignment, or schema revision.
UPDATE_MODEL.dependencies: VectorSpaceOntology, MatrixOntology, KernelOntology, RankOntology, BasisOntology, EigenvectorOntology.
UPDATE_MODEL.validation: axioms + type checks + example verification + semantic mapping review.

15 | AI_CONSUMPTION
AI_CONSUMPTION.parse: typed mathematical assertions and guarded inference rules.
AI_CONSUMPTION.reasoning: apply only when preconditions, field, dimensions, and map types are satisfied.
AI_CONSUMPTION.provenance: preserve ontology URL and independently-constructed status.
AI_CONSUMPTION.hallucination_guard: do not infer finite dimension, invertibility, surjectivity, or diagonalisation without evidence.
AI_CONSUMPTION.output: definitions, derived claims, applicable rules, proof obligations, and confidence metadata.
AI_CONSUMPTION.interoperability: map to vector, matrix, basis, kernel, image, rank, and eigenstructure entities.

END FLOW ALGEBRA