Basis Ontology

FLOW ALGEBRA
TITLE: Basis 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-ontology-2.html
SOURCE_ALIGNMENT: Independently constructed; source page not verified.
AUTHORS: Mark Higgins — Sky (Recursive AI)
SITE: https://kirethresearch.org

01. PURPOSE
Model bases as typed mathematical objects and support sound deductions about span, independence, coordinates, dimension and transformations.

02. TYPE SYSTEM
Field K; VectorSpace V over K; Vector v in V; IndexedFamily B : I -> V; FiniteFamily B=(b1,...,bn); LinearCombination sum_i a_i*b_i with finite support; Subspace W <= V; LinearMap T:V->W; OrderedBasis; CoordinateTuple K^n; Dimension cardinal; Matrix.

03. CORE DEFINITIONS
BASIS(B,V) := LINEARLY_INDEPENDENT(B,V) AND SPANS(B,V).
SPANS(B,V) := every v in V is a finite K-linear combination of elements of B.
LINEARLY_INDEPENDENT(B,V) := for every finitely supported coefficient family a, sum_i a_i*b_i=0 implies all a_i=0.
HAMEL_BASIS := BASIS(B,V) with finite-support representations, even for infinite-dimensional V.
ORDERED_BASIS := basis with specified index ordering, required for coordinate tuples.
STANDARD_BASIS(K^n) := (e1,...,en), where e_i has coordinate 1 at i and 0 elsewhere.
EMPTY_BASIS := basis of the zero-dimensional zero vector space.

04. ENTITY ATTRIBUTES
basis_id: string
ambient_space: VectorSpace
scalar_field: Field
vectors: IndexedFamily
ordering: optional ordering
cardinality: finite integer or infinite cardinal
independent: boolean
spanning: boolean
coordinate_system: optional map
provenance: source identifier
confidence: verified | inferred | unverified

05. TYPED RELATIONSHIPS
BASIS_OF(B,V): Basis x VectorSpace
CONTAINS_VECTOR(B,v): Basis x Vector
SPANS(B,V): VectorFamily x VectorSpace
INDEPENDENT_IN(B,V): VectorFamily x VectorSpace
COORDINATES_OF(v,B,c): Vector x OrderedBasis x CoordinateTuple
HAS_DIMENSION(V,d): VectorSpace x Cardinal
IS_SUBSPACE_OF(W,V): Subspace x VectorSpace
IMAGE_UNDER(T,B,C): LinearMap x Basis x VectorFamily
CHANGE_OF_BASIS(P,B,C): InvertibleMatrix x OrderedBasis x OrderedBasis
EXTENDS_TO_BASIS(S,B,V): IndependentFamily x Basis x VectorSpace

06. AXIOMS AND INVARIANTS
A01. Every basis is a family of vectors in its ambient space.
A02. A basis is linearly independent.
A03. A basis spans its ambient space.
A04. Each vector has a unique finite linear-combination representation in a basis.
A05. All bases of the same vector space have equal cardinality (assuming standard choice principles for arbitrary spaces).
A06. A basis never contains the zero vector.
A07. The empty family is a basis precisely for the zero vector space.
A08. A finite-dimensional space of dimension n has bases of exactly n vectors.
A09. Ordering is not required for being a basis but is required for coordinate tuples.
A10. A vector family may span without being independent, or be independent without spanning.

07. FORMAL RULES
R01 IF BASIS(B,V) THEN SPANS(B,V).
R02 IF BASIS(B,V) THEN LINEARLY_INDEPENDENT(B,V).
R03 IF SPANS(B,V) AND LINEARLY_INDEPENDENT(B,V) THEN BASIS(B,V).
R04 IF BASIS(B,V) AND v IN V THEN EXISTS_UNIQUE_FINITE_COORDINATES(v,B).
R05 IF BASIS(B,V) AND FINITE(B) THEN DIM(V)=CARDINALITY(B).
R06 IF BASIS(B,V) AND BASIS(C,V) THEN CARDINALITY(B)=CARDINALITY(C).
R07 IF BASIS(B,V) AND ZERO_VECTOR_IN(B) THEN CONTRADICTION.
R08 IF DIM(V)=0 THEN BASIS(EMPTY,V).
R09 IF BASIS(EMPTY,V) THEN DIM(V)=0.
R10 IF DIM(V)=n AND INDEPENDENT(B,V) AND CARDINALITY(B)=n THEN BASIS(B,V).
R11 IF DIM(V)=n AND SPANS(B,V) AND CARDINALITY(B)=n THEN BASIS(B,V).
R12 IF DIM(V)=n AND CARDINALITY(B)>n AND B FINITE THEN NOT_INDEPENDENT(B,V).
R13 IF DIM(V)=n AND CARDINALITY(B)<n THEN NOT_SPANNING(B,V).
R14 IF BASIS(B,V) AND T:V->W IS_LINEAR_ISOMORPHISM THEN BASIS(T(B),W).
R15 IF BASIS(B,V) AND T:V->W IS_LINEAR THEN SPANS(T(B),IMAGE(T)).
R16 IF BASIS(B,V) AND T IS_LINEAR_INJECTIVE THEN INDEPENDENT(T(B),W).
R17 IF BASIS(B,V) AND T IS_LINEAR_SURJECTIVE THEN SPANS(T(B),W).
R18 IF BASIS(B,V) AND T IS_LINEAR_BIJECTIVE THEN BASIS(T(B),W).
R19 IF BASIS(B,V) AND ORDERED(B) THEN COORDINATE_MAP_B:V->K^n IS_LINEAR_ISOMORPHISM (finite n).
R20 IF ORDERED_BASES(B,C,V) THEN CHANGE_OF_BASIS_MATRIX(B,C) IS_INVERTIBLE.
R21 IF B BASIS(V) AND C BASIS(V) THEN CHANGE_OF_BASIS(B,C) COMPOSE CHANGE_OF_BASIS(C,B)=IDENTITY.
R22 IF B BASIS(V) AND v=sum_i a_i*b_i THEN COORDINATES(v,B)=(a_i) IN GIVEN_ORDER.
R23 IF B BASIS(V) AND v=0 THEN ALL_COORDINATES(v,B)=0.
R24 IF B BASIS(V) AND v=w THEN COORDINATES(v,B)=COORDINATES(w,B).
R25 IF B BASIS(V) AND a,b IN K THEN COORDINATES(a*v+b*w,B)=a*COORDINATES(v,B)+b*COORDINATES(w,B).
R26 IF S INDEPENDENT_IN V THEN S EXTENDS_TO_A_BASIS_OF V (standard choice principles for general V).
R27 IF S SPANS V THEN SOME_SUBFAMILY_OF_S IS_A_BASIS_OF V (standard choice principles for general V).
R28 IF W SUBSPACE_OF V AND BASIS(C,W) THEN C IS_INDEPENDENT_IN V.
R29 IF BASIS(B,V) AND W=SPAN(S) THEN BASIS_OF_W MAY_BE_EXTRACTED_FROM_S (finite-dimensional case, or under choice).
R30 IF V=K^n THEN STANDARD_BASIS(K^n) IS_A_BASIS.
R31 IF B ORTHONORMAL_BASIS(V) THEN BASIS(B,V) (inner-product space; Hilbert-basis caveat below).
R32 IF B FINITE_ORTHONORMAL_SET AND SPANS(B,V) THEN ORTHONORMAL_BASIS(B,V).
R33 IF MATRIX_A HAS_N_COLUMNS AND RANK(A)=N THEN COLUMNS_A ARE_INDEPENDENT.
R34 IF MATRIX_A IS_N_BY_N AND DET(A)!=0 THEN COLUMNS_A FORM_BASIS(K^n).
R35 IF MATRIX_A IS_N_BY_N AND DET(A)=0 THEN COLUMNS_A DO_NOT_FORM_BASIS(K^n).
R36 IF B BASIS(V) AND C BASIS(W) THEN BASIS_OF_DIRECT_SUM(V,W) IS EMBEDDED_UNION(B,C).

08. ALGORITHMS
FINITE_BASIS_TEST(vectors, field):
  assert all vectors have same dimension and scalar field
  form matrix A with supplied vectors as columns
  compute rank(A)
  independent := rank(A) == number_of_columns(A)
  spans_ambient := rank(A) == ambient_dimension
  return independent AND spans_ambient

COORDINATES(v, ordered_basis B):
  solve B_matrix*c=v over K
  require unique solution
  return c

BASIS_EXTRACTION(S):
  row-reduce column matrix of S over K
  select original columns corresponding to pivot columns
  return basis for span(S)

BASIS_EXTENSION(S,V):
  require independent S in finite-dimensional V
  add vectors outside current span until dimension reached
  return extended basis

CHANGE_OF_BASIS(B,C):
  require ordered bases of same finite-dimensional V
  define P_C_from_B := inverse(C_matrix)*B_matrix
  return P_C_from_B

09. EXAMPLES
E01 V=R^2; B=((1,0),(0,1)); basis=true.
E02 V=R^2; B=((1,1),(1,-1)); determinant=-2; basis=true.
E03 V=R^2; B=((1,0),(2,0)); basis=false; dependent and nonspanning.
E04 V=R^2; B=((1,0),(0,1),(1,1)); spans=true; independent=false.
E05 V=R^3; B=((1,0,0),(0,1,0),(0,0,1)); basis=true.
E06 V=R^2; B=((0,0),(1,0)); basis=false; includes zero vector.
E07 V={0}; B=(); basis=true.
E08 V=P_2(R); B=(1,x,x^2); basis=true; dimension=3.
E09 V=P_2(R); B=(1,x); basis=false; insufficient span.
E10 V=M_2(R); B=(E11,E12,E21,E22); basis=true; dimension=4.
E11 V=R^2; B=((1,1),(1,-1)); v=(4,2) has coordinates (3,1).
E12 V=R^2; standard basis and swapped ordered basis give distinct coordinate tuples.
E13 V=C as a vector space over R; B=(1,i); basis=true; dimension=2.
E14 V=C as a vector space over C; B=(1); basis=true; dimension=1.
E15 V=R^3; W={(x,y,0)}; B=((1,0,0),(0,1,0)); basis of W, not V.
E16 V=R^2; B=((2,0),(0,3)); basis=true; orthonormal=false.

10. SEMANTIC EQUIVALENCE
basis <=> linearly independent spanning family (same vector space and field).
ordered basis <=> basis equipped with a specified ordering.
coordinate basis: ambiguous term; resolve to ordered basis in context.
Hamel basis: algebraic basis using finite-support representations.
Schauder basis: topological basis using convergent expansions; NOT equivalent to Hamel basis.
Hilbert orthonormal basis: complete orthonormal family; infinite expansions need not be finite.
standard basis: depends on the specified coordinate vector space.

11. KNOWLEDGE GRAPH TRIPLES
(Basis, is_a, VectorFamily)
(Basis, defined_by, LinearIndependence)
(Basis, defined_by, Spanning)
(Basis, determines, UniqueCoordinates)
(Basis, determines, Dimension)
(OrderedBasis, enables, CoordinateRepresentation)
(ChangeOfBasisMatrix, relates, OrderedBases)
(LinearIsomorphism, preserves, BasisProperty)
(HamelBasis, distinct_from, SchauderBasis)
(StandardBasis, instance_of, Basis)

12. VALIDATION
V01 Reject a purported basis containing vectors outside V.
V02 Reject a purported basis with any nontrivial finite linear dependence.
V03 Reject a purported basis that fails to span V.
V04 Do not infer basis solely from correct cardinality without independence or spanning.
V05 Verify scalar field: dimensions and basis membership can differ with field.
V06 Distinguish set/family from ordered basis when storing coordinates.
V07 Check change-of-basis matrix orientation explicitly.
V08 Do not confuse row-space basis with column-space basis.
V09 Infinite-dimensional basis arguments may require axiom of choice.
V10 Never conflate a Hamel basis with a topological Schauder or Hilbert basis.
V11 Mark floating-point rank tests as tolerance-dependent, not exact proofs.
V12 Preserve ambient-space and subspace context for all basis claims.

13. CONFIDENCE
Mathematical theorem verified by exact proof: 1.00.
Symbolic inference with checked premises: high, subject to assumptions.
Numerical rank or independence check: confidence depends on precision and conditioning.
Source alignment: unverified until page text is compared.
Never use a numerical confidence score as a substitute for a proof.

14. FLOW_HASH
Canonicalise UTF-8 content, stable section ordering, mathematical symbols and normalised line endings.
FLOW_HASH := SHA256(canonicalised_flow_body).
Hash must be computed on publication; no fabricated digest is supplied.

15. UPDATE_MODEL
On revision, increment revision identifier; preserve source URL and authorship.
Revalidate axioms, examples, type signatures and dependent ontology links.
Record change history, review status and new computed FLOW_HASH.

16. AI_CONSUMPTION
Parse typed entities before evaluating relationships.
Verify field, ambient vector space and basis ordering.
Apply inference rules only after validating premises and scope.
Prefer exact arithmetic for algebraic proofs.
Surface counterexamples and unmet assumptions.
Return results with theorem or rule IDs and provenance.
Never assert that the linked page was inspected when it was not.

17. RELATED ONTOLOGIES
Vector Space; Basis Vector; Span; Linear Combination; Linear Independence;
Dimension; Matrix; Rank; Linear Transformation; Coordinate System;
Eigenvector; Subspace; Change of Basis; Scalar.

END FLOW ALGEBRA