Rational dependence¶

The property that a finite collection of numbers satisfies a nontrivial linear relation with rational coefficients.

Version: v1 · 2026-09-08 · History
Domain-specific #: 6401
Origin domain: linear algebra over fields
Subdomain: linear algebra over fields

Core Idea¶

The coefficient field is specifically the rationals, zero coefficients are allowed but not all may vanish, dependence of numbers as vectors over Q differs from algebraic dependence by polynomial relations and pairwise irrational ratios do not guarantee larger-set independence. The numbers are treated as vectors in the real or complex numbers viewed as a vector space over Q; an integer or rational coefficient vector in the kernel of their linear-combination map witnesses dependence. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

Scope of Application¶

Rational dependence belongs to linear algebra over fields and is useful where the analyst can specify the typed linear algebra over fields carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, then evaluate the finite or declared infinite collection of real or complex numbers, scalar field Q, rational coefficient tuple, not-all-zero condition, vanishing linear combination, equivalent integer relation after clearing denominators, witness and minimal relation, rational span and rank, independence negation and distinction from algebraic and real-linear dependence are explicit.

Clarity¶

The abstraction clarifies a crowded vocabulary by making the finite or declared infinite collection of real or complex numbers, scalar field Q, rational coefficient tuple, not-all-zero condition, vanishing linear combination, equivalent integer relation after clearing denominators, witness and minimal relation, rational span and rank, independence negation and distinction from algebraic and real-linear dependence are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test.

Manages Complexity¶

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Rational dependence. Rational dependence compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning¶

Identify the carrier. State what the elements, states, objects, or observations are: the typed linear algebra over fields carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the finite or declared infinite collection of real or complex numbers, scalar field Q, rational coefficient tuple, not-all-zero condition, vanishing linear combination, equivalent integer relation after clearing denominators, witness and minimal relation, rational span and rank, independence negation and distinction from algebraic and real-linear dependence are explicit independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of linear algebra over fields because they reuse the typed linear algebra over fields carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, The numbers are treated as vectors in the real or complex numbers viewed as a vector space over Q; an integer or rational coefficient vector in the kernel of their linear-combination map witnesses dependence., and type the carrier, state every parameter and convention in the definition, test that the finite or declared infinite collection of real or complex numbers, scalar field Q, rational coefficient tuple, not-all-zero condition, vanishing linear combination, equivalent integer relation after clearing denominators, witness and minimal relation, rational span and rank, independence negation and distinction from algebraic and real-linear dependence are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Rational dependence Domain-specific

Parents (1) — more general patterns this builds on

Rational dependence is a kind of Relation Prime

The proposed strict upward parent is prime:relation.

Hierarchy path (1) — routes to 1 parentless root

Rational dependence → Relation

Neighborhood in Abstraction Space¶

Rational dependence sits in a crowded region of the domain-specific corpus (20^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Matrix Structure & Linear Maps (48 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-09-08