{ "model": { "rank": 7048, "code": "thing-q207643", "model_id": "vr.tr.linear-map", "name": "linear map", "purpose": "Let an agent handle linear maps by definition, representation as matrices, properties and applications.", "family": "Thing Registry", "category": "Cross-cutting context", "status": "research-draft", "kind": "thing", "plane": "XCT", "domain": "XCT.QLT", "industry": "", "version": "", "url": "/models/thing/q207643/", "tier": 2, "score": 57, "payload": { "layer": "wikidata", "aliases": [ "quantum Fourier transform", "Householder transformation", "integral transform", "Hecke operator", "Hodge star operator", "pseudo-differential operator", "linear operator", "affine tensor", "unitary transformation", "linear functional", "transpose of a linear map", "linear endomorphism", "Weingarten operator", "orthogonal transformation", "Delta operator", "pseudoreflection", "squeeze mapping", "linear isomorphism", "vectorization", "sine and cosine transforms", "continuous wavelet transform", "Jacobi transform", "oscillatory integral operator", "integral operator", "Linear canonical transformation", "fractional derivative", "shape operator", "algebraic operator", "Eigenoperator", "discontinuous linear operator", "antisymmetrizer", "composition operator", "densely defined operator", "multiplication operator", "positive linear functional", "singular integral operator on a closed curve", "closed operator" ], "aliasCount": 37, "merged": 37, "knownIn": 57, "facets": null, "markers": [], "lexicalClass": "", "senseRank": null, "alsoRegisteredAs": null, "source": { "dataset": "wikidata", "item": "Q207643", "url": "https://www.wikidata.org/wiki/Q207643", "license": "CC0 1.0" } }, "research": { "vercy": "1.0-draft", "publication": { "status": "research-draft", "adjudicationStatus": "unreviewed", "publishableCanonical": false, "generatedAt": "2026-09-11T11:45:32Z", "providers": [ "Claude" ], "breadth": "written by Claude from model knowledge without web access - no source was read, every claim is a lead to verify", "pass": 2, "wave": 1, "engine": "claude" }, "metaModel": { "id": "THING-Q207643", "registryId": "vr.tr.linear-map", "name": "linear map", "version": "0.2.0-wave.1", "entryKind": "thing", "family": "Thing Registry", "domain": [ "XCT.QLT" ], "status": "research-draft" }, "canonicalUrl": "https://ver.cy/models/thing/q207643/", "model": { "registry_id": "vr.tr.linear-map", "name": "linear map", "purpose": "Let an agent handle linear maps by definition, representation as matrices, properties and applications.", "definition": "A function between vector spaces that preserves addition and scalar multiplication, such as rotations, projections and matrix transformations, and in analysis operators such as integral transforms and the Fourier transform.", "what_it_is_for": "Linear algebra, physics, engineering, computer graphics and data science.", "affordances": [ "test whether a map is linear", "represent linear maps as matrices", "compute kernels, images and eigenvalues", "apply transformations in graphics and signals" ], "distinguishing_features": [ "Preserves linear structure", "Represented by matrices in finite dimensions", "Has kernel and image", "Central to linear algebra" ], "appearance": "Written as functions or matrices.", "visual_identification": [ "T(u + v) = T(u) + T(v) and T(cu) = cT(u)", "Matrix representations", "Affine maps add a translation and are not linear" ], "physical_properties": [], "families_and_kinds": [ "linear transformations of finite-dimensional spaces", "linear operators", "integral transforms", "projections and reflections", "Householder transformations" ], "related_models": [ { "relation": "is a kind of", "target": "function and additive map", "why": "category" }, { "relation": "is represented by", "target": "matrix", "why": "representation" }, { "relation": "has", "target": "eigenvalue and kernel", "why": "properties" }, { "relation": "is used in", "target": "computer graphics and quantum mechanics", "why": "applications" } ], "identifiers": [], "standards_and_regulation": [ "ISO 80000-2 notation" ], "failure_modes_and_hazards": [ "Confusing linear and affine maps", "Basis-dependent errors" ], "in_scope": [], "out_of_scope": [], "characteristics": [] }, "sources": [], "structure": { "bundles": [ { "id": "definition", "name": "Definition", "description": "Is it linear.", "rationale": "Tests decide.", "layers": [ { "id": "test", "name": "Test", "description": "Linearity.", "findings": [ { "id": "test-finding", "name": "Test", "description": "Linearity test.", "questions": [ { "text": "Does the map preserve addition and scalar multiplication?", "kind": "boundary" }, { "text": "Can a counterexample be given?", "kind": "action" } ] } ] }, { "id": "spaces", "name": "Spaces", "description": "Domain and codomain.", "findings": [ { "id": "spaces-finding", "name": "Spaces", "description": "Vector spaces.", "questions": [ { "text": "Between which vector spaces does it map?", "kind": "definition" }, { "text": "What are their dimensions?", "kind": "measurement" } ] } ] } ] }, { "id": "representation", "name": "Representation", "description": "Matrices.", "rationale": "Bases matter.", "layers": [ { "id": "matrix", "name": "Matrix", "description": "In a basis.", "findings": [ { "id": "matrix-finding", "name": "Matrix", "description": "Matrix representation.", "questions": [ { "text": "What is the matrix of the map in this basis?", "kind": "measurement" }, { "text": "How does it change with a change of basis?", "kind": "definition" } ] } ] }, { "id": "invariants", "name": "Invariants", "description": "Rank and determinant.", "findings": [ { "id": "invariants-finding", "name": "Invariants", "description": "Invariants.", "questions": [ { "text": "What are its rank, determinant and trace?", "kind": "measurement" }, { "text": "What do they tell us?", "kind": "definition" } ] } ] } ] }, { "id": "structure", "name": "Structure", "description": "Kernel and eigenvalues.", "rationale": "Structure reveals behaviour.", "layers": [ { "id": "kernel", "name": "Kernel", "description": "Kernel and image.", "findings": [ { "id": "kernel-finding", "name": "Kernel", "description": "Kernel and image.", "questions": [ { "text": "What are the kernel and image?", "kind": "definition" }, { "text": "Is the map injective or surjective?", "kind": "boundary" } ] } ] }, { "id": "eigen", "name": "Eigen", "description": "Eigenvalues.", "findings": [ { "id": "eigen-finding", "name": "Eigen", "description": "Eigenvalues.", "questions": [ { "text": "What are its eigenvalues and eigenvectors?", "kind": "measurement" }, { "text": "Is it diagonalisable?", "kind": "boundary" } ] } ] } ] }, { "id": "applications", "name": "Applications", "description": "Uses.", "rationale": "Linear maps are everywhere.", "layers": [ { "id": "graphics", "name": "Graphics", "description": "Transformations.", "findings": [ { "id": "graphics-finding", "name": "Graphics", "description": "Graphics.", "questions": [ { "text": "How are rotations and scalings used in computer graphics?", "kind": "definition" }, { "text": "Why are homogeneous coordinates used for translations?", "kind": "definition" } ] } ] }, { "id": "signals", "name": "Signals", "description": "Transforms.", "findings": [ { "id": "signals-finding", "name": "Signals", "description": "Signal processing.", "questions": [ { "text": "How is the Fourier transform a linear map?", "kind": "definition" }, { "text": "What does that imply?", "kind": "definition" } ] } ] } ] } ] }, "openQuestions": [ "Should linear operators be a separate entry?", "How should matrix representations be linked?", "How should infinite-dimensional cases be handled?" ], "statistics": { "bundles": 4, "layers": 8, "findings": 8, "questions": 16 } }, "draft": { "generator": "vr.draft.v3", "status": "draft-generated", "researched": false, "archetype": "abstract concept", "method": "Written from the archetype playbook - what this kind of thing needs beyond identity and provenance - and from the structure that recurred across 6,333 models already researched by two engines. Applied to this entry by rule. No source was read for this thing and no claim here is researched. This entry carries no facets of its own, so they were inferred from its domain - a guess about a whole domain applied to one thing.", "facetsInferred": true, "nextPass": "A researcher replaces this draft with a sourced specification. Treat every sentence below as a proposal to argue with.", "purpose": "Give an agent a durable, checkable way to recognise a linear map, record what state it is in, and decide what may be done with it.", "whatItIs": "Let an agent handle linear maps by definition, representation as matrices, properties and applications.", "characteristics": { "substance": "abstract", "origin": "conceptual", "agency": "inert" }, "whatYouCanDoWithIt": [ "observed and measured" ], "distinguishingFeatures": [ "Names folded into this entry, which a task may need to split apart again: quantum Fourier transform, Householder transformation, integral transform, Hecke operator, Hodge star operator, pseudo-differential operator, linear operator, affine tensor, unitary transformation, linear functional, transpose of a linear map, linear endomorphism.", "37 finer distinctions are held as aliases rather than separate entries, because telling them apart needs a task that asks for it.", "Described in 57 Wikipedia languages, which is a measure of how widely the thing is known, not of how important it is." ], "openQuestionsForResearch": [ "Which of the bundles below does a real task actually need, and which are ceremony?", "What does this thing have that the facets do not capture at all?", "Which neighbouring kind is most often confused with a linear map, and on what evidence are they told apart?" ], "whatItIsMadeOf": "an abstraction with no physical instance", "physicalCharacter": [ "Does nothing on its own; everything it does, something else did to it.", "These come from the domain this entry sits in rather than from the entry itself, so treat them as a first guess about the whole domain applied to one thing." ], "whatCanBeDoneWithIt": [ "observe it, measure it, record its state" ], "howItIsRecognised": [ "Nothing to see. What is recognised is an instance of it, and which instances count is exactly what is argued about." ], "relatedModels": [ { "relation": "covers", "note": "Finer kinds folded into this entry because telling them apart needs a task that asks for it. Each is a model waiting to be split out when one does.", "targets": [ "quantum Fourier transform", "Householder transformation", "integral transform", "Hecke operator", "Hodge star operator", "pseudo-differential operator", "linear operator", "affine tensor", "unitary transformation", "linear functional", "transpose of a linear map", "linear endomorphism" ] } ], "standing": "Described in 57 Wikipedia languages, which measures how widely it is written about rather than how important or how common it is. 37 finer distinctions are held inside this entry as names rather than as separate models.", "structure": { "bundles": [ { "id": "identity-and-classification", "name": "Identity, naming and classification", "description": "How an agent tells one linear map from another, and a linear map from things that resemble it.", "rationale": "Recognition comes before every other claim. Without stable identity nothing else in the model can be trusted to be about the same thing twice.", "layers": [ { "id": "naming-and-identifiers", "name": "Names and identifiers", "description": "The names this thing goes by and the identifiers that survive translation and time.", "findings": [ { "id": "preferred-name-and-aliases", "name": "Preferred name, aliases and local names", "description": "Which name to use, which names mean the same thing, and which merely sound similar.", "questions": [ { "id": "preferred-name-and-aliases-q01", "text": "What identifies and describes the name of a linear map, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "preferred-name-and-aliases-q02", "text": "Who or what asserted this about the name of a linear map, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "preferred-name-and-aliases-q03", "text": "What may an agent decide or do once the name of a linear map is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] }, { "id": "stable-identifiers", "name": "Stable identifiers and external keys", "description": "Identifiers that keep pointing at this kind of thing across systems and languages.", "questions": [ { "id": "stable-identifiers-q01", "text": "What identifies and describes an identifier for a linear map, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "stable-identifiers-q02", "text": "Who or what asserted this about an identifier for a linear map, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "stable-identifiers-q03", "text": "What may an agent decide or do once an identifier for a linear map is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "classification-and-granularity", "name": "Classification and granularity", "description": "Where a linear map sits among kinds, and how finely a task needs to cut it.", "findings": [ { "id": "kind-and-parents", "name": "Kind, parents and neighbouring kinds", "description": "The classes this thing belongs to and the ones it is next to.", "questions": [ { "id": "kind-and-parents-q01", "text": "What identifies and describes the kind of a linear map, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "kind-and-parents-q02", "text": "Who or what asserted this about the kind of a linear map, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "kind-and-parents-q03", "text": "What may an agent decide or do once the kind of a linear map is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] }, { "id": "distinguishing-features", "name": "Distinguishing features", "description": "What separates a linear map from the things most often confused with it.", "questions": [ { "id": "distinguishing-features-q01", "text": "What identifies and describes what distinguishes a linear map, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "distinguishing-features-q02", "text": "Who or what asserted this about what distinguishes a linear map, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "distinguishing-features-q03", "text": "What may an agent decide or do once what distinguishes a linear map is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] }, { "id": "state-and-lifecycle", "name": "State and lifecycle", "description": "The states a linear map passes through and the events that move it between them.", "rationale": "Most decisions about a thing depend on what state it is in now, which is a claim with a time on it, not a property.", "layers": [ { "id": "lifecycle-stages", "name": "Lifecycle stages", "description": "From coming into existence to ceasing to be one of these.", "findings": [ { "id": "stages-and-transitions", "name": "Stages and transitions", "description": "The stages worth naming and what moves a linear map between them.", "questions": [ { "id": "stages-and-transitions-q01", "text": "What identifies and describes the lifecycle of a linear map, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "stages-and-transitions-q02", "text": "Who or what asserted this about the lifecycle of a linear map, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "stages-and-transitions-q03", "text": "What may an agent decide or do once the lifecycle of a linear map is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "observations-and-status", "name": "Observations and current status", "description": "What is observed about a linear map, how often and by whom.", "findings": [ { "id": "observation-record", "name": "Observation record", "description": "How an observation of a linear map is recorded so that it can be superseded rather than overwritten.", "questions": [ { "id": "observation-record-q01", "text": "What identifies and describes an observation of a linear map, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "observation-record-q02", "text": "Who or what asserted this about an observation of a linear map, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "observation-record-q03", "text": "What may an agent decide or do once an observation of a linear map is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] }, { "id": "definitions-in-contest", "name": "Definitions and who holds them", "description": "What linear map is taken to mean, and by whom.", "rationale": "When a field disagrees about a concept, the disagreement is the content. A model that picks one definition silently destroys the information.", "layers": [ { "id": "competing-definitions", "name": "Competing definitions", "description": "The main readings and the traditions behind them.", "findings": [ { "id": "definition-map", "name": "Definitions and their holders", "description": "Each definition with the school or body that holds it.", "questions": [ { "id": "definition-map-q01", "text": "Which definitions of linear map are in use, and which tradition or body holds each?", "kind": "definition" }, { "id": "definition-map-q02", "text": "What turns on the difference between them in practice?", "kind": "boundary" } ] } ] }, { "id": "operationalisation", "name": "Operationalisation", "description": "How it is measured or applied when it has to be.", "findings": [ { "id": "operational-record", "name": "Measures and proxies", "description": "Instruments and indicators used to stand in for it.", "questions": [ { "id": "operational-record-q01", "text": "How is linear map operationalised or measured in practice, and by what instrument?", "kind": "measurement" }, { "id": "operational-record-q02", "text": "What does that operationalisation leave out, and when does that matter?", "kind": "boundary" } ] } ] } ] }, { "id": "instances-and-use", "name": "Instances, use and consequence", "description": "What counts as an instance of linear map and what follows from calling something that.", "rationale": "Applying a concept is an act with consequences, so a model must say what the label licenses and what it does not.", "layers": [ { "id": "instances", "name": "What counts as an instance", "description": "Clear cases, borderline cases and non-cases.", "findings": [ { "id": "instance-tests", "name": "Tests for an instance", "description": "What would settle whether something falls under it.", "questions": [ { "id": "instance-tests-q01", "text": "What would settle whether something is an instance of linear map?", "kind": "boundary" }, { "id": "instance-tests-q02", "text": "Which borderline cases are argued about, and on what grounds?", "kind": "definition" } ] } ] }, { "id": "consequence", "name": "Consequence of application", "description": "Rights, duties or decisions that follow from the label.", "findings": [ { "id": "consequence-record", "name": "What the label licenses", "description": "What an agent may do once something is classified this way.", "questions": [ { "id": "consequence-record-q01", "text": "What follows practically once something is treated as linear map?", "kind": "action" }, { "id": "consequence-record-q02", "text": "What must an agent not infer from the label alone?", "kind": "action" } ] } ] } ] }, { "id": "provenance-and-evidence", "name": "Provenance, evidence and time", "description": "Where every claim about a linear map came from and when it held.", "rationale": "A claim without a source and a time cannot be superseded, only overwritten, and an agent that overwrites loses the ability to explain itself.", "layers": [ { "id": "source-and-authority", "name": "Source and authority", "description": "Who said it, on what evidence, and how strongly.", "findings": [ { "id": "claim-provenance", "name": "Claim provenance and confidence", "description": "The authority behind each claim about a linear map and how confident it is.", "questions": [ { "id": "claim-provenance-q01", "text": "What identifies and describes a claim about a linear map, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "claim-provenance-q02", "text": "Who or what asserted this about a claim about a linear map, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "claim-provenance-q03", "text": "What may an agent decide or do once a claim about a linear map is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "time-and-versions", "name": "Time, versions and supersession", "description": "When a claim was true, when it was learnt, and what replaced it.", "findings": [ { "id": "validity-and-supersession", "name": "Validity period and supersession", "description": "How an old claim about a linear map is retired without being erased.", "questions": [ { "id": "validity-and-supersession-q01", "text": "What identifies and describes the validity of a claim about a linear map, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "validity-and-supersession-q02", "text": "Who or what asserted this about the validity of a claim about a linear map, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "validity-and-supersession-q03", "text": "What may an agent decide or do once the validity of a claim about a linear map is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] } ] }, "statistics": { "bundles": 5, "layers": 10, "findings": 12, "questions": 32 } } } }