# Linear map between real vector spaces

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

A map between real vector spaces is linear when it preserves linear combinations for all vectors and real scalars.

mko-general-linear-map · definition · v0.10.0



## Mathematical statement

A real linear map T:V→W satisfies T(alpha*u+beta*v)=alpha*T(u)+beta*T(v) for all u,v and real alpha,beta. A finite-dimensional matrix formula supplies a universal coordinate certificate; a nonzero affine translation fails linearity.





## Representative formula (exact TeX)

T(v)





## Assumptions


- Exercises explicitly give a matrix or matrix-plus-bias formula from R^n to R^m.
- Matrix distributivity supplies the universal certificate; finitely many sampled inputs do not prove an arbitrary black-box map linear.





## Symbol scope


- T — real linear map
- V — source space
- W — target space





## Intuition, definition, and misconceptions

One identity encompasses both additivity and real homogeneity.

Matrix multiplication gives a linear map in finite-dimensional coordinates.

The affine formula Ax+b qualifies as linear exactly when b=0.

An arbitrary affine map, a black box passing only a few samples, or an additive function without real homogeneity need not be real-linear.





## Adopted derivation guide

[Linear maps between real vector spaces: a universal certificate for a matrix map, with affine and black-box counterexamples](https://ocme.evemisslab.com/en/derivations/derivation-mko-general-linear-map-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Real vector space](https://ocme.evemisslab.com/en/mko/mko-real-vector-space/) — Both source and target are real vector spaces.
- [Function mapping](https://ocme.evemisslab.com/en/mko/mko-function-mapping/) — T must be an explicit total map.
- [Linear combinations and span](https://ocme.evemisslab.com/en/mko/mko-linear-combination-span/) — A linear map preserves finite linear combinations.





## Verification boundary

The coordinate-matrix certificate covers only the explicit finite-dimensional formula; it does not certify arbitrary black boxes or make separate boundedness, continuity, injectivity or surjectivity claims.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Add bounded linear operators on normed spaces.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/general-linear-map.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-general-linear-map/index.md)