# Image of a linear map

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

The image of a linear map is the set of all outputs T(x); for a matrix map it is the full column span.

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



## Mathematical statement

This slice uses exact 2-by-2 matrices and symbolic column-span certificates. Membership is exact real consistency of A*x=y. A bounded fiber search supplies witnesses or bounded nonexistence only, not global nonmembership in the real image. The image must not be conflated with the entire declared codomain or a sampled output list.





## Representative formula (exact TeX)

I(T)





## Assumptions


- The complete 2-by-2 matrix defines T:R²→R².
- Global membership uses a symbolic linear system; bounded search has a separate explicit boundary.





## Symbol scope


- I — linear-map image
- T — specified linear map





## Intuition, definition, and misconceptions

The image is the full real span of the matrix columns.

Membership is equivalent to existence of a real solution of A*x=y.

Failure of bounded search is not global nonmembership.

The image need not equal the codomain, and finitely many sampled inputs do not establish the complete output range.





## Adopted derivation guide

[The full image from consistency](https://ocme.evemisslab.com/en/derivations/derivation-mko-linear-map-image-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Linear map between real vector spaces](https://ocme.evemisslab.com/en/mko/mko-general-linear-map/) — The map T is fixed.
- [Real vector subspace](https://ocme.evemisslab.com/en/mko/mko-vector-subspace/) — The image is a subspace of the target.
- [Function mapping](https://ocme.evemisslab.com/en/mko/mko-function-mapping/) — Image and codomain have different roles.





## Verification boundary

Only exact 2-by-2 matrices and explicit symbolic certificates are handled.

not_formalized

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





## Next obligation

Add rank and rank-nullity.





## 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/linear-map-image.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-linear-map-image/index.md)