# Kernel 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 kernel of T consists of all source vectors x satisfying T(x)=0 and forms a subspace of the source.

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



## Mathematical statement

This slice binds an exact 2-by-2 integer matrix to T:R²→R². Symbolic null equations or exact row reduction describe the full kernel. A bounded coordinate search covers only its box and cannot replace the whole real kernel. A nonzero target fiber is an affine solution set, not the kernel.





## Representative formula (exact TeX)

K(T)





## Assumptions


- The complete 2-by-2 matrix defines the usual real linear map on R².
- The kernel target is fixed as the zero vector and all divisions are exact rational operations.





## Symbol scope


- K — linear-map kernel
- T — specified linear map





## Intuition, definition, and misconceptions

Check membership by evaluating A*x=0 exactly.

Describe the complete kernel with null equations or a basis.

Failure of a bounded search does not establish global absence in R²; a nonzero target fiber is not the kernel.

Do not call a nonzero fiber A*x=b a kernel or infer the whole kernel from a finite grid.





## Adopted derivation guide

[The full kernel of a two-by-two map](https://ocme.evemisslab.com/en/derivations/derivation-mko-linear-map-kernel-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 and its linearity obligations are fixed.
- [Real vector subspace](https://ocme.evemisslab.com/en/mko/mko-vector-subspace/) — The kernel forms a source subspace.
- [Equation](https://ocme.evemisslab.com/en/mko/mko-equation/) — Kernel vectors solve A*x=0.





## Verification boundary

Only exact 2-by-2 matrices are handled; every bounded result retains its explicit range.

not_formalized

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





## Next obligation

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