# Rank-nullity theorem

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

A linear map T:V→W with finite-dimensional source satisfies dim(V)=dim(ker T)+rank(T).

mko-rank-nullity-theorem · theorem · v0.10.0



## Mathematical statement

An explicit m-by-n matrix represents T:R^n→R^m. Exact RREF, a full kernel basis and column-space rank verify n=nullity+rank. Nullity is kernel dimension, not the number of solutions; sampled vectors do not establish the identity in general.





## Representative formula (exact TeX)

N(T)





## Assumptions


- The source is finite-dimensional and the matrix shape is explicit.
- All rank, kernel and image certificates use exact rational arithmetic.





## Symbol scope


- N — rank-nullity record
- T — linear map





## Intuition, definition, and misconceptions

Source dimension is the column count n.

Nullity is the number of vectors in a kernel basis.

Complete symbolic certificates verify the identity for the map; sampling is insufficient.

Nullity is not the number of solutions, and the direction of an m-by-n matrix map must not be reversed.





## Adopted derivation guide

[Rank and nullity for three source coordinates](https://ocme.evemisslab.com/en/derivations/derivation-mko-rank-nullity-theorem-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Kernel of a linear map](https://ocme.evemisslab.com/en/mko/mko-linear-map-kernel/) — Kernel dimension gives nullity.
- [Image of a linear map](https://ocme.evemisslab.com/en/mko/mko-linear-map-image/) — Image dimension gives rank.
- [Matrix rank](https://ocme.evemisslab.com/en/mko/mko-matrix-rank/) — The matrix has an exact rank certificate.
- [Dimension of a finite-dimensional vector space](https://ocme.evemisslab.com/en/mko/mko-vector-space-dimension/) — Source and kernel dimensions are explicit.





## Verification boundary

Only small exact matrices are handled; finitely sampled vectors do not prove the general theorem.

not_formalized

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





## Next obligation

Add the first isomorphism theorem.





## 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/rank-nullity-theorem.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-rank-nullity-theorem/index.md)