# Real vector subspace

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

A nonempty subset of a real vector space is a subspace when it contains zero and is closed under vector addition and multiplication by every real scalar.

mko-vector-subspace · definition · v0.10.0



## Mathematical statement

This slice handles the zero space, the whole space, lines through the origin and explicit finite counterexamples in R². Positive cases use universal algebraic certificates from homogeneous linear equations or spans. Finitely many successful samples do not prove a subspace.





## Representative formula (exact TeX)

U(V)





## Assumptions


- The ambient space is fixed as R² with the usual vector operations.
- Positive certificates cover addition and every real scalar universally; an affine line not through zero is not a subspace.





## Symbol scope


- U — candidate subspace
- V — ambient real vector space





## Intuition, definition, and misconceptions

Containing the zero vector is necessary.

A homogeneous line through the origin has a universal closure certificate from its linear equation.

One counterexample refutes a subspace; finitely many positive samples do not prove it.

An arbitrary affine line, or finitely sampled closure, is not automatically a subspace certificate.





## Adopted derivation guide

[Universal closure of a line through zero](https://ocme.evemisslab.com/en/derivations/derivation-mko-vector-subspace-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/) — The parent supplies R² vector operations and real scalars.
- [Subset](https://ocme.evemisslab.com/en/mko/mko-subset/) — A subspace is a subset of the ambient space.
- [Quantifiers and explicit domains](https://ocme.evemisslab.com/en/mko/mko-quantifier/) — Closure is a universal obligation.





## Verification boundary

Recomputation covers only R² and the explicit certificates, not arbitrary dimensions or unknown operations.

not_formalized

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





## Next obligation

Add kernels, images, dimension and quotient vector 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/vector-subspace.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-vector-subspace/index.md)