# Quotient vector space

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

Cosets of a subspace W form the quotient vector space V/W, whose operations must be independent of representative choices.

mko-quotient-vector-space · definition · v0.10.0



## Mathematical statement

Vectors v,u name the same coset exactly when v−u lies in W. Quotient elements are cosets, not unique representatives. Check representative independence of [v]+[u]=[v+u] and scalar*[v]=[scalar*v]. In finite dimension dim(V/W)=dim(V)−dim(W), and the zero coset is W.





## Representative formula (exact TeX)

Q(V,W)





## Assumptions


- V=R^n and W has a complete exact independent basis.
- Membership, rank and dimension certificates use exact rational arithmetic.





## Symbol scope


- V/W — quotient vector space
- W — subspace being quotiented





## Intuition, definition, and misconceptions

[v]=[u] exactly when v−u lies in W.

Well-defined operations give the same result coset after changing representatives.

The dimension formula is finite-dimensional and the zero element is the whole coset W.

A representative vector is not a unique quotient-element name, and a quotient space is not numerical division.





## Adopted derivation guide

[Quotient by the diagonal line](https://ocme.evemisslab.com/en/derivations/derivation-mko-quotient-vector-space-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 subspace](https://ocme.evemisslab.com/en/mko/mko-vector-subspace/) — W must be a subspace.
- [Quotient set](https://ocme.evemisslab.com/en/mko/mko-quotient-set/) — Cosets form an equivalence-class carrier.
- [Real vector space](https://ocme.evemisslab.com/en/mko/mko-real-vector-space/) — Vector operations descend to cosets.
- [Equivalence class](https://ocme.evemisslab.com/en/mko/mko-equivalence-class/) — Representatives are not unique.
- [Dimension of a finite-dimensional vector space](https://ocme.evemisslab.com/en/mko/mko-vector-space-dimension/) — The dimension formula is finite-dimensional.





## Verification boundary

Only finite-dimensional real coordinate quotients are handled; topological and Banach quotients are separate.

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 and exact sequences.





## 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/quotient-vector-space.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-quotient-vector-space/index.md)