# Dimension of a finite-dimensional vector space

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

The dimension of a finite-dimensional vector space is the cardinality of any finite basis; different bases have the same cardinality.

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



## Mathematical statement

This slice handles complete coordinate families in R² or R³. Check exact linear independence and spanning before using basis size as dimension. An independent family supplies only a lower bound and a spanning family only an upper bound; basis-cardinality invariance invokes the finite-dimensional theorem.





## Representative formula (exact TeX)

D(V)





## Assumptions


- The ambient space is R² or R³ and every vector is fully listed.
- Rank uses exact rational Gaussian elimination, not floating-point approximation.





## Symbol scope


- D — dimension record
- V — finite-dimensional real vector space





## Intuition, definition, and misconceptions

Read dimension from a family only after it qualifies as a basis.

An independent family has size at most the dimension.

A spanning family has size at least the dimension; invariance of basis cardinality is a theorem obligation.

The size of an arbitrary independent or spanning family is not automatically the dimension.





## Adopted derivation guide

[Prove a basis before reading dimension](https://ocme.evemisslab.com/en/derivations/derivation-mko-vector-space-dimension-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 coordinates and vector operations.
- [Linear independence and bases](https://ocme.evemisslab.com/en/mko/mko-linear-independence-basis/) — A basis is both independent and spanning.
- [Cardinality comparison of sets](https://ocme.evemisslab.com/en/mko/mko-cardinality-comparison/) — Basis size gives the dimension and related bounds.





## Verification boundary

Only exact R²/R³ matrix instances are recomputed; a few examples do not prove general basis invariance.

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, nullity and the rank-nullity 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/vector-space-dimension.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-vector-space-dimension/index.md)