# Linear independence and bases

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

Linear independence excludes nontrivial zero-sum coefficient relations; a basis is both independent and spanning for the declared space.

mko-linear-independence-basis · concept · v0.10.0



## Mathematical statement

A finite vector family is linearly independent if a weighted sum equal to zero requires every coefficient to be zero. To be a basis of a space V, it must be independent and have span equal to V. Neither obligation replaces the other.





## Representative formula (exact TeX)

B(V)





## Assumptions


- Exercises use finite integer-coordinate vector families, with rank computed by exact rational elimination.
- Determinants and coordinate uniqueness are used only for ordered square R² bases, not extended to non-square families.





## Symbol scope


- B — basis-audit record
- V — declared vector space





## Intuition, definition, and misconceptions

Independence asks whether the zero vector has only the trivial coefficient representation.

Being a basis is the conjunction of independence and spanning.

Only after an ordered basis has been established does every R² target have unique coordinates in it.

Do not call a family a basis merely because it spans or merely because it is independent.





## Adopted derivation guide

[Linear independence and bases: proving an ordered basis of ℝ² by exact elimination, with two counterexamples that separate the two obligations](https://ocme.evemisslab.com/en/derivations/derivation-mko-linear-independence-basis-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Linear combinations and span](https://ocme.evemisslab.com/en/mko/mko-linear-combination-span/) — A basis must span the declared space.
- [Linear systems in two variables and degenerate constraints](https://ocme.evemisslab.com/en/mko/mko-system-linear-equations/) — Independence and coordinates reduce to homogeneous or inhomogeneous systems.
- [Determinants of real 2×2 matrices](https://ocme.evemisslab.com/en/mko/mko-determinant/) — A nonzero determinant checks independence and spanning for a square R² family.





## Verification boundary

Rank and determinant conclusions are restricted to the explicitly declared finite coordinate model.

not_formalized

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





## Next obligation

Add dimension, change of basis and matrices of linear maps.





## 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-independence-basis.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-linear-independence-basis/index.md)