# Linear combinations and span

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

A linear combination is a scalar-weighted sum of finitely many vectors; the span collects every finite linear combination.

mko-linear-combination-span · definition · v0.10.0



## Mathematical statement

For a vector family in a real vector space, a finite linear combination is a finite sum weighted by real coefficients. Its span is the set of all finite-length linear combinations and does not include undefined infinite series.





## Representative formula (exact TeX)

S(V)





## Assumptions


- Exercises use integer coordinate vectors in a fixed dimension and integer or exact rational coefficients.
- Each question audits only its explicitly supplied finite generator family and candidate witness; a fixed arity does not exhaust the abstract definition of span.





## Symbol scope


- S — span operator
- V — finite generator family





## Intuition, definition, and misconceptions

A weighted sum requires matching vector dimensions and the correct number of coefficients.

Supplied coefficients can reconstruct a target directly, providing a membership witness.

Two independent generators in R² permit unique coordinates to be found using their nonzero determinant.

Do not call an infinite sum a finite linear combination or infer membership in a span from dimension alone.





## Adopted derivation guide

[Linear combinations and span: deciding membership using finite coefficient lists as witnesses](https://ocme.evemisslab.com/en/derivations/derivation-mko-linear-combination-span-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/) — Linear combinations use vector addition and real scalar multiplication.
- [Linear systems in two variables and degenerate constraints](https://ocme.evemisslab.com/en/mko/mko-system-linear-equations/) — Coordinate membership can be expressed as a linear system.





## Verification boundary

Only finite coordinate witnesses are audited; a general span still quantifies over all finite combinations.

not_formalized

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





## Next obligation

Add linear independence, bases and dimension.





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