# Direct sum of subspaces

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

The sum of two subspaces is an internal direct sum when every vector in that sum has a unique decomposition; covering an ambient space is a separate spanning obligation.

mko-direct-sum · definition · v0.10.0



## Mathematical statement

Use complete bases and exact ranks in a finite-dimensional real ambient space. U∩W={0} ensures uniqueness in U+W, while V=U⊕W also needs U+W=V. For three or more subspaces pairwise zero intersections do not replace global uniqueness; an external direct sum uses a new product-type carrier.





## Representative formula (exact TeX)

U+W





## Assumptions


- Every subspace is represented by a complete independent basis and exact rational rank.
- Check internal uniqueness, ambient coverage, many-summand directness and the external carrier separately.





## Symbol scope


- U — first subspace
- W — second subspace





## Intuition, definition, and misconceptions

For two subspaces, unique decomposition in their sum is equivalent to zero intersection.

Coverage of V is separate; uniqueness alone concerns U+W.

Three or more summands need global independence of all basis columns; an external sum is not an internal-subspace assertion.

Zero intersection alone does not prove ambient coverage, and pairwise zero intersections do not prove a direct sum of three subspaces.





## Adopted derivation guide

[Unique decomposition and coverage in R3](https://ocme.evemisslab.com/en/derivations/derivation-mko-direct-sum-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/) — The subspaces lie in a common ambient vector space.
- [Set operations](https://ocme.evemisslab.com/en/mko/mko-set-operations/) — Intersection and sum have distinct roles.
- [Linear independence and bases](https://ocme.evemisslab.com/en/mko/mko-linear-independence-basis/) — Complete basis columns give rank certificates.





## Verification boundary

Only finite-dimensional real coordinates and exact ranks are handled; no infinite or topological direct-sum claim is made.

not_formalized

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





## Next obligation

Add quotient spaces and splitting of short 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/direct-sum.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-direct-sum/index.md)