Direct sum of subspaces

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

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 — The subspaces lie in a common ambient vector space.
• Set operations — Intersection and sum have distinct roles.
• Linear independence and bases — 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.