Real vector subspace

Mathematical statement

This slice handles the zero space, the whole space, lines through the origin and explicit finite counterexamples in R². Positive cases use universal algebraic certificates from homogeneous linear equations or spans. Finitely many successful samples do not prove a subspace.

Representative formula (exact TeX)

U(V)

Assumptions

• The ambient space is fixed as R² with the usual vector operations.
• Positive certificates cover addition and every real scalar universally; an affine line not through zero is not a subspace.

Symbol scope

• U — candidate subspace
• V — ambient real vector space

Intuition, definition, and misconceptions

Containing the zero vector is necessary.

A homogeneous line through the origin has a universal closure certificate from its linear equation.

One counterexample refutes a subspace; finitely many positive samples do not prove it.

An arbitrary affine line, or finitely sampled closure, is not automatically a subspace certificate.

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 space — The parent supplies R² vector operations and real scalars.
• Subset — A subspace is a subset of the ambient space.
• Quantifiers and explicit domains — Closure is a universal obligation.

Verification boundary

Recomputation covers only R² and the explicit certificates, not arbitrary dimensions or unknown operations.

not_formalized

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

Next obligation

Add kernels, images, dimension and quotient vector spaces.

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.