Quotient vector space

Mathematical statement

Vectors v,u name the same coset exactly when v−u lies in W. Quotient elements are cosets, not unique representatives. Check representative independence of [v]+[u]=[v+u] and scalar*[v]=[scalar*v]. In finite dimension dim(V/W)=dim(V)−dim(W), and the zero coset is W.

Representative formula (exact TeX)

Q(V,W)

Assumptions

• V=R^n and W has a complete exact independent basis.
• Membership, rank and dimension certificates use exact rational arithmetic.

Symbol scope

• V/W — quotient vector space
• W — subspace being quotiented

Intuition, definition, and misconceptions

[v]=[u] exactly when v−u lies in W.

Well-defined operations give the same result coset after changing representatives.

The dimension formula is finite-dimensional and the zero element is the whole coset W.

A representative vector is not a unique quotient-element name, and a quotient space is not numerical division.

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 — W must be a subspace.
• Quotient set — Cosets form an equivalence-class carrier.
• Real vector space — Vector operations descend to cosets.
• Equivalence class — Representatives are not unique.
• Dimension of a finite-dimensional vector space — The dimension formula is finite-dimensional.

Verification boundary

Only finite-dimensional real coordinate quotients are handled; topological and Banach quotients are separate.

not_formalized

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

Next obligation

Add the first isomorphism theorem and 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.