Dimension of a finite-dimensional vector space

Mathematical statement

This slice handles complete coordinate families in R² or R³. Check exact linear independence and spanning before using basis size as dimension. An independent family supplies only a lower bound and a spanning family only an upper bound; basis-cardinality invariance invokes the finite-dimensional theorem.

Representative formula (exact TeX)

D(V)

Assumptions

• The ambient space is R² or R³ and every vector is fully listed.
• Rank uses exact rational Gaussian elimination, not floating-point approximation.

Symbol scope

• D — dimension record
• V — finite-dimensional real vector space

Intuition, definition, and misconceptions

Read dimension from a family only after it qualifies as a basis.

An independent family has size at most the dimension.

A spanning family has size at least the dimension; invariance of basis cardinality is a theorem obligation.

The size of an arbitrary independent or spanning family is not automatically the dimension.

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 coordinates and vector operations.
• Linear independence and bases — A basis is both independent and spanning.
• Cardinality comparison of sets — Basis size gives the dimension and related bounds.

Verification boundary

Only exact R²/R³ matrix instances are recomputed; a few examples do not prove general basis invariance.

not_formalized

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

Next obligation

Add rank, nullity and the rank-nullity theorem.

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.