Rank-nullity theorem

Mathematical statement

An explicit m-by-n matrix represents T:R^n→R^m. Exact RREF, a full kernel basis and column-space rank verify n=nullity+rank. Nullity is kernel dimension, not the number of solutions; sampled vectors do not establish the identity in general.

Representative formula (exact TeX)

N(T)

Assumptions

• The source is finite-dimensional and the matrix shape is explicit.
• All rank, kernel and image certificates use exact rational arithmetic.

Symbol scope

• N — rank-nullity record
• T — linear map

Intuition, definition, and misconceptions

Source dimension is the column count n.

Nullity is the number of vectors in a kernel basis.

Complete symbolic certificates verify the identity for the map; sampling is insufficient.

Nullity is not the number of solutions, and the direction of an m-by-n matrix map must not be reversed.

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

• Kernel of a linear map — Kernel dimension gives nullity.
• Image of a linear map — Image dimension gives rank.
• Matrix rank — The matrix has an exact rank certificate.
• Dimension of a finite-dimensional vector space — Source and kernel dimensions are explicit.

Verification boundary

Only small exact matrices are handled; finitely sampled vectors do not prove the general theorem.

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.

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.