Public Mathlib landmark · existing upstream theorem · read-only
ProofAtlas

Linear algebra · existing Mathlib theorem

Rank–Nullity Theorem

A linear map collapses the directions in its kernel and carries the surviving quotient directions into its range. Those two cardinal-rank contributions recover the rank of the original domain.

Four separate status axes

Upstream indexedPinned source bytes verified locally
Locally reproducedExact upstream declaration replayed
Reviewed page2 of 3 presentation reviews recorded
Accepted Atlas resultNot recorded for the preferred artifact

These states distinguish upstream identity, local reproduction, review, and Atlas acceptance. This page is part of the public, read-only Mathlib landmark collection.

Formal orientation

Statement map

Statement map for Rank–Nullity TheoremFor a linear map under Mathlib's rank-nullity hypothesis, the cardinal ranks of the range and kernel add to the rank of the domain. The pinned upstream declaration is LinearMap.rank_range_add_rank_ker. The exact checked statement is theorem LinearMap.rank_range_add_rank_ker {R M M₁ : Type*} [Ring R] [AddCommGroup M] [AddCommGroup M₁] [Module R M] [Module R M₁] [HasRankNullity R] (f : M →ₗ[R] M₁) : Module.rank R f.range + Module.rank R f.ker = Module.rank R M.Mathematical readingFor a linear map underMathlib's rank-nullityhypothesis, the cardinalranks of the range andkernel add to the rankof the domain.Pinned declarationmathlib ·LinearMap.rank_range_add_rank_kerExact checked formtheoremLinearMap.rank_range_add_rank_ker{R M M₁ : Type*} [RingR] [AddCommGroup M][AddCommGroup M₁][Module R M] [Module RM₁] [HasRankNullity R](f : M →ₗ[R] M₁) :Module.rank R f.range +Module.rank R f.ker =Module.rank R MStatement map for Rank–Nullity TheoremFor a linear map under Mathlib's rank-nullity hypothesis, the cardinal ranks of the range and kernel add to the rank of the domain. The pinned upstream declaration is LinearMap.rank_range_add_rank_ker. The exact checked statement is theorem LinearMap.rank_range_add_rank_ker {R M M₁ : Type*} [Ring R] [AddCommGroup M] [AddCommGroup M₁] [Module R M] [Module R M₁] [HasRankNullity R] (f : M →ₗ[R] M₁) : Module.rank R f.range + Module.rank R f.ker = Module.rank R M.Mathematical readingFor a linear map underMathlib's rank-nullityhypothesis, the cardinalranks of the range andkernel add to the rankof the domain.Pinned declarationmathlib ·LinearMap.rank_range_add_rank_kerExact checked formtheoremLinearMap.rank_range_add_rank_ker{R M M₁ : Type*} [RingR] [AddCommGroup M][AddCommGroup M₁][Module R M] [Module RM₁] [HasRankNullity R](f : M →ₗ[R] M₁) :Module.rank R f.range +Module.rank R f.ker =Module.rank R M

This deterministic map orients the reader from the mathematical summary to the pinned declaration and exact checked form. The HTML statement below is authoritative.

The theorem at a glance

Rank–Nullity Theorem at a glance

A source-bound editorial overview of the exact Mathlib formulation. The image is explanatory; the statement map, formal type, and pinned source remain authoritative.

The range and kernel ranks account for the cardinal rank of the domain. Generated explanation only; the exact HTML statement and pinned source are authoritative.
Detailed visual description

The poster separates collapsed kernel directions from the surviving quotient, connects that quotient to the range, and states the exact cardinal-rank theorem without replacing it by a finite matrix example.

Kernel directions collapse while the quotient by the kernel recovers the range, and their cardinal ranks recover the domain rank. Generated explanation only; the exact HTML statement and pinned source are authoritative.
Detailed visual description

Parallel kernel-direction fibers foliate the entire domain rather than splitting it into complementary regions. Collapsing each full coset produces the quotient; a central gold equivalence carries those classes to the range, while the encompassing brace relates the kernel and range contributions back to the domain without assuming finite dimension.

Complementary intuition

A mathematical landmark

Rank–nullity is the structural accounting principle behind linear equations, kernels, images, quotient spaces, and dimension arguments. Mathlib's selected declaration records the stronger cardinal-rank form rather than only the familiar finite-dimensional formula.

The low-text schematic gives the theorem a visual identity; it does not replace the source-bound poster or exact statement.

Authoritative formal type

Exact Mathlib statement

theorem LinearMap.rank_range_add_rank_ker {R M M₁ : Type*} [Ring R] [AddCommGroup M] [AddCommGroup M₁] [Module R M] [Module R M₁] [HasRankNullity R] (f : M →ₗ[R] M₁) : Module.rank R f.range + Module.rank R f.ker = Module.rank R M
Selected evidence declaration text
theorem LinearMap.rank_range_add_rank_ker {R M M₁ : Type*} [Ring R] [AddCommGroup M] [AddCommGroup M₁] [Module R M] [Module R M₁] [HasRankNullity R] (f : M →ₗ[R] M₁) : Module.rank R f.range + Module.rank R f.ker = Module.rank R M

Claim boundary

No new theorem is claimed

This target indexes Mathlib's cardinal-rank identity for a linear map under its HasRankNullity hypothesis: the ranks of the range and kernel add to the rank of the domain. It is not merely the finite-dimensional natural-number formula and does not claim a new proof.

Provenance

Origin and evidence stay separate

Existing declaration
LinearMap.rank_range_add_rank_ker in mathlib
Relationship
The checked artifact is the upstream declaration itself
Preferred checked artifact
artifact.library.mathlib.rank-nullity-theorem.v001
Source handling
Verified upstream reference; no mirrored source package