CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-11

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The rank of a matrix equals the rank of the linear map $x \mapsto A x$

Statement

For $A \in M_{m \times n} (F)$ , the matrix rank of $A$ equals the rank of the linear map $L_{A} : F^{n} \to F^{m}$ defined by $L_{A} (x) = A x$ .

Facts & Assumptions

Given: A matrix $A \in M_{m \times n} (F)$ and its associated linear map $L_{A}$ .

[L1]

Matrix rank is row rank, which equals column rank (Row space, column space, nullspace, row rank, column rank and matrix rank).

[L2]

Row rank equals column rank (Row rank equals column rank, and both equal the number of pivots).

[L3]

The rank of a linear map is the dimension of its image (Rank and nullity of a linear map with finite-dimensional domain).

[L4]

Matrix multiplication is given entrywise by $(A x)_{i} = \sum_{j < n} a_{i j} x_{j}$ (Rectangular matrix multiplication and the identity matrix $I_{n}$ , including zero-sized shapes).

[L5]

Matrix multiplication distributes over addition and is compatible with scalar multiplication (Matrix multiplication is associative, unital, distributive, and compatible with scalar multiplication).

Proof

technique · direct

1.1L4L5algebra

Fact [L5] shows that $L_{A} (x) = A x$ is linear. By [L4], every value $A x$ is a linear combination of the columns of $A$ , with coefficients $x_{j}$ , and every such combination is $A x$ for that coefficient column. Hence $im L_{A} = Col (A)$ .

2.1step 1.1L1L2L3∎

Taking dimensions and using [L1]–[L3] gives $rank L_{A} = \dim Col (A) = rank A$ .

Depends on

Dependency tree · 2 levels

Sources

D. Margalit and J. Rabinoff, Interactive Linear Algebra, §3.5

The rank of a matrix equals the rank of the linear map x↦Axx↦Ax