CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11

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

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 $L_A(x)=Ax$ .

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 $(Ax)_i=\sum_{j<n}a_{ij}x_j$ (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

1.1

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

L4L5algebra

2.1

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

step 1.1L1L2L3∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 63 results over 18 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

D. Margalit and J. Rabinoff, Interactive Linear Algebra, §3.5 (standard reference, not scraped)

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