##  [Rank](/rank-2) 

 Definition

A nonnegative integer (or cardinal) describing the maximal number of linearly independent rows or columns of a matrix, equivalently the dimension of the image (column space) of a linear map; it quantifies the effective linear degrees of freedom of the operator or matrix.

 

 

 

 

 

 





## Principle

Principle

Rank equals the dimension of the image of a linear transformation and is invariant under change of basis; in finite dimensions it satisfies the rank–nullity theorem: dim(domain) = rank + nullity, and full rank characterizes invertibility for square matrices.

 

 

 

 

 





## Demonstration

Demonstration

A 3×3 identity matrix has rank 3; a 3×3 matrix with two independent rows has rank 2 and maps R^3 onto a 2-dimensional subspace; the linear map T: R^4 → R^3 given by a 3×4 matrix of rank 3 has image dimension 3 and nullity 1.

 

 

 

 

## Misapplication

Misapplication

Confusing matrix rank with tensor rank or multilinear rank, computing rank over the wrong field (for example using real arithmetic when the problem is over a finite field), or assuming numerical rank computed approximately equals exact algebraic rank without stability checks.

 

 

 

 

 





## Consequence

Consequence

Rank determines solvability of linear systems (a system has solutions constrained by rank conditions), invertibility of square matrices (full rank), and the dimension of images and cokernels; it also controls dimension counts in linear algebraic geometry and determines the number of independent constraints.

 

 

 

 

## Reversal

Reversal

Nullity (the dimension of the kernel) measures the deficiency of rank; one may instead focus on cokernel dimension or corank (codimension of the image) which quantifies constraints unmet by the map's image.

 

 

 

 

 





## Boundary

Boundary

Over general rings rank can be ambiguous (several non-equivalent notions such as Smith rank or free rank exist); for infinite-dimensional operators one may speak of finite rank, infinite rank, or Fredholm index instead of a simple integer rank.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Rank is easily conflated with related invariants (tensor rank, analytic rank, numerical rank, or free rank over rings); specifying the underlying field or ring and whether one means matrix/operator/tensor rank prevents misinterpretation.

 

 

 

 

 





## Synthesis

Synthesis

Rank measures the size of the image of a linear map — the number of independent output directions — and together with nullity provides the fundamental partition of domain dimension that governs solvability, invertibility, and constraint counts in linear settings.