##  [Corank](/corank-0) 

 Definition

The codimension of the image (range) of a linear map between finite-dimensional vector spaces; equivalently the dimension of the codomain minus the rank of the map, measuring how far the map is from being surjective.

 

 

 

 

 

 





## Principle

Principle

Corank = dim W − rank(T) for T: V → W. In finite-dimensional duality contexts, corank equals the nullity of the transpose (or adjoint) mapping on the dual space.

 

 

 

 

 





## Demonstration

Demonstration

Example: T: R^5 → R^7 with rank 4 has corank 7−4=3; three independent target directions are not attained by T, so the image is a 4-dimensional subspace of R^7 with codimension 3.

 

 

 

 

## Misapplication

Misapplication

Using corank as a synonym for nullity (kernel dimension) of the same map, or applying the finite-dimensional formula in infinite-dimensional settings without considering closure of the image or continuous duals.

 

 

 

 

 





## Consequence

Consequence

Knowing corank quantifies the obstruction to surjectivity: corank=0 iff T is onto. It guides construction of right inverses and controls solvability conditions for inhomogeneous linear equations.

 

 

 

 

## Reversal

Reversal

Reverse focus to rank: instead of counting missed target dimensions, count achieved target dimensions. Alternatively, consider nullity of the map to measure injectivity failure rather than surjectivity failure.

 

 

 

 

 





## Boundary

Boundary

Defined for linear maps between finite-dimensional vector spaces; in infinite-dimensional contexts corank can be infinite or depends on topological closures and dual spaces; for non-linear maps analogues require different invariants.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Tension between corank and nullity: both measure 'deficiency' but in dual positions (codomain vs domain). Also tension with related invariants like cokernel dimension or Fredholm index which combine nullity and corank.

 

 

 

 

 





## Synthesis

Synthesis

Corank is the integer codimension of the image of a linear operator: dim(codomain)−rank, a precise measure of how many target directions the map fails to reach.