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

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

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

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

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

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

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

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

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.