Cokernel

In linear algebra, the cokernel of a linear map f: V → W between vector spaces over a field F is the quotient space W / Im(f). It measures how far f is from being surjective; the cokernel is zero if and only if f is surjective. For finite-dimensional spaces, dim(coker f) = dim(W) − rank(f).

Intuition and universal property: Let π: W → W/Im(f) be the natural projection. Then Im(f) = Ker(π). The cokernel is characterized by a universal property: any linear map g: W → X with g ∘ f = 0 factors uniquely through π; equivalently, there exists a unique h: W/Im(f) → X with g = h ∘ π.

Examples: If f is the zero map, Im(f) = {0} and coker(f) ≅ W. If f: R^2 → R^3 is given by f(x, y) = (x, y, 0), then Im(f) is the plane {(a, b, 0)}, so coker(f) ≅ R, a one-dimensional space.

Generalizations: The concept extends to modules over a ring: for a homomorphism f: M → N of modules, the cokernel is N / Im(f). In abelian categories (such as modules), the cokernel is defined by the same universal-property idea: it is a morphism B → C that vanishes on Im(f) and is universal with respect to this property. The cokernel complements the kernel in the study of exact sequences; for example, 0 → Ker(f) → M → Im(f) → 0 and 0 → Im(f) → N → Coker(f) → 0 relate kernels, images, and cokernels.