Cokernel
The cokernel of a linear mapping of vector spaces is the quotient space of the codomain of by the image of . The dimension of the cokernel is called the corank of . Cokernels are dual to the kernels of category theory, hence the name: the kernel is a subobject of the domain (it maps to the domain), while the cokernel is a quotient object of the codomain (it maps from the codomain). Intuitively, given an equation that one is seeking to solve, the cokernel measures the constraints that must satisfy for this equation to have a solution – the obstructions to a solution – while the kernel measures the degrees of freedom in a solution, if one exists. This is elaborated in intuition, below. More generally, the cokernel of a morphism in some category (e.g. a homomorphism between groups or a bounded linear operator between Hilbert spaces) is an object and a morphism such that the composition is the zero morphism of the category, and furthermore is universal with respect to this property. Often the map is understood, and itself is called the cokernel of . In many situations in abstract algebra, such as for abelian groups, vector spaces