Editing Cokernel (section)

{{Short description|Quotient space of a codomain of a linear map by the map's image}}
{{Redirect|Coker (mathematics)|other uses|Coker (disambiguation)}}
{{no footnotes|date=February 2013}}

The '''cokernel''' of a [[linear mapping]] of [[vector spaces]] {{math|''f'' : ''X'' → ''Y''}} is the [[quotient space (linear algebra)|quotient space]] {{math|''Y'' / im(''f'')}} of the [[codomain]] of {{mvar|f}} by the image of {{mvar|f}}. The dimension of the cokernel is called the ''corank'' of {{mvar|f}}.

Cokernels are [[dual (category theory)|dual]] to the [[kernel (category theory)|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 {{math|1=''f''(''x'') = ''y''}} that one is seeking to solve, the cokernel measures the ''constraints'' that {{mvar|y}} 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|intuition]], below.

More generally, the cokernel of a [[morphism]] {{math|''f'' : ''X'' → ''Y''}} in some [[category theory|category]] (e.g. a [[group homomorphism|homomorphism]] between [[group (mathematics)|group]]s or a [[bounded linear operator]] between [[Hilbert space]]s) is an object {{mvar|Q}} and a morphism {{math|''q'' : ''Y'' → ''Q''}} such that the composition {{math|''q f''}} is the [[zero morphism]] of the category, and furthermore {{mvar|q}} is [[universal mapping property|universal]] with respect to this property. Often the map {{mvar|q}} is understood, and {{mvar|Q}} itself is called the cokernel of {{mvar|f}}.

In many situations in [[abstract algebra]], such as for [[abelian group]]s, [[vector space]]s or [[module (mathematics)|module]]s, the cokernel of the [[homomorphism]] {{math|''f'' : ''X'' → ''Y''}} is the [[quotient set|quotient]] of {{mvar|Y}} by the [[Image (mathematics)|image]] of {{mvar|f}}. In [[topology|topological]] settings, such as with bounded linear operators between Hilbert spaces, one typically has to take the [[closure (mathematics)|closure]] of the image before passing to the quotient.