Quotient space of a codomain of a linear map by the map's image
"Coker (mathematics)" redirects here. For other uses, see Coker (disambiguation).
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The cokernel of a linear mapping of vector spaces f : X → Y is the quotient space Y / im(f) of the codomain of f by the image of f. The dimension of the cokernel is called the corank of f.
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 f(x) = y that one is seeking to solve, the cokernel measures the constraints that 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, below.
More generally, the cokernel of a morphism f : X → Y in some category (e.g. a homomorphism between groups or a bounded linear operator between Hilbert spaces) is an object Q and a morphism q : Y → Q such that the composition q f is the zero morphism of the category, and furthermore q is universal with respect to this property. Often the map q is understood, and Q itself is called the cokernel of f.
In many situations in abstract algebra, such as for abelian groups, vector spaces or modules, the cokernel of the homomorphism f : X → Y is the quotient of Y by the image of f. In topological settings, such as with bounded linear operators between Hilbert spaces, one typically has to take the closure of the image before passing to the quotient.
The cokernel of a linear mapping of vector spaces f : X → Y is the quotient space Y / im(f) of the codomain of f by the image of f. The dimension of the...
the kernel of some morphism, and an epimorphism is conormal if it is the cokernel of some morphism. A category C is binormal if it's both normal and conormal...
B, while the cokernel of f is the coequaliser of f and this zero morphism. Unlike with products and coproducts, the kernel and cokernel of f are generally...
invariant of a linear transformation f : V → W {\textstyle f:V\to W} is the cokernel, which is defined as coker ( f ) := W / f ( V ) = W / im ( f ) . {\displaystyle...
in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example...
particularly simple. It is just the factor group Y / im(f – g). (This is the cokernel of the morphism f – g; see the next section). In the category of topological...
kernel ker T {\displaystyle \ker T} and finite-dimensional (algebraic) cokernel coker T = Y / ran T {\displaystyle \operatorname {coker} T=Y/\operatorname...
of algebra ker f = {x in G | f(x) = e}), and also a category-theoretic cokernel (given by the factor group of H by the normal closure of f(G) in H). Unlike...
linear map, the dimension of the map's kernel minus the dimension of its cokernel Index of a matrix Index of a real quadratic form Index, the winding number...
the cokernel is nontrivial, in which case U {\displaystyle \mathbf {U} } is padded with m − n {\displaystyle m-n} orthogonal vectors from the cokernel. Conversely...
the dimension of the left nullspace of a matrix, the dimension of the cokernel of a linear transformation of a vector space, or the number of elements...
notion of an exact sequence makes sense in any category with kernels and cokernels, and more specially in abelian categories, where it is widely used. To...
I_{X}(x)=x} for all x in X. (An operator is Fredholm if its kernel and cokernel are finite-dimensional.) The essential spectrum is always closed, and it...
equivalently characterized as operators with a finite dimensional kernel and cokernel. The index of a Fredholm operator T is defined by index T = dim ker...
This is a list of homological algebra topics, by Wikipedia page. Cokernel Exact sequence Chain complex Differential module Five lemma Short five lemma...
z} and f z = f c ∘ h {\displaystyle f_{z}=f_{c}\circ h} Quotient object Cokernel Mitchell, Barry (1965). Theory of categories. Pure and applied mathematics...
case striking down the death sentence of a defendant convicted of rape Cokernel, also referred to as the coker, a concept in mathematics Cocker (disambiguation)...
infinite) of the quotient space V/W, which is more abstractly known as the cokernel of the inclusion. For finite-dimensional vector spaces, this agrees with...
fiber. More abstractly, the codimension of a map is the dimension of the cokernel, while the relative dimension of a map is the dimension of the kernel....
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