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...

pre-abelian category is an additive category that has all kernels and cokernels. Spelled out in more detail, this means that a category C is pre-abelian...

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...

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...

zero object. Then there is an exact sequence relating the kernels and cokernels of a, b, and c: ker ⁡ a   ⟶   ker ⁡ b   ⟶   ker ⁡ c   ⟶ d   coker ⁡ a...

The dual concept to that of kernel is that of cokernel. That is, the kernel of a morphism is its cokernel in the opposite category, and vice versa. As...

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...

in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example...

operator is semi-Fredholm if its range is closed and either its kernel or cokernel (or both) is finite-dimensional.) Example 1: λ = 0 ∈ σ e s s , 1 ( A )...

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...

theory of triangulated categories it is a kind of combined kernel and cokernel: if the chain complexes take their terms in an abelian category, so that...

is again a kernel and, dually, the pullback of a cokernel along arbitrary morphisms is again a cokernel. A quasi-abelian category is an exact category.[citation...

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...

defined as the equalizer ( I m , m ) {\displaystyle (Im,m)} of the so-called cokernel pair ( Y ⊔ X Y , i 1 , i 2 ) {\displaystyle (Y\sqcup _{X}Y,i_{1},i_{2})}...

\operatorname {rank} (A)+\operatorname {nullity} (A)=n.} The left null space, or cokernel, of a matrix A consists of all column vectors x such that xTA = 0T, where...

they are closed under operations such as taking kernels, images, and cokernels. The quasi-coherent sheaves are a generalization of coherent sheaves and...

with the inclusion homomorphism i : K → A. The same is true for cokernels; the cokernel of f is the quotient group C = B / f(A) together with the natural...

circle. The kernel of an operator consists of left zero modes, and the cokernel consists of the right zero modes. Vaughn, Michael T. (2008). Introduction...

that: C ≅ B/ker r ≅ B/q(A) (i.e., C isomorphic to the coimage of r or cokernel of q) to: B = q(A) ⊕ u(C) ≅ A ⊕ C where the first isomorphism theorem is...

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...

right-singular vectors of singular value 0 comprise all unit vectors in the cokernel and kernel, respectively, of ⁠ M , {\displaystyle \mathbf {M} ,} ⁠ which...

the kernel of D (solutions of Df = 0), and the (finite) dimension of the cokernel of D (the constraints on the right-hand-side of an inhomogeneous equation...

category has zero morphisms, then one can define the notions of kernel and cokernel for any morphism in that category. Section 1.7 of Pareigis, Bodo (1970)...

Inn(G). By the first isomorphism theorem we get, G/Z(G) ≃ Inn(G). The cokernel of this map is the group Out(G) of outer automorphisms, and these form...

additive category. If all morphisms have a kernel and a cokernel, and all epimorphisms are cokernels and all monomorphisms are kernels, then we speak of an...

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...

This is a list of homological algebra topics, by Wikipedia page. Cokernel Exact sequence Chain complex Differential module Five lemma Short five lemma...

(the nullity of T) plus the dimension of the image (the rank of T). The cokernel of a linear operator T : V → W is defined to be the quotient space W/im(T)...