Kernmenge

Kernmenge is a term used in German-language mathematics and related disciplines to denote the kernel of a function, morphism, or related structure. The exact meaning depends on context, but it generally refers to the set of elements that map to a distinguished value, such as the identity element or zero.

In algebra, the kernel of a homomorphism f from a structure G to H is defined as Ker(f) = {g in G | f(g) = e_H}, where e_H is the identity in H. The kernel is a normal subgroup of G and measures the non-injectivity of f. In linear algebra, for a linear map T: V -> W, the kernel is Ker(T) = {v in V | T(v) = 0}, the null space of T, which is a subspace of V. The rank-nullity theorem relates the dimension of the kernel to the dimension of V and the rank of T.

In other branches of mathematics, the concept of a kernel often extends as the preimage of a particular value under a map, and kernels play a role in constructing quotient structures or analyzing fibers of maps.

Outside pure mathematics, Kernmenge can appear in algorithmic and data-analytic contexts to denote a core set or core-set, a small, representative subset of a larger dataset used to approximate computations such as clustering, regression, or other optimization tasks. The exact terminology and usage can vary between fields and authors, and in some cases related concepts are described with different terms.

Examples illustrating the idea include the kernel of a matrix, the set of solutions to Ax = 0 for a given matrix A, and the interpretation of the kernel as the space of directions in which a linear transformation has no effect. See also Kernel, Null space, Core-set.