KernelBeispiele

KernelBeispiele is a collection of worked examples that illustrate the concept of a kernel across mathematical and computational disciplines. The term appears in German-language teaching materials and software documentation as a way to organize demonstrations that connect theory with computation.

In linear algebra, the kernel of a linear map T from a vector space V to W is the set of vectors that are mapped to the zero vector. For a matrix A, the kernel consists of all solutions x to Ax = 0, forming a subspace whose dimension relates to the rank by the rank-nullity theorem.

In analysis, a kernel is a function K(x, y) that defines an integral operator (Tf)(x) = ∫ K(x, y) f(y) dy. Common kernels include the Gaussian (radial basis) kernel, polynomial kernels, and sine or exponential variants. Kernels may satisfy symmetry and positive semidefiniteness properties crucial for convergence and representation theorems such as Mercer.

In machine learning, kernels enable the kernel trick, allowing inner products in high-dimensional feature spaces without explicit mapping. KernelBeispiele presents typical kernels, including linear, polynomial, Gaussian (RBF), and sigmoid variants, along with simple calculations such as evaluating k(x, z) for specified x and z and describing the induced feature spaces.

The structure of KernelBeispiele collections is to provide a short theory note, one or more worked examples, and optional exercises. They aim to clarify how the choice of kernel affects results, interpretation, and computational complexity, and to show connections across domains.