Kernel (algebra)

In algebra, the kernel of a homomorphism (function that preserves the structure) is generally the of 0 (except for groups whose operation is denoted multiplicatively, where the kernel is the inverse image of 1). An important special case is the kernel of a linear map. The kernel of a matrix, also called the null space, is the kernel of the linear map defined by the matrix. The kernel of a homomorphism is reduced to 0 (or 1) if and only if the homomorphism is injective, that is if the inverse image of every element consists of a single element. This means that the kernel can be viewed as a measure of the degree to which the homomorphism fails to be injective. For some types of structure, such as abelian groups and vector spaces, the possible kernels are exactly the substructures of the same type. This is not always the case, and, sometimes, the possible kernels have received a special name, such as normal subgroup for groups and two-sided ideals for rings. Kernels allow defining quotient objects (also called quotient algebras in universal algebra, and cokernels in ). For many types of algebraic structure, the fundamental theorem on homomorphisms (or first isomorphism theorem) states that of a homomorphism is isomorphic to the quotient by the kernel. The concept of a kernel has been extended to structures such that the inverse image of a single element is not sufficient for deciding whether a homomorphism is injective. In these cases, the kernel is a congruence relation. This article is a survey for some important types of kernels in algebraic structures. Kernel (linear algebra) Let V and W be vector spaces over a field (or more generally, modules over a ring) and let T be a linear map from V to W. If 0W is the zero vector of W, then the kernel of T is the of the zero subspace {0W}; that is, the subset of V consisting of all those elements of V that are mapped by T to the element 0W. The kernel is usually denoted as ker T, or some variation thereof: Since a linear map preserves zero vectors, the zero vector 0V of V must belong to the kernel.