Kernel (algebra)

related topics
{math, number, function}

In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective.[1][2] An important special case is the kernel of a matrix, also called the null space.

The definition of kernel takes various forms in various contexts. But in all of them, the kernel of a homomorphism is trivial (in a sense relevant to that context) if and only if the homomorphism is injective. The fundamental theorem on homomorphisms (or first isomorphism theorem) is a theorem, again taking various forms, that applies to the quotient algebra defined by the kernel.

In this article, we first survey kernels for some important types of algebraic structures; then we give general definitions from universal algebra for generic algebraic structures.

Contents

Survey of examples

Linear operators

Let V and W be vector spaces and let T be a linear transformation from V to W. If 0W is the zero vector of W, then the kernel of T is the preimage of the singleton set {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 transformation preserves zero vectors, the zero vector 0V of V must belong to the kernel. The transformation T is injective if and only if its kernel is only the singleton set {0V }.

Full article ▸

related documents
Cardinal number
Gaussian elimination
Numerical analysis
Denotational semantics
Huffman coding
Sequence alignment
Complete lattice
Infinity
Lua (programming language)
Hash function
Stone–Čech compactification
Interval (mathematics)
Entropy (information theory)
Absolute value
Logic programming
Series (mathematics)
Simplex
Ruby (programming language)
RSA
Newton's method
Integration by parts
Functor
Direct sum of modules
Boolean satisfiability problem
Proofs of Fermat's little theorem
Riemann zeta function
List of trigonometric identities
Russell's paradox
Pascal's triangle
Compass and straightedge constructions