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 0_{W} is the zero vector of W, then the kernel of T is the preimage of the singleton set {0_{W} }; that is, the subset of V consisting of all those elements of V that are mapped by T to the element 0_{W}. The kernel is usually denoted as "ker T ", or some variation thereof:
Since a linear transformation preserves zero vectors, the zero vector 0_{V} of V must belong to the kernel. The transformation T is injective if and only if its kernel is only the singleton set {0_{V} }.
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