Search results
Results From The WOW.Com Content Network
If 0 S is the zero element of S, then the kernel of f is its kernel as linear map over the integers, or, equivalently, as additive groups. It is the preimage of the zero ideal {0 S}, which is, the subset of R consisting of all those elements of R that are mapped by f to the element 0 S. The kernel is usually denoted ker f (or a variation). In ...
The kernel of a m × n matrix A over a field K is a linear subspace of K n. That is, the kernel of A, the set Null(A), has the following three properties: Null(A) always contains the zero vector, since A0 = 0. If x ∈ Null(A) and y ∈ Null(A), then x + y ∈ Null(A). This follows from the distributivity of matrix multiplication over addition.
As mentioned above, a kernel is a type of binary equaliser, or difference kernel. Conversely, in a preadditive category, every binary equaliser can be constructed as a kernel. To be specific, the equaliser of the morphisms f and g is the kernel of the difference g − f. In symbols: eq (f, g) = ker (g − f).
In set theory, the kernel of a function (or equivalence kernel [1]) may be taken to be either the equivalence relation on the function's domain that roughly expresses the idea of "equivalent as far as the function f {\displaystyle f} can tell", [ 2 ] or
For example, convolution of digit sequences is the kernel operation in multiplication of multi-digit numbers, which can therefore be efficiently implemented with transform techniques (Knuth 1997, §4.3.3.C; von zur Gathen & Gerhard 2003, §8.2). Eq.1 requires N arithmetic operations per output value and N 2 operations for N outputs. That can be ...
The function : is often referred to as a kernel or a kernel function. The word "kernel" is used in mathematics to denote a weighting function for a weighted sum or integral . Certain problems in machine learning have more structure than an arbitrary weighting function k {\displaystyle k} .
Kernel (linear algebra) or null space, a set of vectors mapped to the zero vector; Kernel (category theory), a generalization of the kernel of a homomorphism; Kernel (set theory), an equivalence relation: partition by image under a function; Difference kernel, a binary equalizer: the kernel of the difference of two functions
In operator theory, a branch of mathematics, a positive-definite kernel is a generalization of a positive-definite function or a positive-definite matrix. It was first introduced by James Mercer in the early 20th century, in the context of solving integral operator equations .