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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.
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 ...
In other words, the solution of equation 2, u(x), can be determined by the integration given in equation 3. Although f ( x ) is known, this integration cannot be performed unless G is also known. The problem now lies in finding the Green's function G that satisfies equation 1 .
In mathematics, and specifically in potential theory, the Poisson kernel is an integral kernel, used for solving the two-dimensional Laplace equation, given Dirichlet boundary conditions on the unit disk. The kernel can be understood as the derivative of the Green's function for the Laplace equation.
the kernel is the space of solutions to the homogeneous equation T(v) = 0, and its dimension is the number of degrees of freedom in solutions to T(v) = w, if they exist; the cokernel is the space of constraints on w that must be satisfied if the equation is to have a solution, and its dimension is the number of independent constraints that must ...
In mathematics, the Fredholm integral equation is an integral equation whose solution gives rise to Fredholm theory, the study of Fredholm kernels and Fredholm operators. The integral equation was studied by Ivar Fredholm. A useful method to solve such equations, the Adomian decomposition method, is due to George Adomian.
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