Kernél (matematika)

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Kecap kernel mibanda sababaraha harti dina matematik, kadang pakait jeung hal séjén, kadangkala heunteu. Di handap ieu sababaraha artikel ngeunaan kernel leuwih jéntré dina : kernel (algebra), kernel of a function.

Operator Kernel

Dina analisa, tempo hiji operator integral T nu mindahkeun fungsi f kana fungsi Tf dina rumus integral

$(Tf)(x):=\int _{-\infty }^{\infty }K(x,y)f(y)\,dy.$

The function K that appéars in this formula is the kernel of the operator T. This usage applies also to convolution operators such as the Dirichlet kernel. This type of kernel is used in the kernel trick which has applications in several fields of applied mathematics.

Kernels in algebra and category theory

Unrelated to this, if f is any function in any context, then the kernel of f is a certain equivalence relation on the domain of f which is defined in terms of f. For more on this in general, see kernel of a function.

This notion is used héavily in abstract algebra. But in the case of Mal'cev algebras, it can be replaced by a simpler definition; the kernel of a homomorphism f is the preimage under f of the zero element of the codomain. For more on this, see kernel (algebra).

Finally, for this last notion of kernel is generalised in a certain sense in category theory; the kernel of a morphism f is the difference kernel of f and the corresponding zero morphism (if this exists). For more on this, see kernel (category theory).

Kernel of a linear map

Perhaps the best-known case of the concept of kernel in algebra is that of the kernel of a linear map. The kernel of a linéar map is the same thing as its null space.