Kernel (EP)

Kernel is an EP by indie rock band Seam. It was released in February 15, 1993 through Touch and Go Records. It contains two new song, an alternate take of Shame and a cover of Breaking Circus.

Track listing

Personnel

References

External links

Kernel

Kernel may refer to:

Computing

Mathematics

Objects

Functions

Kernel (category theory)

In category theory and its applications to other branches of mathematics, kernels are a generalization of the kernels of group homomorphisms, the kernels of module homomorphisms and certain other kernels from algebra. Intuitively, the kernel of the morphism f : X → Y is the "most general" morphism k : K → X that yields zero when composed with (followed by) f.

Note that kernel pairs and difference kernels (aka binary equalisers) sometimes go by the name "kernel"; while related, these aren't quite the same thing and are not discussed in this article.

Definition

Let C be a category. In order to define a kernel in the general category-theoretical sense, C needs to have zero morphisms. In that case, if f : X → Y is an arbitrary morphism in C, then a kernel of f is an equaliser of f and the zero morphism from X to Y. In symbols:

To be more explicit, the following universal property can be used. A kernel of f is an object K together with a morphism k : K → X such that:

Kernel (linear algebra)

In mathematics, and more specifically in linear algebra and functional analysis, the kernel (also known as null space or nullspace) of a linear map L : V → W between two vector spaces V and W, is the set of all elements v of V for which L(v) = 0, where 0 denotes the zero vector in W. That is, in set-builder notation,

Properties of the Kernel

The kernel of L is a linear subspace of the domain V. In the linear map L : V → W, two elements of V have the same image in W if and only if their difference lies in the kernel of L:

It follows that the image of L is isomorphic to the quotient of V by the kernel:

This implies the rank–nullity theorem:

where, by “rank” we mean the dimension of the image of L, and by “nullity” that of the kernel of L.

When V is an inner product space, the quotient V / ker(L) can be identified with the orthogonal complement in V of ker(L). This is the generalization to linear operators of the row space, or coimage, of a matrix.

Application to modules

The notion of kernel applies to the homomorphisms of modules, the latter being a generalization of the vector space over a field to that over a ring. The domain of the mapping is a module, and the kernel constitutes a "submodule". Here, the concepts of rank and nullity do not necessarily apply.