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.