In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the part of the domain which is mapped to the zero vector of the co-domain; the kernel is always a linear subspace of the domain.[1] That is, given a linear map L : V → W between two vector spacesV and W, the kernel of L is the vector space of all elements v of V such that L(v) = 0, where 0 denotes the zero vector in W,[2] or more symbolically:
Properties
The kernel of L is a linear subspace of the domain V.[3][2]In the linear map two elements of V have the same image in W if and only if their difference lies in the kernel of L, that is,
From this, it follows by the first isomorphism theorem that the image of L is isomorphic to the quotient of V by the kernel:In the case where V is finite-dimensional, this implies the rank–nullity theorem:where the term rank refers to the dimension of the image of L, while nullity refers to the dimension of the kernel of L, [4]
That is,so that the rank–nullity theorem can be restated as
The notion of kernel also makes sense for homomorphisms of modules, which are generalizations of vector spaces where the scalars are elements of a ring, rather than a field. The domain of the mapping is a module, with the kernel constituting a submodule. Here, the concepts of rank and nullity do not necessarily apply.
In functional analysis
If V and W are topological vector spaces such that W is finite-dimensional, then a linear operator L: V → W is continuous if and only if the kernel of L is a closed subspace of V.
Representation as matrix multiplication
Consider a linear map represented as a m × n matrix A with coefficients in a fieldK (typically or ), that is operating on column vectors x with n components over K.
The kernel of this linear map is the set of solutions to the equation Ax = 0, where 0 is understood as the zero vector. The dimension of the kernel of A is called the nullity of A. In set-builder notation,The matrix equation is equivalent to a homogeneous system of linear equations:Thus the kernel of A is the same as the solution set to the above homogeneous equations.
Subspace properties
The kernel of a m × n matrix A over a field K is a linear subspace of Kn. 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 x ∈ Null(A) and c is a scalarc ∈ K, then cx ∈ Null(A), since A(cx) = c(Ax) = c0 = 0.
The row space of a matrix
The product Ax can be written in terms of the dot product of vectors as follows: