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. That is, given a linear map L : V → W between two vector spaces V 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, or more symbolically:
ker ( L ) = { v ∈ V ∣ L ( v ) = 0 } = L − 1 ( 0 ) . {\displaystyle \ker(L)=\left\{\mathbf {v} \in V\mid L(\mathbf {v} )=\mathbf {0} \right\}=L^{-1}(\mathbf {0} ).}
Properties
The kernel of L is a linear subspace of the domain V. In the linear map L : V → W , {\displaystyle L:V\to W,} two elements of V have the same image in W if and only if their difference lies in the kernel of L, that is,
L ( v 1 ) = L ( v 2 ) ⟺ L ( v 1 − v 2 ) = 0 . {\displaystyle L\left(\mathbf {v} _{1}\right)=L\left(\mathbf {v} _{2}\right)\quad \iff \quad L\left(\mathbf {v} _{1}-\mathbf {v} _{2}\right)=\mathbf {0} .}
From this, it follows by the first isomorphism theorem that the image of L is isomorphic to the quotient of V by the kernel:
im ( L ) ≅ V / ker ( L ) . {\displaystyle \operatorname {im} (L)\cong V/\ker(L).}
In the case where V is finite-dimensional, this implies the rank–nullity theorem:
dim ( ker L ) + dim ( im L ) = dim ( V ) . {\displaystyle \dim(\ker L)+\dim(\operatorname {im} L)=\dim(V).}
where the term rank refers to the dimension of the image of L, dim ( im L ) , {\displaystyle \dim(\operatorname {im} L),} while nullity refers to the dimension of the kernel of L, dim ( ker L ) . {\displaystyle \dim(\ker L).} That is,
Rank ( L ) = dim ( im L ) and Nullity ( L ) = dim ( ker L ) , {\displaystyle \operatorname {Rank} (L)=\dim(\operatorname {im} L)\qquad {\text{ and }}\qquad \operatorname {Nullity} (L)=\dim(\ker L),}
so that the rank–nullity theorem can be restated as
Rank ( L ) + Nullity ( L ) = dim ( domain L ) . {\displaystyle \operatorname {Rank} (L)+\operatorname {Nullity} (L)=\dim \left(\operatorname {domain} L\right).}
When V is an inner product space, the quotient V / ker ( L ) {\displaystyle V/\ker(L)} can be identified with the orthogonal complement in V of ker ( L ) {\displaystyle \ker(L)} . This is the generalization to linear operators of the row space, or coimage, of a matrix.
Generalization to modules
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 field K (typically R {\displaystyle \mathbb {R} } or C {\displaystyle \mathbb {C} } ), 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,
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