# Kernel (linear operator)

{{#invoke:main|main}} In linear algebra and functional analysis, the kernel of a linear operator L is the set of all operands v for which L(v) = 0. That is, if LV → W, then

${\displaystyle \ker(L)=\left\{v\in V:L(v)=0\right\}{\text{,}}}$

where 0 denotes the null vector in W. The kernel of L is a linear subspace of the domain V.

The kernel of a linear operator Rm → Rn is the same as the null space of the corresponding n × m matrix. Sometimes the kernel of a linear operator is referred to as the null space of the operator, and the dimension of the kernel is referred to as the operator's nullity.

## Examples

1. If LRm → Rn, then the kernel of L is the solution set to a homogeneous system of linear equations. For example, if L is the operator:
${\displaystyle L(x_{1},x_{2},x_{3})=(2x_{1}+5x_{2}-3x_{3},\;4x_{1}+2x_{2}+7x_{3})}$
then the kernel of L is the set of solutions to the equations
{\displaystyle {\begin{alignedat}{7}2x_{1}&&\;+\;&&5x_{2}&&\;-\;&&3x_{3}&&\;=\;&&0\\4x_{1}&&\;+\;&&2x_{2}&&\;+\;&&7x_{3}&&\;=\;&&0\end{alignedat}}{\text{.}}}
2. Let C[0,1] denote the vector space of all continuous real-valued functions on the interval [0,1], and define LC[0,1] → R by the rule
${\displaystyle L(f)=f(0.3){\text{.}}\,}$
Then the kernel of L consists of all functions f ∈ C[0,1] for which f(0.3) = 0.
3. Let C(R) be the vector space of all infinitely differentiable functions R → R, and let DC(R) → C(R) be the differentiation operator:
${\displaystyle D(f)={\frac {df}{dx}}{\text{.}}}$
Then the kernel of D consists of all functions in C(R) whose derivatives are zero, i.e. the set of all constant functions.
4. Let R be the direct product of infinitely many copies of R, and let sR → R be the shift operator
${\displaystyle s(x_{1},x_{2},x_{3},x_{4},\ldots )=(x_{2},x_{3},x_{4},\ldots ){\text{.}}}$
Then the kernel of s is the one-dimensional subspace consisting of all vectors (x1, 0, 0, ...). Note that s is onto, despite having nontrivial kernel.
5. If V is an inner product space and W is a subspace, the kernel of the orthogonal projection V → W is the orthogonal complement to W in V.

## Properties

If LV → W, then two elements of V have the same image in W if and only if their difference lies in the kernel of L:

${\displaystyle L(v)=L(w)\;\;\;\;\Leftrightarrow \;\;\;\;L(v-w)=0{\text{.}}}$

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

${\displaystyle {\text{im}}(L)\cong V/\ker(L){\text{.}}}$

This implies the rank-nullity theorem:

${\displaystyle \dim(\ker L)+\dim({\text{im}}\,L)=\dim(V){\text{.}}\,}$

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 of a matrix.

## Kernels in functional analysis

If V and W are topological vector spaces (and W is finite-dimensional) then a linear operator LV → W is continuous if and only if the kernel of L is a closed subspace of V.