Definitions

# Kernel (linear operator)

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
$ker\left(L\right) = left\left\{ vin V : L\left(v\right)=0 right\right\}text\left\{,\right\}$
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 general linear operator is referred to as the null space of the operator.

## 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:
$L\left(x_1,x_2,x_3\right) = \left(2x_1 + 5x_2 - 3x_3,; 4x_1 + 2x_2 + 7x_3\right)$
then the kernel of L is the set of solutions to the equations
begin\left\{alignat\right\}\left\{7\right\}
`2x_1 &&; + ;&& 5x_2 &&; - ;&& 3x_3 &&; = ;&& 0 `
`4x_1 &&; + ;&& 2x_2 &&; + ;&& 7x_3 &&; = ;&& 0`
end{alignat}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
$L\left(f\right) = f\left(0.3\right)text\left\{.\right\},$
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:
$D\left(f\right) = frac\left\{df\right\}\left\{dx\right\}text\left\{.\right\}$
` 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 sum of infinitely many copies of R, and let sR → R be the shift operator
$s\left(x_1,x_2,x_3,x_4,ldots\right) = \left(x_2,x_3,x_4,ldots\right)text\left\{.\right\}$
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:
$L\left(v\right) = L\left(w\right);;;;Leftrightarrow;;;;L\left(v-w\right)=0text\left\{.\right\}$
It follows that the image of L is isomorphic to the quotient of V by the kernel:
$text\left\{im\right\}\left(L\right) cong V / ker\left(L\right)text\left\{.\right\}$
When V is finite dimensional, this implies the rank-nullity theorem:
$dim\left(ker L\right) + dim\left(text\left\{im\right\},L\right) = dim\left(V\right)text\left\{.\right\},$
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 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.