# Kernel (linear operator)

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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 *L*: *V* → *W*, then

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 **R**^{m} → **R**^{n} 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

- If
*L*:**R**^{m}→**R**^{n}, then the kernel of*L*is the solution set to a homogeneous system of linear equations. For example, if*L*is the operator: then the kernel of*L*is the set of solutions to the equations - Let
*C*[0,1] denote the vector space of all continuous real-valued functions on the interval [0,1], and define*L*:*C*[0,1] →**R**by the rule Then the kernel of*L*consists of all functions*f*∈*C*[0,1] for which*f*(0.3) = 0. - Let
*C*^{∞}(**R**) be the vector space of all infinitely differentiable functions**R**→**R**, and let*D*:*C*^{∞}(**R**) →*C*^{∞}(**R**) be the differentiation operator: Then the kernel of*D*consists of all functions in*C*^{∞}(**R**) whose derivatives are zero, i.e. the set of all constant functions. - Let
**R**^{∞}be the direct product of infinitely many copies of**R**, and let*s*:**R**^{∞}→**R**^{∞}be the shift operator Then the kernel of*s*is the one-dimensional subspace consisting of all vectors (*x*_{1}, 0, 0, ...). Note that*s*is onto, despite having nontrivial kernel. - 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 *L*: *V* → *W*, then 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:

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 *L*: *V* → *W* is continuous if and only if the kernel of *L* is a closed subspace of *V*.

## See also

- Kernel (mathematics)
- Null space
- Vector space
- Linear subspace
- Linear operator
- Function space
- Fredholm alternative

## References

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