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 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.
- If L: Rm → Rn, 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 (x1, 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.
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.
uk:Ядро та образ лінійного оператора