In linear algebra
and functional analysis
, the kernel
of a linear operator L
is the set of all operands v
for which L
) = 0. That is, if L
where 0 denotes the null vector
. 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.
- 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
2x_1 &&; + ;&& 5x_2 &&; - ;&& 3x_3 &&; = ;&& 0
4x_1 &&; + ;&& 2x_2 &&; + ;&& 7x_3 &&; = ;&& 0
- 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 sum 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.
, then two elements of V
have the same image
if and only if their difference lies in the kernel of L
It follows that the image of L
to the quotient
by the kernel:
is finite dimensional, this implies the rank-nullity theorem
is an inner product space
, the quotient V
) can be identified with the orthogonal complement
). This is is the generalization to linear operators of the row space
of a matrix.
Kernels in functional analysis
are topological vector spaces
is finite-dimensional) then a linear operator L
if and only if the kernel of L
is a closed
subspace of V