Kernel (linear operator)

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Main article: Kernel (mathematics)

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 → 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.

Contents 1 Examples 2 Properties 3 Kernels in functional analysis 4 See also

[edit] Examples

1. If L: R^m → 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

   

2. 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.
3. 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.
4. 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.
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.

[edit] 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:

When V is finite dimensional, 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 is the generalization to linear operators of the row space of a matrix.

[edit] 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.

[edit] See also

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