linear-operator

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 L: V → W, then

ker(L) = left{ vin V : L(v)=0 right}text{,}

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.

Examples

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:
$L(x_1,x_2,x_3) = (2x_1 + 5x_2 - 3x_3,; 4x_1 + 2x_2 + 7x_3)$
then the kernel of L is the set of solutions to the equations
$begin{alignat}{7}$
```
2x_1 &&; + ;&& 5x_2 &&; - ;&& 3x_3 &&; = ;&& 0 
```
```
4x_1 &&; + ;&& 2x_2 &&; + ;&& 7x_3 &&; = ;&& 0
```
end{alignat}text{.}
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
$L(f) = f(0.3)text{.},$
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:
$D(f) = frac{df}{dx}text{.}$
```
 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
$s(x_1,x_2,x_3,x_4,ldots) = (x_2,x_3,x_4,ldots)text{.}$
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.

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:

L(v) = L(w);;;;Leftrightarrow;;;;L(v-w)=0text{.}

It follows that the image of L is isomorphic to the quotient of V by the kernel:

text{im}(L) cong V / ker(L)text{.}

When V is finite dimensional, this implies the rank-nullity theorem:

dim(ker L) + dim(text{im},L) = dim(V)text{.},

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