Kernel (linear algebra)

In linear algebra and functional analysis, the kernel of a linear map L: V → W between two vector spaces or two modules V and W is the set of all elements v of V for which L(v) = 0. That is

$\ker(L) = \left\{ v\in V : L(v)=0 \right\}\text{,}$

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 map R^m → Rⁿ, or, more generally of a linear map between finite dimensional spaces equipped with bases is the same as the null space of the corresponding n × m matrix. Sometimes the kernel of a linear map is referred to as the null space of the map, and the dimension of the kernel is referred to as the map's nullity.

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

$\begin{alignat}{7} 2x_1 &&\; + \;&& 5x_2 &&\; - \;&& 3x_3 &&\; = \;&& 0 \\ 4x_1 &&\; + \;&& 2x_2 &&\; + \;&& 7x_3 &&\; = \;&& 0 . \end{alignat}$

• 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 product 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, ...).

• 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:

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

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

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

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 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 (matrix)
• Vector space
• Linear subspace
• Linear operator
• Function space
• Fredholm alternative

References

• Serge Lang (1987). Linear Algebra. Springer. p. 59. ISBN 9780387964126.