Kernel (linear algebra)

thumb|300x300px|An example for a kernel- the linear operator <math> L : (x,y) \longrightarrow (x, x)</math> transforms all points on the <math> (x=0, y)</math> line to the zero point <math> (0,0)</math>, thus they form the kernel for the linear operator

In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the part of the domain which is mapped to the zero vector of the co-domain; the kernel is always a linear subspace of the domain. That is, given a linear map between two vector spaces and , the kernel of is the vector space of all elements of such that , where denotes the zero vector in , or more symbolically:

<math display="block">\ker(L) = \left\{ \mathbf{v} \in V \mid L(\mathbf{v})=\mathbf{0} \right\} = L^{-1}(\mathbf{0}).</math>

Properties

thumb|300px|Kernel and image of a linear map from to

The kernel of is a linear subspace of the domain .

That is,

<math display=block>\operatorname{Rank}(L) = \dim(\operatorname{im} L) \qquad \text{ and } \qquad \operatorname{Nullity}(L) = \dim(\ker L),</math>

so that the rank–nullity theorem can be restated as

<math display=block>\operatorname{Rank}(L) + \operatorname{Nullity}(L) = \dim \left(\operatorname{domain} L\right).</math>

When is an inner product space, the quotient <math>V / \ker(L)</math> can be identified with the orthogonal complement in of <math>\ker(L)</math>. This is the generalization to linear operators of the row space, or coimage, of a matrix.

Generalization to modules

The notion of kernel also makes sense for homomorphisms of modules, which are generalizations of vector spaces where the scalars are elements of a ring, rather than a field. The domain of the mapping is a module, with the kernel constituting a submodule. Here, the concepts of rank and nullity do not necessarily apply.

In functional analysis

If and are topological vector spaces such that is finite-dimensional, then a linear operator is continuous if and only if the kernel of is a closed subspace of .

Representation as matrix multiplication

Consider a linear map represented as a matrix with coefficients in a field (typically <math>\mathbb{R}</math> or <math>\mathbb{C}</math>), that is operating on column vectors with components over .

The kernel of this linear map is the set of solutions to the equation , where is understood as the zero vector. The dimension of the kernel of A is called the nullity of A. In set-builder notation,

<math display="block">\operatorname{N}(A) = \operatorname{Null}(A) = \operatorname{ker}(A) = \left\{ \mathbf{x}\in K^n \mid A\mathbf{x} = \mathbf{0} \right\}.</math>

The matrix equation is equivalent to a homogeneous system of linear equations:

<math display="block">A\mathbf{x}=\mathbf{0} \;\;\Leftrightarrow\;\;

\begin{alignat}{7}

a_{11} x_1 &&\; + \;&& a_{12} x_2 &&\; + \;\cdots\; + \;&& a_{1n} x_n &&\; = \;&&& 0 \\

a_{21} x_1 &&\; + \;&& a_{22} x_2 &&\; + \;\cdots\; + \;&& a_{2n} x_n &&\; = \;&&& 0 \\

&& && && && &&\vdots\ \;&&& \\

a_{m1} x_1 &&\; + \;&& a_{m2} x_2 &&\; + \;\cdots\; + \;&& a_{mn} x_n &&\; = \;&&& 0\text{.} \\

\end{alignat}</math>

Thus the kernel of A is the same as the solution set to the above homogeneous equations.

Subspace properties

The kernel of a matrix over a field is a linear subspace of . That is, the kernel of , the set , has the following three properties:

always contains the zero vector, since .
If and , then . This follows from the distributivity of matrix multiplication over addition.
If and is a scalar , then , since .

The row space of a matrix

The product Ax can be written in terms of the dot product of vectors as follows:

<math display="block">A\mathbf{x} = \begin{bmatrix} \mathbf{a}_1 \cdot \mathbf{x} \\ \mathbf{a}_2 \cdot \mathbf{x} \\ \vdots \\ \mathbf{a}_m \cdot \mathbf{x} \end{bmatrix}.</math>

Here, denote the rows of the matrix . It follows that is in the kernel of , if and only if is orthogonal (or perpendicular) to each of the row vectors of (since orthogonality is defined as having a dot product of 0).

The row space, or coimage, of a matrix is the span of the row vectors of . By the above reasoning, the kernel of is the orthogonal complement to the row space. That is, a vector lies in the kernel of , if and only if it is perpendicular to every vector in the row space of .

The dimension of the row space of is called the rank of A, and the dimension of the kernel of is called the nullity of . These quantities are related by the rank–nullity theorem

Even for a well conditioned full rank matrix, Gaussian elimination does not behave correctly: it introduces rounding errors that are too large for getting a significant result. As the computation of the kernel of a matrix is a special instance of solving a homogeneous system of linear equations, the kernel may be computed with any of the various algorithms designed to solve homogeneous systems. A state of the art software for this purpose is the Lapack library.

Notes and references

Bibliography

External links

Khan Academy, Introduction to the Null Space of a Matrix