Null space

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In mathematics, the null space (also nullspace) of an operator A is the set of all operands v which solve the equation Av = 0. It is also called the kernel of A. In set-builder notation,

$\hbox{Null}(A) = \{\mathbf{v} \in V : A\mathbf{v} = \mathbf{0} \}.$

While the term kernel is used more often, the term null space is sometimes used in the context in which one wants to avoid confusion with integral kernel. One should not confuse the null space with the zero vector space, which is a space of the only zero vector.

If the operator is a linear operator on a vector space, the null space is a linear subspace. Hence the null space is a vector space.

[edit] Example

Consider the function $f (x, y) = x - y$ defined for $x$ and $y$ real numbers, which is linear as $f (x + z, y + w) = (x + z) - (y + w) = f (x, y) + f (z, w)$ . Its null space consists of vectors whose first and second coordinates coincide, that is the set { $(x, x)$ : $x$ is a real number} describing a straight line.

Fix a real vector $y$ and define a linear map $f$ as the dot product of a real vector $x$ and $y$ . Its null space then consists of the zero vector and other vectors orthogonal to $y$ .

[edit] Properties

If A is a matrix, this is a linear subspace of the space of all vectors. The dimension of this linear subspace is called the nullity of A. This can be calculated as the number of columns that don't contain pivots in the row echelon form of the matrix A. The rank-nullity theorem states that the rank of any matrix plus its nullity equals the number of columns of that matrix.

The right singular vectors of A corresponding to zero singular values form a basis for the null space of A.

The null space of A can be used to find and express all solutions (the complete solution) of the equation Ax = b. If x₁ solves this equations it is called a particular solution. The complete solution of the equation is equal to the particular solution added to any vectors from the null space. The particular solutions vary according to b, while the null space vectors do not.

To show how this works we consider each direction. In one direction, if Ay = b, and Av = 0, then it is clear that A(y+v) = Ay+Av = b+0 = b. So y+v is also a solution of Ax=b. In the other direction, if we have another solution z to Ax=b, then A(z−y) = Az−Ay = b−b = 0. Thus the vector u = z−y is in the null space of A and z = y+u. So any other solution can be found by adding a vector from the null space to the single particular solution y.

If a linear mapping A is an isomorphism or equivalently, its determinant when defined is nonzero, then its null space is zero. This is so since conversely, if the null space is nonzero, the map is not one-to-one. The identity map is an example of such A.

If the map is a zero map, then the null space is the same as the domain of the map.

[edit] Finding the null space of a matrix

Consider the matrix

$A=\begin{bmatrix}-2 & -4 & 4 \\ 2 & -8 & 0 \\ 8 & 4 & -12\end{bmatrix}.$

To find its null space, one should find all vectors $v$ such that $A v = 0$ . One proceeds by transforming $A$ to row echelon form

$E=\begin{bmatrix}1 & 0 & -4/3 \\ 0 & 1 & -1/3 \\ 0 & 0 & 0\end{bmatrix}.$

One has that $A v = 0$ if and only if $E v = 0.$ Using the notation $v = [x, y, z] T,$ the latter equality becomes

$\begin{bmatrix}1 & 0 & -4/3 \\ 0 & 1 & -1/3 \\ 0 & 0 & 0\end{bmatrix}\begin{bmatrix}x \\ y \\ z\end{bmatrix}=\begin{bmatrix}0 \\ 0 \\ 0\end{bmatrix}; \begin{bmatrix}x-4z/3 \\ y-z/3 \\ 0\end{bmatrix}=\begin{bmatrix}0 \\ 0 \\ 0\end{bmatrix}; \begin{bmatrix}x=4z/3 \\ y=z/3 \\ 0=0\end{bmatrix}.$