null space

Null space

The null space of a matrix $A$, denoted $\ker (A)$ or $\text{null}(A)$ or $N(A)$, is the set of all vectors $x$ such that $Ax=0$. It is the set of all solutions to the homogeneous equation $Ax=0$.

$$\ker (A)=\{x\in {\mathbb{K}}^n\mid Ax=0\}$$

Where $\mathbb{K}$ is a field like $\mathbb{R}$ or $\mathbb{C}$, $A\in {\mathbb{K}}^{m\times n}$, and $x\in {\mathbb{K}}^n$ and $\ker (A)\subseteq {\mathbb{K}}^n$ (subspace of the domain of $A$).
The dimension of $\ker (A)$ is $n-\text{rank}(A)$.

A null space vector is any $v$ with $Av=0$. If the null space has dimension $k$, it can be written as $\text{span}\{v_1,\dots ,v_k\}$ for $k$ linearly independent null space vectors. In linear systems of equations, these vectors appear in the general solution: adding any linear combination of them to a particular solution yields another solution.

$\ker (A)$ is orthogonal to the row space of $A$ and $\ker (A^T)$ to the column space of $A$, also called the orthogonal complements.

EXAMPLE

E.g. for

$$\begin{array}{r}A=\left[\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\end{array}\right]\end{array}$$

the vector $[1-1{]}^T$ is in the null space of $A$.

Example

$$A=\left[\begin{array}{ccc}1& 2& 3\\ 2& 4& 6\end{array}\right]$$

$\ker (A)$ contains vectors in ${\mathbb{R}}^3$ and has dimension $2$, since $n=3$ and the rank of $A$ is $1$.
$\ker (A^T)$ contains vectors in ${\mathbb{R}}^2$ and has dimension $1$, since $m=2$ and the rank of $A^T$ is $1$.

Example

$$A=\left[\begin{array}{cc}1& 2\\ 2& 4\end{array}\right],\ker (A)=\text{span}\left\{\left[\begin{array}{c}-2\\ 1\end{array}\right]\right\}$$

Connection to kernel from homomorphism

The kernel of a homomorphism $f:G\to H$ is the set of elements mapping to the identity element of $H$.
A linear transformation $T:V\to W$ is a homomorphism of vector spaces, where the identity is the zero vector.
So $\ker (T)=\{v:T(v)=0\}$. The null space of $A$ is the kernel of the linear transformation $x\mapsto Ax$.