Null space

Definition

Let $F$ be a field and let $A$ be an $m \times n$ matrix over $F$. The null space of $A$ is $$\operatorname{Null}(A) = \{x \in F^n : Ax = 0\}.$$ It is the kernel of the linear map $x \mapsto Ax$.

Remarks

The null space is a subspace of $F^n$, of dimension $n - \rank A$, the nullity of $A$. Row equivalent matrices have the same null space. The null space of $\begin{pmatrix} 1 & 1 \end{pmatrix}$ is the line spanned by $(1, -1)^T$. Each vector of the null space is orthogonal to each row of $A$, for the dot product.

Explanations

L1 human

The null space of a matrix is the set of solutions of $Ax = 0$. We find a basis of it by row reduction: each free variable gives one basis vector. It is the kernel of the matrix, viewed as a linear map.