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# kernel of a linear mapping

Let $T\colon V\to W$ be a linear mapping between vector spaces.

The set of all vectors in $V$ that $T$ maps to $0$
is called the *kernel* (or *nullspace*) of $T$,
and is denoted $\ker T$. So

$\ker T=\{\,x\in V\mid T(x)=0\,\}.$ |

The kernel is a vector subspace of $V$, and its dimension is called the nullity of $T$.

The function $T$ is injective if and only if $\ker T=\{0\}$ (see the attached proof). In particular, if the dimensions of $V$ and $W$ are equal and finite, then $T$ is invertible if and only if $\ker T=\{0\}$.

If $U$ is a vector subspace of $V$, then we have

$\ker T|_{U}=U\cap\ker T,$ |

where $T|_{U}$ is the restriction of $T$ to $U$.

Related:

LinearTransformation, ImageOfALinearTransformation, Nullity, RankNullityTheorem

Synonym:

nullspace, null-space, kernel

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

15A04*no label found*

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## Attached Articles

## Corrections

Four Things by antizeus ✓

Definition of Nullity? by NeuRet ✘

two additions by matte ✓

ker =0 <=> invertible by matte ✓

Definition of Nullity? by NeuRet ✘

two additions by matte ✓

ker =0 <=> invertible by matte ✓