# kernel of a linear mapping

Let $T:V\to W$ 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 $\mathrm{ker}T$. So

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

The kernel is a vector subspace of $V$,
and its dimension^{} (http://planetmath.org/Dimension2) is called the nullity^{} of $T$.

The function $T$ is injective if and only if $\mathrm{ker}T=\{0\}$
(see the attached proof (http://planetmath.org/OperatornamekerL0IfAndOnlyIfLIsInjective)).
In particular, if the dimensions of $V$ and $W$ are equal and finite,
then $T$ is invertible^{} if and only if $\mathrm{ker}T=\{0\}$.

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

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

where ${T|}_{U}$ is the restriction (http://planetmath.org/RestrictionOfAFunction) of $T$ to $U$.

When the linear mappings are given by means of matrices, the kernel of the matrix $A$ is

$$\mathrm{ker}A=\{x\in V\mid Ax=0\}.$$ |

Title | kernel of a linear mapping |

Canonical name | KernelOfALinearMapping |

Date of creation | 2013-03-22 11:58:22 |

Last modified on | 2013-03-22 11:58:22 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 20 |

Author | yark (2760) |

Entry type | Definition |

Classification | msc 15A04 |

Synonym | nullspace |

Synonym | null-space |

Synonym | kernel |

Related topic | LinearTransformation |

Related topic | ImageOfALinearTransformation |

Related topic | Nullity |

Related topic | RankNullityTheorem |