# rank-nullity theorem

The sum of the rank and the nullity^{} of a linear mapping gives
the dimension^{} of the mapping’s domain. More precisely, let
$T:V\to W$ be a linear mapping. If $V$ is a
finite-dimensional, then

$$dimV=dimKerT+dimImgT.$$ |

The intuitive content of the Rank-Nullity theorem^{} is the principle that

Every independent

^{}linear constraint takes away one degree of freedom.

The rank is just the number of independent linear constraints on $v\in V$ imposed by the equation

$$T(v)=0.$$ |

The dimension of $V$ is the number of unconstrained degrees of freedom. The nullity is the degrees of freedom in the resulting space of solutions. To put it yet another way:

The number of variables minus the number of independent linear constraints equals the number of linearly independent

^{}solutions.

Title | rank-nullity theorem |

Canonical name | RanknullityTheorem |

Date of creation | 2013-03-22 12:24:09 |

Last modified on | 2013-03-22 12:24:09 |

Owner | rmilson (146) |

Last modified by | rmilson (146) |

Numerical id | 8 |

Author | rmilson (146) |

Entry type | Theorem |

Classification | msc 15A03 |

Classification | msc 15A06 |

Related topic | Overdetermined |

Related topic | Underdetermined |

Related topic | RankLinearMapping |

Related topic | Nullity |

Related topic | UnderDetermined |

Related topic | FiniteDimensionalLinearProblem |