# rank-nullity theorem

Let $V$ and $W$ be vector spaces^{} over the same field.
If $\varphi :V\to W$ is a linear mapping, then

$$\mathrm{dim}V=\mathrm{dim}(\mathrm{ker}\varphi )+\mathrm{dim}(\mathrm{im}\varphi ).$$ |

In other words, the dimension^{} of $V$
is equal to the sum (http://planetmath.org/CardinalArithmetic)
of the rank (http://planetmath.org/RankLinearMapping) and nullity^{} of $\varphi $.

Note that if $U$ is a subspace^{} of $V$, then this
(applied to the canonical mapping $V\to V/U$) says that

$$\mathrm{dim}V=\mathrm{dim}U+\mathrm{dim}(V/U),$$ |

that is,

$$\mathrm{dim}V=\mathrm{dim}U+\mathrm{codim}U,$$ |

where $\mathrm{codim}$ denotes codimension.

An alternative way of stating the rank-nullity theorem^{} is
by saying that if

$$0\to U\to V\to W\to 0$$ |

is a short exact sequence^{} of vector spaces, then

$$\mathrm{dim}(V)=\mathrm{dim}(U)+\mathrm{dim}(W).$$ |

In fact, if

$$0\to {V}_{1}\to \mathrm{\cdots}\to {V}_{n}\to 0$$ |

is an exact sequence^{} of vector spaces, then

$$\sum _{i=1}^{\lfloor n/2\rfloor}{V}_{2i}=\sum _{i=1}^{\lceil n/2\rceil}{V}_{2i-1},$$ |

that is, the sum of the dimensions of even-numbered terms is the same as the sum of the dimensions of the odd-numbered terms.

Title | rank-nullity theorem |
---|---|

Canonical name | RanknullityTheorem |

Date of creation | 2013-03-22 16:35:40 |

Last modified on | 2013-03-22 16:35:40 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 7 |

Author | yark (2760) |

Entry type | Theorem |

Classification | msc 15A03 |

Related topic | RankLinearMapping |

Related topic | Nullity |