# dimension (vector space)

Let $V$ be a vector space^{} over a field $K$. We say that $V$ is
*finite-dimensional* if there exists a finite basis of $V$. Otherwise we
call $V$ *infinite-dimensional*.

It can be shown that every basis of $V$ has the same cardinality. We call this cardinality the *dimension* of $V$. In particular, if
$V$ is finite-dimensional, then every basis of $V$ will consist of a finite set^{} ${v}_{1},\mathrm{\dots},{v}_{n}$. We then call the natural number^{} $n$ the *dimension* of $V$.

Next, let $U\subset V$ a subspace^{}. The dimension of the quotient
vector space^{} $V/U$ is called the *codimension* of $U$ relative to $V$.

In circumstances where the choice of field is ambiguous, the dimension of a vector space depends on the choice of field. For example, every complex vector space is also a real vector space, and therefore has a real dimension, double its complex dimension.

Title | dimension (vector space) |
---|---|

Canonical name | DimensionvectorSpace |

Date of creation | 2013-03-22 12:42:31 |

Last modified on | 2013-03-22 12:42:31 |

Owner | rmilson (146) |

Last modified by | rmilson (146) |

Numerical id | 13 |

Author | rmilson (146) |

Entry type | Definition |

Classification | msc 15A03 |

Related topic | dimension3 |

Defines | dimension |

Defines | codimension |

Defines | finite-dimensional |

Defines | infinite-dimensional |