# coordinate vector

Let $V$ be a vector space^{} of dimension^{} $n$ over a field $K$. If $({b}_{1},\mathrm{\dots},{b}_{n})$ is a basis of $V$, then any element $v$ of $V$ can be uniquely expressed in the form

$$v={\xi}_{1}{b}_{1}+\mathrm{\dots}+{\xi}_{n}{b}_{n}$$ |

with ${\xi}_{1},\mathrm{\dots},{\xi}_{n}\in K$. The $n$-tuplet (http://planetmath.org/OrderedTuplet) $({\xi}_{1},\mathrm{\dots},{\xi}_{n})$ is called the *coordinate vector* of $v$ with respect to the basis in question. The scalars ${\xi}_{i}$ are the *coordinates* (or the *components* of $v$).

It’s evident that the correspondence

$$v\mapsto ({\xi}_{1},\mathrm{\dots},{\xi}_{n})$$ |

provides a linear isomorphism between the vector space $V$ and the vector space formed by the Cartesian product ${K}^{n}$.

Title | coordinate vector |
---|---|

Canonical name | CoordinateVector |

Date of creation | 2013-03-22 19:02:16 |

Last modified on | 2013-03-22 19:02:16 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 6 |

Author | pahio (2872) |

Entry type | Definition |

Classification | msc 15A03 |

Related topic | ListVector |

Defines | coordinates |

Defines | components |