# standard basis

If $R$ is a division ring, then the direct sum^{} (http://planetmath.org/DirectSum) of $n$ copies of $R$,

$${R}^{n}=R\oplus \mathrm{\cdots}\oplus R\text{(n times),}$$ |

is a vector space^{}.

The *standard basis for ${R}^{n}$* consists of $n$ elements

$${e}_{1}=(1,0,\mathrm{\dots},0),{e}_{2}=(0,1,0,\mathrm{\dots},0),\mathrm{\dots}\mathit{\hspace{1em}}{e}_{n}=(0,\mathrm{\dots},0,1)$$ |

where each ${e}_{i}$ has $1$ for its $i$th component^{} and $0$ for every other component.
The ${e}_{i}$ are called the *standard basis vectors*.

Title | standard basis |
---|---|

Canonical name | StandardBasis |

Date of creation | 2013-03-22 14:20:07 |

Last modified on | 2013-03-22 14:20:07 |

Owner | Mathprof (13753) |

Last modified by | Mathprof (13753) |

Numerical id | 9 |

Author | Mathprof (13753) |

Entry type | Definition |

Classification | msc 15A03 |

Related topic | BasalUnits |

Defines | standard basis vectors |