# zero vector in a vector space is unique

Theorem The zero vector^{} in a vector space is unique.

*Proof.* Suppose $0$ and $\stackrel{~}{0}$ are zero vectors
in a vector space $V$. Then both
$0$ and $\stackrel{~}{0}$ must satisfy
axiom 3 (http://planetmath.org/VectorSpace),
i.e., for all $v\in V$,

$v+0$ | $=$ | $v,$ | ||

$v+\stackrel{~}{0}$ | $=$ | $v.$ |

Setting $v=\stackrel{~}{0}$ in the first equation, and $v=0$ in the second yields $\stackrel{~}{0}+0=\stackrel{~}{0}$ and $0+\stackrel{~}{0}=0$. Thus, using axiom 2 (http://planetmath.org/VectorSpace),

$0$ | $=\stackrel{~}{0}+0$ | |||

$=0+\stackrel{~}{0}$ | ||||

$=\stackrel{~}{0},$ |

and $0=\stackrel{~}{0}$. $\mathrm{\square}$

Title | zero vector in a vector space is unique |
---|---|

Canonical name | ZeroVectorInAVectorSpaceIsUnique |

Date of creation | 2013-03-22 13:37:16 |

Last modified on | 2013-03-22 13:37:16 |

Owner | matte (1858) |

Last modified by | matte (1858) |

Numerical id | 7 |

Author | matte (1858) |

Entry type | Theorem |

Classification | msc 16-00 |

Classification | msc 13-00 |

Classification | msc 20-00 |

Classification | msc 15-00 |

Related topic | IdentityElementIsUnique |