vector space

Let F be a field (or, more generally, a division ring). A vector spaceMathworldPlanetmath V over F is a set with two operations, +:V×VV and :F×VV, such that

  1. 1.

    (𝐮+𝐯)+𝐰=𝐮+(𝐯+𝐰) for all 𝐮,𝐯,𝐰V

  2. 2.

    𝐮+𝐯=𝐯+𝐮 for all 𝐮,𝐯V

  3. 3.

    There exists an element 𝟎V such that 𝐮+𝟎=𝐮 for all 𝐮V

  4. 4.

    For any 𝐮V, there exists an element 𝐯V such that 𝐮+𝐯=𝟎

  5. 5.

    a(b𝐮)=(ab)𝐮 for all a,bF and 𝐮V

  6. 6.

    1𝐮=𝐮 for all 𝐮V

  7. 7.

    a(𝐮+𝐯)=(a𝐮)+(a𝐯) for all aF and 𝐮,𝐯V

  8. 8.

    (a+b)𝐮=(a𝐮)+(b𝐮) for all a,bF and 𝐮V

Equivalently, a vector space is a module V over a ring F which is a field (or, more generally, a division ring).

The elements of V are called vectors, and the element 𝟎V is called the zero vector of V.

Title vector space
Canonical name VectorSpace
Date of creation 2013-03-22 11:49:10
Last modified on 2013-03-22 11:49:10
Owner djao (24)
Last modified by djao (24)
Numerical id 17
Author djao (24)
Entry type Definition
Classification msc 16-00
Classification msc 13-00
Classification msc 20-00
Classification msc 15-00
Classification msc 70B15
Synonym linear space
Related topic Module
Related topic Vector2
Related topic Vector
Related topic VectorSubspace
Defines zero vector