Vector Spaces and Modules

  • Serge Lang
Part of the Undergraduate Texts in Mathematics book series (UTM)


Let K be a field. A vector space V over the field K is an additive (abelian) group, together with a multiplication of elements of V by elements of K, i.e. an association
$$ \left( {x,v} \right) \mapsto xv $$
of K × V into V, satisfying the following conditions:
  1. VS 1.

    If 1 is the unit element of K, then 1 υ = υ for all υ ∈ V.

  2. VS 2.

    If c ∈ K and υ, w ∈ V, then c(v + w) = cυ + cw.

  3. VS 3.

    If x, y ∈ K and υ ∈ V, then (x + y)υ = xυ + yv.

  4. VS 4.

    If x, y ∈ K and υ ∈ V, then (xy)υ = x(yυ).



Vector Space Abelian Group Additive Group Unit Element Left Ideal 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag New York Inc. 1987

Authors and Affiliations

  • Serge Lang
    • 1
  1. 1.Department of MathematicsYale UniversityNew HavenUSA

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