Generalities on linear representations

  • Jean-Pierre Serre
Part of the Graduate Texts in Mathematics book series (GTM, volume 42)


Let V be a vector space over the field C of complex numbers and let GL(V) be the group of isomorphisms of V onto itself. An element a of GL(V) is, by definition, a linear mapping of V into V which has an inverse a-1; this inverse is linear. When V has a finite basis (e i ) of n elements, each linear map a: V → V is defined by a square matrix (a ij ) of order n. The coefficients a ij are complex numbers; they are obtained by expressing the images a(e j ) in terms of the basis (e i ):
$$ a({e_j}) = \sum\limits_i {{a_{ij}}} {e_i}. $$


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Copyright information

© Springer-Verlag, New York Inc. 1977

Authors and Affiliations

  • Jean-Pierre Serre
    • 1
  1. 1.Chaire d’algèbre et géométrieCollège de FranceParisFrance

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