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Linear Algebra pp 225-243 | Cite as

Multilinear algebra: determinants

  • Larry Smith
Part of the Undergraduate Texts in Mathematics book series (UTM)

Abstract

Let n be a positive integer, and let V1,...,V n ,W be vector spaces. A function
$$ f:{V_1}{\text{x}} \cdots {\text{x}}{V_n} \to W $$
is called a multilinear form iff for each integer i, 1 ≤ in, and each (n − 1)tuple (v1,..., v i +1,..., v n ) the function
$$ F:{V_i} \to W $$
definced by
$$ F(v) = f(v,...,{v_{i - 1}},v,{v_{i + 1}},...,{v_n}) $$
is a linear transformation. For n = 1 a multilinear form is simply a linear transformation. For n = 2 we speak of a bilinear form. Often we just say form for a multilinear form.

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Copyright information

© Springer-Verlag New York Inc. 1984

Authors and Affiliations

  • Larry Smith
    • 1
  1. 1.Mathematisches InstitutUniversität GöttingenGöttingenWest Germany

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