The Exterior Product and Exterior Algebras

  • Igor R. Shafarevich
  • Alexey O. Remizov


The chapter begins with notion of the Plücker coordinates of a subspace in a vector space. Then the Plücker relations are derived, and the Grassmann varieties are described. Then an exterior product of vectors is defined, and the connection between the exterior product and Plücker coordinates is explained: the Plücker relations give necessary and sufficient conditions for an m−vector to be represented as an exterior product of m vectors of the initial vector space. The properties of an exterior product are investigated in greater detail; the notion of exterior algebra is introduced and discussed. Finally, several applications of the obtained theoretical results are considered, for instance, Laplace’s formula for determinant of a square matrix and the Cauchy–Binet formula for the determinant of the matrix product are proved using the exterior product.


Vector Space Projective Space Multiplication Table Homogeneous Element Grassmann Variety 
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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Igor R. Shafarevich
    • 1
  • Alexey O. Remizov
    • 2
  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia
  2. 2.CMAPÉcole Polytechnique CNRSPalaiseau CedexFrance

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