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Elements of Linear Algebra

  • Antonio Romano
  • Addolorata Marasco
Chapter
  • 2.4k Downloads
Part of the Modeling and Simulation in Science, Engineering and Technology book series (MSSET)

Abstract

In this chapter the fundamental concepts of linear algebra are exposed. More precisely, we introduce vector spaces, bases and contravariant components of a vector relative to a base, together with their transformation formulae on varying the basis. Then, Euclidean vector spaces and some fundamental operations in these spaces are analyzed: vector product, mixed product, etc. The elementary definition of n-tensors is given together with elements of tensor algebra. The problem of eigenvalues of symmetric 2-tensor and orthogonal 2-tensors is widely discussed and Cauchy’s polar decomposition theorem is proved. Finally, the Euclidean point spaces are introduced. The last sections contain exercises and detailed descriptions of some packages written with Mathematica® [69], which allows the user to solve by computer many of the problems considered in this chapter.

Keywords

Euclidean Point Space Orthogonal Tensor Contravariant Components Polar Decomposition Theorem Transformation Formula 
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.

References

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    A. Lichnerowicz, Algèbra et Analyse Linéaires (Masson Paris, 1947)Google Scholar
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    G. Mostov, J. Sampson, J. Meyer, Fundamental Structures of Algebra (Mcgraw-Hill, 1963)Google Scholar
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    A. Romano, Elementi di Algebra Lineare e Geometria Differenziale, Liguori Editore Napoli, (1996)Google Scholar
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    A. Romano, Meccanica Razionale con Elementi di Meccanica Statistica (Liguori Editore Napoli, 1996)Google Scholar
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    C. Tolotti, Lezioni di Meccanica Razionale (Liguori Editore Napoli, 1965)Google Scholar
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    S. Wolfram, Mathematica ®. A System for Doing Mathematics by Computer (Addison-Wesley Redwood City, California, 1991)Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Antonio Romano
    • 1
  • Addolorata Marasco
    • 1
  1. 1.Department of Mathematics and Applications “R. Caccioppoli”University of Naples Federico IINaplesItaly

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