Vector Space and Linear Maps

  • Antonio Romano
Part of the Modeling and Simulation in Science, Engineering and Technology book series (MSSET)


Vectors are usually introduced with a comment that many physical quantities (e.g., displacements, velocities, accelerations, forces, and torques) are conveniently described by oriented segments characterized by intensity, direction, and versus. Then, a vector space E is defined as the set of the oriented segments starting from a given point O. Algebraic operations are introduced in the set E as the addition x + y of two vectors x, yE, the multiplication a x of a real number a by a vector x, the scalar product x ⋅ y, and the cross or vector product x ×y.


Vector Space Basis Change Inverse Matrix Vector Subspace Algebraic Operation 
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.


  1. 1.
    Abraham, R., Marsden, J.E.: Foundation of Mechanics. Benjamin Cummings, Reading, Mass. (1978)Google Scholar
  2. 2.
    Arnold, V.L.: Equazioni Differenziali Ordinarie. Mir, Moscow (1979)Google Scholar
  3. 3.
    Arnold, V.L.: Metodi Matematici della Meccanica Classica. Editori Riuniti, Rome (1979)MATHGoogle Scholar
  4. 4.
    Barger, V., Olsson, M.: Classical Mechanics: A Modern Perspective. McGraw-Hill, New York (1995)Google Scholar
  5. 5.
    Bellomo, N., Preziosi, L., Romano, A.: Mechanics and Dynamical Systems with Mathematica. Birkhäuser, Basel (2000)MATHCrossRefGoogle Scholar
  6. 6.
    Birkhoff, G., Rota, G.C.: Ordinary Differential Equations. Wiley, New York (1989)Google Scholar
  7. 7.
    Bishop, R.L., Goldberg, S.I.: Tensor Analysis on Manifolds. MacMillan, New York (1968)MATHGoogle Scholar
  8. 8.
    Boothby, W.M.: An Introduction to Differential Manifolds and Riemannian Geometry. Academic, San Diego (1975)Google Scholar
  9. 9.
    Borrelli, R.L., Coleman, C.S.: Differential Equations: A Modeling Approach. Prentice-Hall, Englewood Cliffs, NJ (1987)Google Scholar
  10. 10.
    Cercignani, C., Illner, R., Pulvirenti, M.: The Mathematical Theory of Diluite gases. Springer, Berlin Heidelberg New York (1994)Google Scholar
  11. 11.
    Choquet-Bruhat, Y.: Gèometrie Diffèrentielle et Systèmes Extèrieurs. Dunod, Paris (1968)MATHGoogle Scholar
  12. 12.
    Chow, T.L.: Classical Mechanics. Wiley, New York (1995)Google Scholar
  13. 13.
    Crampin, M., Pirani, F.: Applicable Differential Geometry. Cambridge University Press, Cambridge (1968)Google Scholar
  14. 14.
    de Carmo, Manfredo P.: Differential Geometry of Curves and Surfaces. Prentice-Hall, Englewood Cliffs, NJ (1976)Google Scholar
  15. 15.
    Dixon, W.G.: Special Relativity: The Foundations of Macroscopic Physics. Cambridge University Press, Cambridge (1978)Google Scholar
  16. 16.
    Dubrovin, B.A., Novikov, S.P., Fomenko, A.T.: Geometria delle Superfici, dei Gruppi di Trasformazioni e dei Campi. Editori Riuniti, Rome (1987)Google Scholar
  17. 17.
    Dubrovin, B.A., Novikov, S.P., Fomenko, A.T.: Geometria e Topologia delle Varietà. Editori Riuniti, Rome (1988)Google Scholar
  18. 18.
    Dubrovin, B.A., Novikov, S.P., Fomenko, A.T.: Geometria Contemporanea. Editori Riuniti, Rome (1989)Google Scholar
  19. 19.
    Fasano, A., Marmi, S.: Meccanica Analitica. Bollati Boringhieri, Turin (1994)Google Scholar
  20. 20.
    Fawles, G., Cassidy, G.: Analytical Dynamics. Saunders, Ft. Worth, TX (1999)Google Scholar
  21. 21.
    Flanders, H.: Differential Forms with Applications to the Physical Sciences. Academic, New York (1963)MATHGoogle Scholar
  22. 22.
    Fock, V.: The Theory of Space, Time and Gravitation, 2nd edn. Pergamon, Oxford (1964)Google Scholar
  23. 23.
    Goldstein, H.: Classical Mechanics, 2nd edn. Addison-Wesley, Reading, MA (1981)Google Scholar
  24. 24.
    Goldstein, H., Poole, C., Safko, J.: Classical Mechanics, 3rd edn. Addison Wesley, Reading, MA (2002)Google Scholar
  25. 25.
    Hale, J., Koçak, H.: Dynamics and Bifurcations. Springer, Berlin Heidelberg New York (1991)MATHCrossRefGoogle Scholar
  26. 26.
    Hill, R.: Principles of Dynamics. Pergamon, Oxford (1964)MATHGoogle Scholar
  27. 27.
    Hutter, K., van der Ven, A.A., Ursescu, A.: Electromagnetic Field Matter Interactions in Thermoelastic Solids and Viscous Fluids. Springer, Berlin Heidelberg New York (2006)Google Scholar
  28. 28.
    Jackson, E.A.: Perspectives of Nonlinear Dynamic, vol. I. Cambrige University Press, Cambridge (1990)CrossRefGoogle Scholar
  29. 29.
    Jackson, E.A:, Perspectives of Nonlinear Dynamic, vol. II. Cambrige University Press, Cambridge (1990)Google Scholar
  30. 30.
    Josè, J.V., Saletan, E.: Classical Dynamics: A Contemporary Approach. Cambridge University Press, Cambridge (1998)MATHCrossRefGoogle Scholar
  31. 31.
    Khinchin, A.I.: Mathematical Foundations of Statistical Mechanics. Dover, New York (1949)MATHGoogle Scholar
  32. 32.
    Landau, L., Lifshitz, E.: Mechanics, 3rd edn. Pergamon, Oxford (1976)Google Scholar
  33. 33.
    MacMillan, W.D.: Dynamics of Rigid Bodies. Dover, New York (1936)MATHGoogle Scholar
  34. 34.
    Marasco, A.: Lindstedt-Poincarè method and mathematica applied to the motion of a solid with a fixed point. Int. J. Comput. Math. Appl. 40, 333 (2000)MATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Marasco, A., Romano, A.: Scientific Computing with Mathematica. Birkhauser, Basel (2001)MATHCrossRefGoogle Scholar
  36. 36.
    Möller, C.: The Theory of Relativity, 2nd edn. Clarendon, Gloucestershire, UK (1972)Google Scholar
  37. 37.
    Norton, J.D.: General covariance and the foundations of general relativity: eight decades of dispute. Rep. Prog. Phys. 56, 791–858 (1993)CrossRefMathSciNetGoogle Scholar
  38. 38.
    Panowsky, W., Phyllips, M.: Classical Electricity and Magnetism. Addison-Wesley, Reading, MA (1962)Google Scholar
  39. 39.
    Penfield, P., Haus, H.: Electrodynamics of Moving Media. MIT Press, Cambridge, MA (1967)Google Scholar
  40. 40.
    Petrovski, I.G.: Ordinary Differential Equations. Prentice-Hall, Englewood Cliffs, NJ (1966)MATHGoogle Scholar
  41. 41.
    Resnick, R.: Introduction to Special Relativity. Wiley, New York (1971)Google Scholar
  42. 42.
    Rindler, W.: Introduction to Special Relativity. Clarendon Press, Gloucestershire, UK (1991)MATHGoogle Scholar
  43. 43.
    Romano, A., Lancellotta, R., Marasco, A.: Continuum Mechanics using Mathematica, Fundamentals, Applications, and Scientific Computing. Birkhauser, Basel (2006)MATHGoogle Scholar
  44. 44.
    Romano, A., Marasco, A.: Continuum Mechanics, Advanced Topics and Research Trends. Birkhauser, Basel (2010)MATHCrossRefGoogle Scholar
  45. 45.
    Saletan, E.J., Cromer, A.H.: Theoretical Mechanics. Wiley, New York (1971)MATHGoogle Scholar
  46. 46.
    Scheck, F.: Mechanics: From Newton’s Laws to Deterministic Chaos. Springer, Berlin Heidelberg New York (1990)Google Scholar
  47. 47.
    Stratton, J.A.: Electromagnetic Theory. McGraw-Hill, New York (1952)Google Scholar
  48. 48.
    Synge, J.: Relativity: The Special Theory, 2nd edn. North-Holland, Amsterdam (1964) (Reprint 1972)Google Scholar
  49. 49.
    Synge, J.L., Griffith, B.A.: Principles of Mechanics. McGraw-Hill, New York (1959)Google Scholar
  50. 50.
    Thomson, C.J.: Mathematical Statistical Mechanics. Princeton University Press, Princeton, NJ (1972)Google Scholar
  51. 51.
    Truesdell, C., Noll, W.: The Nonlinear Field Theories of Mechanics. Handbuch der Physik, vol. III/3. Springer, Berlin Heidelberg New York (1965)Google Scholar
  52. 52.
    von Westenholz, C.: Differential Forms in Mathematical Physics. North-Holland, New York (1981)Google Scholar
  53. 53.
    Wang, K.: Statistical Mechanics, 2nd edn. Wiley, New York (1987)Google Scholar
  54. 54.
    Whittaker, E.T.: A Treatise on the Analytical Dynamics of Particles and Rigid Bodies. Cambridge University Press, Cambridge (1989)Google Scholar
  55. 55.
    Zhang, W-B.: Differential Equations, Bifurcations, and Chaos in Economy. World Scientific, Singapore (2005)Google Scholar
  56. 56.
    Zhong Zhang, Y.: Special Relativity and Its Experimental Foundations. Advanced Series on Theoretical Physical Science, vol. 4. World Scientific, Singapore (1997)Google Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • Antonio Romano
    • 1
  1. 1.Dipartimento di Matematica e ApplicazioniUniversità degli Studi di NapoliNapoliItalia

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