Advertisement

Linear Algebra pp 199-225 | Cite as

Linear Transformations

  • Belkacem Said-Houari
Chapter
Part of the Compact Textbooks in Mathematics book series (CTM)

Abstract

In Chap.  1, we defined a matrix as a rectangular array of numbers (Definition  1.1.5). In this chapter, we give the mathematical definition of matrices through linear transformations. We will see that the multiplication of two matrices is equivalent to the composition of two linear transformations. One of the important properties of linear transformations is that they carry some algebraic properties from one vector space to another. Sometimes, this will provide us with the necessary knowledge of some vector spaces, without even studying them in detail, but by rather seeing them as the result of a linear transformation of other well-known vector spaces.

References

  1. 1.
    H. Anton, C. Rorres, Elementary Linear Algebra: with Supplemental Applications, 11th edn. (Wiley, Hoboken, 2011)zbMATHGoogle Scholar
  2. 2.
    M. Artin, Algebra, 2nd edn. (Pearson, Boston, 2011)zbMATHGoogle Scholar
  3. 3.
    S. Axler, Linear Algebra Done Right. Undergraduate Texts in Mathematics, 2nd edn. (Springer, New York, 1997)Google Scholar
  4. 4.
    E.F. Beckenbach, R. Bellman, Inequalities, vol. 30 (Springer, New York, 1965)CrossRefzbMATHGoogle Scholar
  5. 5.
    F. Boschet, B. Calvo, A. Calvo, J. Doyen, Exercices d’algèbre, 1er cycle scientifique, 1er année (Librairie Armand Colin, Paris, 1971)Google Scholar
  6. 6.
    L. Brand, Eigenvalues of a matrix of rank k. Am. Math. Mon. 77(1), 62 (1970)Google Scholar
  7. 7.
    G.T. Gilbert, Positive definite matrices and Sylvester’s criterion. Am. Math. Mon. 98(1), 44–46 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    R. Godement, Algebra (Houghton Mifflin Co., Boston, MA, 1968)zbMATHGoogle Scholar
  9. 9.
    J. Grifone, Algèbre linéaire, 4th edn. (Cépaduès–éditions, Toulouse, 2011)zbMATHGoogle Scholar
  10. 10.
    G.N. Hile, Entire solutions of linear elliptic equations with Laplacian principal part. Pac. J. Math 62, 127–140 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    R.A. Horn, C.R. Johnson, Matrix Analysis, 2nd edn. (Cambridge University Press, Cambridge, 2013)zbMATHGoogle Scholar
  12. 12.
    D. Kalman, J.E. White, Polynomial equations and circulant matrices. Am. Math. Mon. 108(9), 821–840 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    P. Lancaster, M. Tismenetsky, The Theory of Matrices, 2nd edn. (Academic Press, Orlando, FL, 1985)zbMATHGoogle Scholar
  14. 14.
    S. Lang, Linear Algebra. Undergraduate Texts in Mathematics, 3rd edn. (Springer, New York, 1987)Google Scholar
  15. 15.
    L. Lesieur, R. Temam, J. Lefebvre, Compléments d’algèbre linéaire (Librairie Armand Colin, Paris, 1978)zbMATHGoogle Scholar
  16. 16.
    H. Liebeck, A proof of the equality of column and row rank of a matrix. Am. Math. Mon. 73(10), 1114 (1966)Google Scholar
  17. 17.
    C.D. Meyer, Matrix Analysis and Applied Linear Algebra (SIAM, Philadelphia, PA, 2000)CrossRefGoogle Scholar
  18. 18.
    D.S. Mitrinović, J.E. Pečarić, A.M. Fink, Classical and New Inequalities in Analysis. Mathematics and Its Applications (East European Series), vol. 61 (Kluwer Academic, Dordrecht, 1993)Google Scholar
  19. 19.
    C. Moler, C. Van Loan, Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Rev. 45(1), 3–49 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    J.M. Monier, Algèbre et géométrie, PC-PST-PT, 5th edn. (Dunod, Paris, 2007)Google Scholar
  21. 21.
    P.J. Olver, Lecture notes on numerical analysis, http://www.math.umn.edu/~olver/num.html. Accessed Sept 2016
  22. 22.
    F. Pécastaings, Chemins vers l’algèbre, Tome 2 (Vuibert, Paris, 1986)Google Scholar
  23. 23.
    M. Queysanne, Algebre, 13th edn. (Librairie Armand Colin, Paris, 1964)zbMATHGoogle Scholar
  24. 24.
    J. Rivaud, Algèbre linéaire, Tome 1, 2nd edn. (Vuibert, Paris, 1982)zbMATHGoogle Scholar
  25. 25.
    S. Roman, Advanced Linear Algebra. Graduate Texts in Mathematics, vol. 135 (Springer, New York, 2008)Google Scholar
  26. 26.
    H. Roudier, Algèbre linéaire: cours et exercices, 3rd edn. (Vuibert, Paris, 2008)zbMATHGoogle Scholar
  27. 27.
    B. Said-Houari, Differential Equations: Methods and Applications. Compact Textbook in Mathematics (Springer, Cham, 2015)Google Scholar
  28. 28.
    D. Serre, Matrices. Theory and Applications. Graduate Texts in Mathematics, vol. 216, 2nd edn. (Springer, New York, 2010)Google Scholar
  29. 29.
    G. Strang, Linear Algebra and Its Applications, 3rd edn. (Harcourt Brace Jovanovich, San Diego, 1988)zbMATHGoogle Scholar
  30. 30.
    V. Sundarapandian, Numerical Linear Algebra (PHI Learning Pvt. Ltd., New Delhi, 2008)zbMATHGoogle Scholar
  31. 31.
    H. Valiaho, An elementary approach to the Jordan form of a matrix. Am. Math. Mon. 93(9), 711–714 (1986)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Belkacem Said-Houari
    • 1
  1. 1.Department of Mathematics, College of SciencesUniversity of SharjahSharjahUnited Arab Emirates

Personalised recommendations