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Linear Vector Space

  • K. N. Srinivasa Rao
Part of the Texts and Readings in Physical Sciences book series

Abstract

For convenience, we begin by listing the axioms of a linear vector space once again. Let V be a set containing elements 0, x, y,… and Δ a field with elements 0, a, b, ⋯ , λ, μ, ⋯ . We say that V is a linear vector space over the ground field Δ if the following axioms are satisfied:
  1. (i)

    xV, yVx + yV

     
  2. (ii)

    x + (y + z) = (x + y) + z

     
  3. (iii)

    x + 0 = x

     
  4. (iv)

    x + (−x) = 0

     
  5. (v)

    λ ∈ Δ, xVλxV

     
  6. (vi)

    (λμ)x = λxμx

     
  7. (vii)

    (λ + μ)x = λ(μx)

     
  8. (viii)

    λ(x + y) = λx + λy

     
  9. (ix)

    1x = x; 1 is the unit of Δ

     
Some obvious consequences may be noted.

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References

  1. 1.
    H. Boerner, Representations of groups; with special consideration for the needs of modern physics, [2d rev. ed.], Amsterdam, North-Holland Pub. Co., 1970.zbMATHGoogle Scholar
  2. 2.
    I.M. Gelfand, Lectures on linear algebra, translated by A. Shenitzer, New York, Dover Publications, 1989, (c1961).Google Scholar
  3. 3.
    A. Kurosh, Higher Algebra, translated from the Russian by George Yankovksy, Mir Publishers, Moscow 1972.zbMATHGoogle Scholar
  4. 4.
    F.D. Murnaghan, The Theory of group representations, New York, Dover Publications, 1963, (cl938).zbMATHGoogle Scholar
  5. 5.
    E.P. Wigner, Group theory and its application to the quantum mechanics of atomic spectra, (Translated from the German by J.J. Griffin), New York, Academic Press, 1959.zbMATHGoogle Scholar

Copyright information

© Hindustan Book Agency 2006

Authors and Affiliations

  • K. N. Srinivasa Rao
    • 1
  1. 1.BangaloreIndia

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