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:
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(i)
x ∈ V, y ∈ V ⇒ x + y ∈ V
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(ii)
x + (y + z) = (x + y) + z
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(iii)
x + 0 = x
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(iv)
x + (−x) = 0
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(v)
λ ∈ Δ, x ∈ V ⇒ λx ∈ V
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(vi)
(λμ)x = λxμx
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(vii)
(λ + μ)x = λ(μx)
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(viii)
λ(x + y) = λx + λy
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(ix)
1x = x; 1 is the unit of Δ
Some obvious consequences may be noted.
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References
H. Boerner, Representations of groups; with special consideration for the needs of modern physics, [2d rev. ed.], Amsterdam, North-Holland Pub. Co., 1970.
I.M. Gelfand, Lectures on linear algebra, translated by A. Shenitzer, New York, Dover Publications, 1989, (c1961).
A. Kurosh, Higher Algebra, translated from the Russian by George Yankovksy, Mir Publishers, Moscow 1972.
F.D. Murnaghan, The Theory of group representations, New York, Dover Publications, 1963, (cl938).
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.
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© 2006 Hindustan Book Agency
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Rao, K.N.S. (2006). Linear Vector Space. In: Linear Algebra and Group Theory for Physicists. Texts and Readings in Physical Sciences. Hindustan Book Agency, Gurgaon. https://doi.org/10.1007/978-93-86279-32-3_3
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DOI: https://doi.org/10.1007/978-93-86279-32-3_3
Publisher Name: Hindustan Book Agency, Gurgaon
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