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Linear Algebra Axioms

  1. Jan 5, 2012 #1
    1. The problem statement, all variables and given/known data

    Show that the axiom [itex]\vec{A} + (\vec{B} + \vec{C}) = (\vec{A} + \vec{B}) + \vec{C}[/itex] holds for polynomials of the form [itex]a_0 + a_1 x + a_2 x^2[/itex]

    3. The attempt at a solution
    I'm pretty new to writing proofs for linear algebra so my first question is should I be treating the polynomials as the vectors? That is, should I write something like,

    [itex]a^A_0 + a^A_1 x + a^A_2 x^2 + (a^B_0 + a^B_1 x + a^B_2 x^2 + a^C_0 + a^C_1 x + a^C_2 x^2) = (a^A_0 + a^A_1 x + a^A_2 x^2 + a^B_0 + a^B_1 x + a^B_2 x^2) + a^C_0 + a^C_1 x + a^C_2 x^2[/itex]

    ?
    I don't think this is correct since the polynomials aren't really vectors (right?). But I'm not sure how else to place these polynomials into the axioms.
     
  2. jcsd
  3. Jan 5, 2012 #2
    Yes, the polynomials under their usual operations are vectors ( since they form a vector space, and this is what you are trying to show ). It's simple: any element of a space that is a vector space, is a vector.
    The key point is that you are proving this with respect to a specific operation, and a specific set of objects ( you already know how to add and subtract these elements, you just have to show that they *also* satisfy these other vector space properties )
     
  4. Jan 5, 2012 #3

    lanedance

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    effectively you can treat [itex] a_0 + a_1 x + a_2 x^2 [/itex] as [itex] (a_0,a_1,a_2)^T [/itex] as the polynomials 1,x,x^2 are linearly independent
     
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