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A Change of Basis Problem

  1. Jul 6, 2008 #1
    The problem statement, all variables and given/known data
    In [itex]\mathcal{P}_3[/itex] with basis [itex]B = \langle 1 + x, 1 - x, x^2 + x^3, x^2 - x^3 \rangle[/itex] we have this representation.

    [tex]\text{Rep}_B(1 - x + 3x^2 - x^3) = \begin{pmatrix} 0 \\ 1 \\ 1 \\ 2 \end{pmatrix}_B[/tex]

    Find a basis [itex]D[/itex] giving this different representation for the same polynomial.

    [tex]\text{Rep}_D(1 - x + 3x^2 - x^3) = \begin{pmatrix} 1 \\ 0 \\ 2 \\ 0 \end{pmatrix}_D[/tex]

    The attempt at a solution
    I've noticed that

    [tex]1 - x + 3x^2 - x^3 = 1 - x + x^2 + x^3 + 2(x^2 - x^3)[/tex]

    so the first and third component of [itex]D[/itex] could be [itex]1 - x + x^2 + x^3[/itex] and [itex]x^2 - x^3[/itex] respectively. I can guess a possible second and fourth component and then check [itex]D[/itex] to determine if it is a basis. Is there an easier way of accomplishing this?
     
  2. jcsd
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