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Change of coordinates resulting in scale

  1. Nov 2, 2011 #1
    1. The problem statement, all variables and given/known data
    Consider q: ℝ[itex]^{2}[/itex]→ℝ
    q(x,y) = [itex]\left[x\ y\right] S \left[\stackrel{x}{y}\right][/itex]

    Show that with the change of coordinates
    [itex]\left[\stackrel{x}{y}\right] = Q\left[\stackrel{x'}{y'}\right][/itex]
    that q(x,y) =[itex]\lambda_{1}x'^{2} + \lambda_{2}y'^{2}[/itex]

    2. Relevant equations
    [itex]S = \left[\stackrel{2}{\sqrt{6}}\ \stackrel{\sqrt{6}}{-3}\right][/itex]
    [itex]Q^{-1}SQ = \left[\stackrel{\lambda_{1}}{0}\ \stackrel{0}{\lambda_{2}}\right][/itex]

    3. The attempt at a solution
    Found the eigen values of -4 and 3. Substituted Q<x', y'> for <x, y> everywhere in q(x,y) and got

    q(x, y) = det(S) * ( λ1 x'2 + λ2 y'2)

    Don't know where the scaling by det(S) comes in.

    P.S.: Sorry, I have no idea how to format matrices with Latex.
  2. jcsd
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