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Volume of a sphere under a linear transformation R3->R4.

  1. Nov 24, 2012 #1
    1. The problem statement, all variables and given/known data
    So there's a linear transformation T: ℝ3 → ℝ4, standard matrix A that satisfies

    det(A e1) = 5, det (A e2) = 4, det (A e3) = 5 and det (A e4) = 5

    If S is the unit sphere, find the 3-dimensional volume of T(S).

    2. Relevant equations
    Volume of sphere = 4/3 * pi * r^3
    Volume based on determinants = det(ATA)


    3. The attempt at a solution
    I know the determinant of a matrix can be seen as the scaling factor for the volume change of a transformation. So the answer will be 4/3 * pi * (something). The (something) will probably be some kind of combination of the determinants above (5,4,5,5), but I have no clue how to find its value. I have spent hours on this question, it's driving me crazy. Help!
     
  2. jcsd
  3. Nov 25, 2012 #2

    haruspex

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    Thinking about the ℝ1 → ℝ2 suggests a root-sum-squares relationship. Can't see how to generalise it yet.
     
  4. Nov 26, 2012 #3

    haruspex

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    Have now proved the following for both ℝ1 → ℝ2 and ℝ2 → ℝ3, but still don't see how to prove it in general.
    Denoting the square matrices formed by appending the basis vectors as Ai, det(ATA) = Ʃ(det(Ai))2.
     
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