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Constructing isomorphisms

  1. Nov 22, 2011 #1
    1. The problem statement, all variables and given/known data
    Construct an isomorphism from a 2 by 2 symmetric matrix to R^3.


    2. Relevant equations
    N/A


    3. The attempt at a solution
    I know that for a transformation to be an isomorphic, it must be one-to-one and onto. Would the transform be T:A->R^3 and I would have to choose a general matrix A to test?

    How would I test it not knowing how the transform is mapped?
     
  2. jcsd
  3. Nov 22, 2011 #2

    micromass

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    An isomorphism between which structure?? Vector spaces??

    Anyway, given a symmetric matrix

    [tex]\left(\begin{array}{cc} a & b\\ b & c \end{array}\right)[/tex]

    what element of [itex]\mathbb{R}^3[/itex] would you associate with this matrix??
     
  4. Nov 22, 2011 #3
    Yes vector spaces. What do you mean by what element? Would R^3 simply be some vector v=(v1, v2, v3)?
     
  5. Nov 22, 2011 #4

    micromass

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    Yes, elements of [itex]\mathbb{R}^3[/itex] would just be vectors (a,b,c).
     
  6. Nov 22, 2011 #5
    So then I would just check the the nullspace and the dimension of the range? What would be the form of my answer? A matrix?
     
  7. Nov 22, 2011 #6

    micromass

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    You still need a suggestion for what your isomorphism actually does. To which matrix would you map (a,b,c)?? That is: if I give you three real numbers, how would you make a symmetric matrix out of it??
     
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