Constructing isomorphisms

  • Thread starter retracell
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  • #1
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Homework Statement


Construct an isomorphism from a 2 by 2 symmetric matrix to R^3.


Homework Equations


N/A


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?
 

Answers and Replies

  • #2
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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??
 
  • #3
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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??
Yes vector spaces. What do you mean by what element? Would R^3 simply be some vector v=(v1, v2, v3)?
 
  • #4
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Yes vector spaces. What do you mean by what element? Would R^3 simply be some vector v=(v1, v2, v3)?

Yes, elements of [itex]\mathbb{R}^3[/itex] would just be vectors (a,b,c).
 
  • #5
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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?
 
  • #6
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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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