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Math Help - Equivalence classes for an particular relation question

  1. #1
    LHS
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    Equivalence classes for an particular relation question

    Hi,

    If anyone could help me with this I would be very glad! I have said that M=(aij) and M^T=M^-1
    therefore if e1 relates v, where v=(x,y,z) then v=(a11,a21,a31) and all of those values can't be simultaneously zero for M^-1 to exist.. can't seem to get any further!

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  2. #2
    Member HappyJoe's Avatar
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    Hello.

    Maybe it helps you, if you recall that orthogonal matrices always preserve the norm of a vector. So if v is some vector, then the lenght of Mv is equal to the lenght of v, when M is orthogonal.

    So certainly the equivalence class of e_1 can only contain vectors of lenght 1. The question remains if the equivalence class contains all vectors of lenght 1.
    Last edited by HappyJoe; April 20th 2011 at 05:33 AM. Reason: Missed a word
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  3. #3
    LHS
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    so would it be ok to say that as MM^T=I then a11^2+a21^2+a31^2=1 therefore [e1]={ (x,y,z) : x^2+y^2+z^2=1 } as a method of proving that is's a sphere of radius 1?
    Then for the others, you can say they are concentric spheres, for for the equivalence class of (a,b,c) is it a sphere with radius (a^2+b^2+c^2)^0.5? Think that's enough justification?
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  4. #4
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    Notice that HappyJoe only said that all vectors in that equivalence class have length 1 (and it follows that all vectors in any given equivalence class have the same length). He then suggested that you determine whether or not all vectors of length 1 are in that equivalence class.
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