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Math Help - Rigid Motions

  1. #16
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    Re: Rigid Motions

    because M/SM is isomorphic to the 2-element group {-1,1}.

    and these are the cosets:

    SM = rotations, SM*R = reflections.

    this is the isomorphism:

    SM<--->1
    SM*R <---> -1

    note that det(reflection*reflection) = det(rotation) = 1 = (-1)(-1) = det(reflection)det(reflection)
    det(rotation*reflection) = det(reflection) = -1 = (1)(-1) = det(rotation)det(reflection)
    det(reflection*rotation) = det(reflection) = -1 = (-1)(1) = det(reflection)det(rotation)
    det(rotation*rotation) = det(rotation) = 1 = (1)(1) = det(rotation)det(rotation)

    that is, the homomorphism det: GL(2,R) ---> R given by det:A--->det(A) yields a surjective homomorphism (which we also call det)

    det:M--->{-1,1} (which sends A-->det(A)), which yields an ISOmorphism from M/SM---->{-1,1},

    the cosets of SM in M are all the elements of M that all have the same determinant, we have only 2, the index of SM in M is 2.

    one of these sets is SM, the rotation subgroup of M, the other is SM*R (where R is any representative reflection), the set of "improper rotations"

    or reflections (these are called reflections because they are orientation-reversing, and any reflection is of order 2:

    R*R = I, the 2x2 identity matrix).
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  2. #17
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    Re: Rigid Motions

    That was a huge help. Thank you.
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