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Thread: Proving ismorphism

  1. #1
    Aug 2012

    Proving ismorphism

    Define on RXR by setting (a,b)(c,d)=(acbd,ad+bc)Show that the function h from system (C,)(RxR,) given by h(a+bi)=(a,b) is a one to one function from the set of complex numbers that is onto RxR and is operation perserving
    I dont really have a background in complex numbers so i am kinda having difficulties
    Last edited by mathrld; Aug 16th 2012 at 07:57 PM.
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  2. #2
    MHF Contributor

    Mar 2011

    Re: Proving ismorphism

    for "operation-preserving" to make sense, you have to say WHICH operation on C, h is preserving. in this particular case, it is clear to me the operation is "complex multiplication".

    in particular, since C is a field, the usual laws of arithmetic apply (we can "FOIL"), so:

    (a+bi)(c+di) = ac + a(di) + (bi)c + (bi)(di) = ac + (ad)i + (bc)i + (bd)i2.

    now in the complex numbers, we have i2 = -1, so:

    ac + (ad)i + (bc)i + (bd)i2 = ac - bd + (ad + bc)i.

    what you have to show is the following:

    h((a+bi)(c+di)) = h(a+bi)⊕h(c+di), that is:

    h(ac-bd + (ad+bc)i) = h(a+bi)⊕h(c+di).

    you will have to use the definition of h, and the definition of ⊕.
    Thanks from mathrld
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