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Math Help - Image and kernel

  1. #1
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    Image and kernel

    C* is the group of all non-zero complex numbers under multiplication

    H: C*---->C*
    z----->zz (second z is complex conjugate of complex number z)

    This is a homomorphism.

    The identity element in (C*,x) is 1, so

    Ker H = {z E C* : H (z) = 1}
    = {z E C* : zz = 1}
    = { z E C* : z = z}
    = R+

    Because z = z (complex conjugate) for each z E C*, the function H is onto and
    Im(H) = C*.

    Is this correct?

    Also I need to find a group with is isomorphic to Ker (H).
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  2. #2
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    Re: Image and kernel

    you can't just SAY H is a homomorphism, you have to offer some evidence.

    if H(z) = z\overline{z}, to show H is a homomorphism, you need to show that:

    for all z,w \in \mathbb{C}^*, H(zw) = H(z)H(w).

    now H(zw) = (zw)(\overline{zw}) = zw(\overline{z})(\overline{w}) = \dots ? (you should finish this....)

    one way to write z\overline{z} is |z|^2. it should be clear that H cannot possibly be onto, because any image H(z) is always a positive real number. so Im(H) is certainly NOT C*.

    also, ker(H) is NOT {1}, and certainly not R+. what is H(2)? what is H(-1), or H(i)? even better, what is H(√2/2 + i√2/2)? what does it mean to say |z|^2 = 1?

    the kernel of H has an easy to remember shape.
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  3. #3
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    Re: Image and kernel

    Thanks

    Would it

    H (zw) H (zw)

    I still dont understand what the kernel would be or the image.
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  4. #4
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    Re: Image and kernel

    since H(z) = |z|^2, any element of the image is real and positive (that's a big hint).

    ker(H) is the pre-image of the identity in C*, which is 1. what complex numbers satisfy |z|^2 = 1?
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  5. #5
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    Re: Image and kernel

    Thanks for your help.

    So Im (H) would be the set R squared.

    Would ker (H) be i(squared).

    By the way how do you go advanced?
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  6. #6
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    Re: Image and kernel

    use the tags "[ tex]" and "[ /tex]" (without the quotes or the spaces).

    Im(H) is not "R squared" it is the set of all non-zero real squares. there is a much simpler description of this set: the positive reals.

    |z| is often called the modulus of z, it is the same thing as the (euclidean) distance of a point in the complex plane from the origin.

    if z = x+iy, then |z|^2 = x^2 + y^2. what do you call the set of points for which x^2 + y^2 = 1?
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