Explore BrainMass
Share

Explore BrainMass

    Hamilton Quaternions : Kernels and Images

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    Show that the map phi: H->M_2(C) defined by
    phi(a+bi+cj+dk)=(a+b(sqrt -1),c+d(sqrt-1);-c+d(sqrt-1),a-b(sqrt-1)) is a ring homomorphism. Calculate its kernel and describe its image.

    Ps. Here H is the ring of integral Hamilton Quaternions and M_2(C) are 2x2 matrices with complex coefficients.
    notation after the equal sign in the definition of phi is a matrix with the first row of two elements a+b(sqrt-1) and c+d(sqrt-1) and second row of two elements
    -c+d(sqrt-1) and a-b(sqrt-1)

    © BrainMass Inc. brainmass.com October 9, 2019, 6:55 pm ad1c9bdddf
    https://brainmass.com/math/linear-transformation/hamilton-quaternions-kernels-images-102197

    Solution Preview

    Please see the attached file for the complete solution.
    Thanks for ...

    Solution Summary

    Hamilton Quaternions, Kernels and Images are investigated. Integral Hamilton Quaternions and complex coefficients are analyzed.

    $2.19