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Math Help - Defining an isomorphism....

  1. #1
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    Defining an isomorphism....

    Prove that the field R of real numbers is isomorphic to the ring of all 2x2 matrics of the form (0 0; 0 a) with a a real number.

    It says to let f(a)=(0 0; 0 a), but I don't know how to show that this is one to one and onto.
    Thanks.
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by zhupolongjoe View Post
    Prove that the field R of real numbers is isomorphic to the ring of all 2x2 matrics of the form (0 0; 0 a) with a a real number.

    It says to let f(a)=(0 0; 0 a), but I don't know how to show that this is one to one and onto.
    Thanks.
    Are you sure you don't mean matrices of the form

    \left( \begin{array}{cc}<br />
a & 0 \\<br />
0 & a \end{array}<br />
\right)?

    As the set you gave has no identity.

    Think about it for a bit - what does surjective mean? take an arbitrary matrix of this form. Is this mapped onto by something? What if two matrices were mapped to by the same element, what would this mean with respect to the elements which were mapping to it?

    Are you confident with the definitons of surjectivity and injectivity?
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  3. #3
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    No, the book says matrices of the form (0 0; 0 a). Doesn't this have identity
    (0 0; 0 1)? Identity doesn't have to be (1 0; 0 1)
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  4. #4
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by zhupolongjoe View Post
    No, the book says matrices of the form (0 0; 0 a). Doesn't this have identity
    (0 0; 0 1)? Identity doesn't have to be (1 0; 0 1)
    Yes, you're right. One day I will learn to think before I type.
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  5. #5
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    When are two matrices equal? I guess answering this question will instantaneously make your map become one-to-one and onto.
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  6. #6
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    Okay, I think I got it actually. Maybe I was just making this problem harder than it really is.
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