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Math Help - generate

  1. #1
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    generate

    Let T: V-> W be a linear map, and let x1,....,xm and y1,...yn be two lists of vector in V. Suppose that

    (a) x1,...xm generate Ker T and
    (b) T(y1),...,T(yn) genterate W.

    Show that the list x1,....,xm, y1,...,yn generates V.
    Pls help
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by mathbeginner View Post
    Pls help
    I helped you with the other one, what have you done for this one?
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  3. #3
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    I know I can use b and show the T is subjective.
    then since T is subjective and T(y1),...(T(yn) generates W then y1,...,yn generates V.
    but I don't know how to show T is subjective.

    and I don't know how to start a yet.
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by mathbeginner View Post
    I know I can use b and show the T is subjective.
    then since T is subjective and T(y1),...(T(yn) generates W then y1,...,yn generates V.
    but I don't know how to show T is subjective.

    and I don't know how to start a yet.
    Firstly, it's [i]surjective[/b], secondly you think that \text{span}\{T(v_1),\cdots,T(v_k)\}=W\implies \text{span}\{v_1,\cdots,v_k\}=V?
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  5. #5
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    Quote Originally Posted by Drexel28 View Post
    Firstly, it's [i]surjective[/b], secondly you think that \text{span}\{T(v_1),\cdots,T(v_k)\}=W\implies \text{span}\{v_1,\cdots,v_k\}=V?
    but how can I show that T is surjective?
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  6. #6
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    I'm working on this question as well. I thought the same but realized that it only proves some vectors in W can be generated by T(b_1*y_1 + ... + b_n*y_n). this is correct, right?

    Do you have any hints? I'm pretty stuck but I've been working on it for a while.

    EDIT> I was thinking of using dimV=dimT(v)+dim(KerT) (v in V), would this help at all?
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