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Homework Help: Show That There Is Only One Linear Transformation Proof Help Please

  1. Feb 6, 2010 #1
    Show That There Is Only One Linear Transformation Proof Help Please!!

    Hi, I have been trying this problem for a couple of days, I have done a proof but I don't know if it makes sense. If you want I can scan it and show it, but if someone can show me how to do it that would be more than amazing, I have midterms coming up soon :confused:

    359eddl.jpg

    Thanks.
     
  2. jcsd
  3. Feb 6, 2010 #2
    Re: Show That There Is Only One Linear Transformation Proof Help Please!!

    Haha... I'm going to assume you are in my class because this is on my assignment for this week!

    I had difficulty with it too but I found this which was immensely helpful.

    Scroll down to proposition 8.3. In the proof they talk about proposition 4.1, which we proved on a previous assignment.

    Hope that helps :)
     

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  4. Feb 6, 2010 #3
    Re: Show That There Is Only One Linear Transformation Proof Help Please!!

    Hey thanks a lot that cleared things up!
     
  5. Feb 7, 2010 #4

    HallsofIvy

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    Re: Show That There Is Only One Linear Transformation Proof Help Please!!

    Suppse there were two such transformations, [itex]T_1[/itex] and [itex]T_2[/itex] [itex]T_1\ne T_2[/itex]. Any vector v can be written in terms of the basis vectors, [itex]v= a_1v_1+ a_2v_2+ \cdot\cdot\cdot+ a_nv_n[/itex].

    Then [itex]T_1(v)=[/itex][itex] T_1(a_1v_1+ a_2v_2+[/itex][itex] \cdot\cdot\cdot+ a_nv_n)= T_1(a_1v_1)+ T_1(a_2v_2)+ \cdot\cdot\cdot+ T_1(a_nv_n)[/itex][itex]a_1T_1(v_1)+ a_2T_1(v_2)+ \cdot\cdot\cdot+ a_nT_1(v_n)[/itex][itex]= a_1w_1+ a_2w_2+ \cdot\cdot\cdot+ a_nw_n[/itex].

    Also [itex]T_2(v)= T_2(a_1v_1+ a_2v_2+[/itex][itex] \cdot\cdot\cdot+ a_nv_n)= T_2(a_1v_1)+ T_2(a_2v_2)+[/itex][itex] \cdot\cdot\cdot+ T_2(a_nv_n)= a_1T_1(v_1)+ a_2T_2(v_2)+ \cdot\cdot\cdot+ a_nT_2(v_n)[/itex][itex]= a_1w_1+ a_2w_2+ \cdot\cdot\cdot+ a_nw_n[/itex].

    That is, for any vector, v, [itex]T_1(v)= T_2(v)[/itex] contradicting the assumption that [itex]T_1\ne T_2[/itex].
     
    Last edited by a moderator: Feb 7, 2010
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