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Linear transformation proof

  1. Aug 15, 2013 #1
    1. The problem statement, all variables and given/known data
    If T is a linear transformation, proof that Tn is a linear transformation (with nEN).


    2. Relevant equations
    I know that T is a linear application if:
    T(u+v) = T(u) + T(v)
    T(au) = aT(u)

    3. The attempt at a solution
    Actually I don't know how to start using these two affirmations. Can anyone help me with it?
    I know how to do this when it has numbers, but then it comes to these kind of proofs, I don't know how to do this.
     
  2. jcsd
  3. Aug 15, 2013 #2

    ehild

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    Start with T2. Is it true that T2(u+v)=T2(u)+T2(v)? Note that T2(u+v) means T(T(u+v)).

    ehild
     
  4. Aug 15, 2013 #3

    HallsofIvy

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    You can do the general proof "by induction".
     
  5. Aug 20, 2013 #4
    I'm trying to solve it by induction.

    For n = 1 ok.

    Assuming that's ok for n = k.

    For n = k+1

    I don't know if I'm doing it right in this part:

    Tk+1 = Tk.T(u+v) = Tk.(T(u+v)) = Tk(T(u)) + Tk(T(v)). Can I just afirm that's ok since T(u+v) is an application and Tk is an application too?
     
  6. Aug 20, 2013 #5

    HallsofIvy

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    I would have put in one more step. Tk(T(u+ v))= Tk(T(u)+ T(v)), using the "given" fact that T is linear, and then "= Tk(T(u))+ Tk(T(v))" using the "induction hypothesis" that Tk is linear.

    And, of course, you now need to prove that Tn(au)= aTn(u) but that can be done the same way.
     
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