Linear Transformations and Isomorphisms

  1. 1. Find out which of the transformations are linear. For those that are linear, determine whether they are isomorphisms. T(f(t)) = f'(t) + t^2 from P2 to P2



    2. To be linear, T(f(t)+g(t))=T(f(t)) + T(g(t)), kT(f(t))=T(f(kt))



    3. After testing for linearity, I am thinking that the equation does not fulfill the requirements and therefore there are no isomorphisms, but I'm not sure if I did it right. First, I said that:
    T(f(t)+g(t))=(f'(t)+g'(t))+ t^2
    and T(f(t) + T(g(t))= f'(t)+t^2+g'(t)+t^2=f'(t)+2(t^2)+g'(t)
    which is not equal to the first, therefore it is not linear. Am I going in the right direction with this?

    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Dick

    Dick 25,852
    Science Advisor
    Homework Helper

    You are doing it right. It's not linear.
     
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