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Show that the T is a linear transformation

  1. May 31, 2016 #1
    1. The problem statement, all variables and given/known data

    T:R2[x] --> R4[x]
    T(f(x)) = (x^3-x)f(x^2)
    2. Relevant equations


    3. The attempt at a solution
    Let f(x) and g(x) be two functions in R2[x].
    T(f(x) + g(x)) = T(f+g(x)) = (x^3-x)(f+g)(x^2) = (x^3-x)f(x^2) + (x^3-x)g(x^2) = T(f(x)) + T(g(x)).
    let a be scalar in R:
    aT(f(x)) = a(x^3-x)f(x^2) = (x^3-x)af(x^2) = T(af(x)).
     
  2. jcsd
  3. May 31, 2016 #2

    Ray Vickson

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    Science Advisor
    Homework Helper

    So, what is your question for us?
     
  4. May 31, 2016 #3
    Title, to show that T is linear transformation, what i did is enough?
     
  5. May 31, 2016 #4

    Ray Vickson

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    I suggest you try to answer that for yourself. Ask yourself: have you verified all the required properties that a linear transformation would have? If you are unsure, then ask yourself which properties you think you have missed.
     
  6. May 31, 2016 #5
    These are the two properties that is required, but the way that it's shown, since it's with a function, is correct?
     
  7. Jun 1, 2016 #6

    Mark44

    Staff: Mentor

    What are R2[x] and R4[x]?
    Are these polynomials with real coefficients of degree less than 2 and 4, respectively?
     
  8. Jun 4, 2016 #7
    that's correct.
     
  9. Jun 4, 2016 #8
    I have been told now it should be R6[x].

    How can i see that it's right?, that's it R6[x] and not R4[x] or i can't see it and it depends on f(x) and it's pre-defined?
     
    Last edited: Jun 4, 2016
  10. Jun 4, 2016 #9

    Mark44

    Staff: Mentor

    Who told you that?
    Since ##f \in R_2[x]##, then f(x) = ax +b, right? According to the formula you posted, what is T[f(x)]?
     
  11. Jun 4, 2016 #10
    Forgot about that simple fact.

    T(f(x)) = T(ax+b)= (x3-x)*(ax2+b) , which is really a polynomial of degree 5 , so the image of transformation is indeed at R6[x] - All the polynomial of degree 5 or less.3
    class teacher.
     
    Last edited: Jun 4, 2016
  12. Jun 4, 2016 #11

    Mark44

    Staff: Mentor

    I got confused since different textbooks describe spaces like R6[x] as "polynomials of degree less than 6" and others describe the as "polynomials of degree less than or equal to 6." Also, the notation I've seen many times is p2[x] to mean the same as your R2[x].

    The work you did in the first post seems OK to me, but I think the intent of the problem is that you should show how things work with the specific spaces that are given. It should be easy to show that T[f + g] = T[f] + T[g] and that T[af] = aT[f], but I believe they want you to use generic functions in R2[x].
     
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