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Another Linear Transformation problem

  1. Jun 26, 2009 #1
    1. The problem statement, all variables and given/known data
    Let F be the vector space of all functions mapping R into R, and letT:F-F be a linear transformationsuch that T(e^2x)=x^2, T(e^3x)= sinx, and T(1)= cos5x. Find the following, if it is determined by this data.


    2. Relevant equations
    a. T(e^5x)
    b. T(3+5e^3x)
    c. T(3e^4x)
    d. T((e^4x + 2e^5x)/e^2x)

    3. The attempt at a solution
    a. T(e^2x)*T(e^3x)= (x^2)sinX?
    b. 3T(1)+5T(e^3x)=3cosx + 5sinx
    c. 3T(e^2x)T(e^2x)= 3x^4
    d. T((e^4x)/(e^2x))+2T((e^5x)/(e^2x))= T(e^2x)+2T(e^3x)= (x^2) + (2sinX)

    Is this right?
     
  2. jcsd
  3. Jun 26, 2009 #2

    Mark44

    Staff: Mentor

    Looks OK. I didn't check the last one very closely, but you have the right idea.
     
  4. Jun 27, 2009 #3

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    I am very hesitant to disagree with Mark44, but generally it is NOT true that T(uv)= T(u)T(v) for a vector space- in fact, the product of two vectors is not part of the definition of "vector space". Is the product of functions somehow being used as the "vector sum"? If so what is the "negative" of the 0 function?
     
  5. Jun 27, 2009 #4
    Solutions a. and c. are incorrect, for the reason cited by HallsOfIvy.

    "linear transformation" does not specify what happens on products.
     
  6. Jun 27, 2009 #5

    Mark44

    Staff: Mentor

    Mea culpa
     
  7. Jun 27, 2009 #6
    So are any of these solveable other than b. based on the given information?
     
  8. Jun 28, 2009 #7

    Mark44

    Staff: Mentor

    Parts b and d can be done with the information given; parts a and c cannot. Your answer for b is partly correct (T(1) = cos(5x), not cos(x)), and your answer for d is correct.
     
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