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Linear Algebra help

  1. Mar 22, 2008 #1
    1. The problem statement, all variables and given/known dataLet F be the vector space of all functions mapping R into R and let T: F -> F be a linear transformation such that T(e^2x) = x^2, T(e^3x) = sinx, and T(1) = cos5x. Find the following, if it is determined by the data.

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




    2. Relevant equationsNone



    3. The attempt at a solutionI know that a linear transformation preserves vector addition and scalar multiplication. To get e^5x, I need to multiply e^2x and e^3x. Would this be scalar multiplication? I figured it would not be because they are vectors in this circumstance. The other problems are similar. Can I multiply them? if so, why?
     
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  3. Mar 22, 2008 #2

    HallsofIvy

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    I see no attempt on your part to solve these problems. Do you want us to do them for you so you can trick your teacher into not actually teachin g you how to do them and then failing the test?
     
  4. Mar 23, 2008 #3
    Wow well clearly I dont understand the concept involved. I tried to say what I knew/understood. I don't want anyone to do the problems for me I just want a hint or some help how to start so I can at least attempt the problem
     
  5. Mar 23, 2008 #4
    Using scalar multiplication and vector addition alone, you should at least be able to work out (c) and (d).
     
  6. Mar 23, 2008 #5
    The exercise is not to calculate them all.
    "Find the following, if it is determined by the data." i.e.
    So when you have to answer when are they determined by data?
    The answer is that they are determined when they can be written as a linear combination of
    [tex]1, exp(2x), exp(3x)[/tex]
    If that is the case you can use the linerarity of T.
     
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