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Linear Algebra Eigenspace Question

  1. Mar 2, 2014 #1
    1. The problem statement, all variables and given/known data
    Let T: C∞(R)→C∞(R) be given by T(f) = f'''' where T sends a function to the fourth derivative.

    a) Find a basis for the 0-eigenspace.
    b) Find a basis for the 1-eigenspace.


    3. The attempt at a solution

    I just want to verify my thought process for this problem. For a), finding the basis for the 0-eigenspace, essentially I needed to find a basis for the vectors v in V such that T(v) = 0v .

    So, would the basis for this 0-eigenspace be all polynomials in P3? If you solve the fourth derivative of any polynomial in P3, you will get 0.

    As for b), when finding the basis for the 1-eigenspace, we need to find a basis for the vectors v in V such that T(v) = 1v, or that after solving the fourth derivative, you get a function that is equal to 1? Is this the correct logic? So would the basis for the 1-eigenspace be any polynomial in P4?

    Thanks much for your help.
     
  2. jcsd
  3. Mar 2, 2014 #2

    PeroK

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    The 1 eigenspace is vectors that map to themselves. Not to the vector 1.
     
  4. Mar 2, 2014 #3
    If by ##C^∞## you mean the space of all infinitely differentiable functions, then there are a lot more than polynomials around.

    Let ##f \in C^∞##. Look at the power series: ##f(x) = ∑_{i=0}^{∞} a_i x^i##. If the fourth derivative of ##f## is 0, then you have that ##a_i = 0## for ##i \geq 4##. Thus the choice of ##a_0, a_1, a_2, a_3## determines ##f## in the 0-eigenspace. Using this, can you come up with a basis? (It will have 4 functions in it).
     
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