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T-cyclic Operator - linear algebra

  1. Mar 17, 2007 #1
    1. The problem statement, all variables and given/known data
    Let T be a linear operator on a vector space V, and suppose that V is a T-cyclic subspace of itself. Prove that if U is a linear operator on V, then UT=TU if and only if U=g(T) for some polynomial g(t).


    2. Relevant equations

    They suggest supposing that V is generated by v.
    and choose g(t) such that g(T)(v) = U(v).

    3. The attempt at a solution
    I could get some coherent stuff for one way..

    => assume U=g(T), prove UT=TU.

    first, {v, T(v), T^2(v), ... , T^n-1(v)} is a basis of V (thus V is n-dimensional).
    U=g(T)
    let y=U(x) choosem g(t) such that y = g(T)(v).

    then UT(x)=U(T(x)) = U(T(a_0v + a_1T(v)+...+a_n-1T^n-1(v)) = U(a_0T(v) + a_1T^2(v) + ... +a_n-1T^n(v))
    = a_0UT(v) + a_1UT^2(v) + ... +a_n-1UT^n(v)

    then TU(x)= T(U(x))

    (where x is a linear comb of basis vectors, x=a_0v + a_1T(v)+...+a_n-1T^n-1(v))

    thats as far as i got.. =(

    <=

    the other direction is more difficult for me.. i'm not really sure where to get the proper expression for g(t) from at all.
     
  2. jcsd
  3. Mar 17, 2007 #2

    AKG

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    Homework Helper

    if U = g(T), then UT = TU is obvious, you don't need to worry about v's or x's or anything, it has nothing to do with V being T-cyclic, you don't use the hint, etc. For the other direction, use the hint. Since V is T-cyclic, there does indeed exist some v such that v generates V. U(v) is an element of V, and since v generates V, U(v) is indeed equal to g(T)(v) for some polynomial g. Let g be this polynomial. Now just show that for any x in V, g(T)(x) = U(x). Remember that in this direction, you may assume UT = TU.
     
  4. Mar 14, 2010 #3
    Hi, sorry for reviving this old thread, but I'm doing the same question.. I know we can assume UT = TU for one direction, but I can't make out what that's supposed to mean..
    The two operators commute, so am I supposed to work something from the fact that they commute? ( some property that makes them commute..)

    thank you
     
  5. Mar 14, 2010 #4
    I tried a couple of promising things, but I'm just blindly stabbing at it . How would I go about coming up with a plan for this? I need to know something about UT = TU... :S

    thanks
     
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