T-cyclic Operator - linear algebra

  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. AKG

    AKG 2,585
    Science Advisor
    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. 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. 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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