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Math Help - Show that T^kU = UT^k

  1. #1
    Newbie Nona's Avatar
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    Show that T^kU = UT^k

    Hello,
    i would like help with this:
    Assume that T : V ....> V is linear and that n and k are positive integers and that a_0,a_1,a_2,.....,a_n are scalars. Let U = a_0IV + a_1T + a_2T^2 +  + a_nT^n. Show that T^kU = UT^k.
    Use from 1 to 6:
    (1) Since
    L(V, V ) is a vector space all of the vector space properties hold for addition and scalar multiplication.
    (2)
    T(U1U2) = (TU1)U2.
    (3) )
    T(U1 + U2) = (TU1) + (TU2).
    (4) (U1 + U2)T = U1T + U2T.
    (5) For any non-negative integers
    i and j, T^iT^j = T^i+j .

    (6) For any scalars
    c and d, (cT )(dU1) = (cd)TU1.

    Thank you
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  2. #2
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    Quote Originally Posted by Nona View Post
    Hello,
    i would like help with this:
    Assume that T : V ....> V is linear and that n and k are positive integers and that a_0,a_1,a_2,.....,a_n are scalars. Let U = a_0IV + a_1T + a_2T^2 + + a_nT^n. Show that T^kU = UT^k.
    Use from 1 to 6:

    (1) Since
    L(V, V ) is a vector space all of the vector space properties hold for addition and scalar multiplication.

    (2)


    T(U1U2) = (TU1)U2.

    (3) )


    T(U1 + U2) = (TU1) + (TU2).

    (4) (U1 + U2)T = U1T + U2T.

    (5) For any non-negative integers




    i and j, T^iT^j = T^i+j .
    (6) For any scalars c and d, (cT )(dU1) = (cd)TU1.


    Thank you


    This is trivial using the given definitions. For example, with k = 2 we get:


    T^2U=T^2(a_0I+a_1T+...+a_nT^n)=T\left[T\left(\sum\limits_{i=0}^na_iT^i\right)\right]= T\left(\sum\limits_{i=0}^na_iT^{i+1}\right)=\sum\l  imits_{i=0}^na_iT^{i+2}

    UT^2=(UT)T=\left[\left(\sum\limits_{i=0}^na_iT^{i}\right)T\right]T =\left[\sum\limits_{i=0}^na_iT^{i+1}\right]T=\sum\limits_{i=0}^na_iT^{i+2}

    Now just apply a simple induction here and you're done.

    Tonio
    Last edited by tonio; November 7th 2009 at 08:14 AM.
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  3. #3
    Newbie Nona's Avatar
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    Thank you very much.
    Is that means: changing 2 to k.
    I have to use all the properties from 1 to 6
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  4. #4
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    Quote Originally Posted by Nona View Post
    Thank you very much.
    Is that means: changing 2 to k.
    I have to use all the properties from 1 to 6

    Well, in fact you need them more than anything else to fully justify the first case which we already did above. For the case k you must assume that it is true for k-1 and then use induction...and yes, still some of the properties 1-6 given.

    Tonio
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  5. #5
    Newbie Nona's Avatar
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    Thank you very much.
    Will try to do it.
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