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Homework Help: Algebra proof!

  1. Nov 6, 2005 #1
    Prove: If V is a finite dimensional vector space and T is in L(V), then there exists a finite list of scalars ao,a1,a2,.....,an, not all 0 such that

    aoX + a1T x + a2T^2 x..... + anT^n x = theata

    for all x in V
    my hint for the question is:
    the powers of T are defined as T^0 = I, T^1 = 1, T^2 = TT, T^3 = T^2T
    consider the sequence I, T, T^2, T3,..... in the finite-dimensional vector space L(V).

    please help, have have no clue what to do. Any help would be greatly appriciated.
     
  2. jcsd
  3. Nov 6, 2005 #2

    HallsofIvy

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    Science Advisor

    "theata"? Do you mean the 0 vector? If that is the case then saying
    aoX + a1T x + a2T^2 x..... + anT^n x = 0 with not all a0,a1,... zero is the same as saying that {x, Tx, T^2x, T^3x, ..., T^n x} are linearly dependent.

    Suppose all of {x, Tx, T^x, ..., T^n x} were distinct for n larger than the dimension of T. What does that say about the linear independence of the set? On the other hand, suppose T^k x= T^j x for some k and j. What does THAT say about the linear independence of the set?
     
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