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T-Cyclic Subspaces

  1. Apr 13, 2010 #1
    1. The problem statement, all variables and given/known data
    For each linear operator T on the vector space V, find an ordered basis for the T-Cyclic subspace generated by the vector z.

    a) V = R4, T(a+b,b-c,a+c,a+d) and z= e1

    2. Relevant equations
    Theorem: Let T be a linear operator on a finite dimensional vector space V, and let W denote the T-cyclic subspace of V generated by a nonzero vector v [tex]\epsilon[/tex] V. Let k = dim(w). Then:

    a) {v, T(v), T2(v),..., Tk-1(v)} is a basis for W.

    3. The attempt at a solution
    v= (1,0,0,0), T(v)= (1,0,1,1), T2(v)= T(T(v))= (1,-1,2,2), T3(v)= T(T2(v)) = (0,-3,3,3)

    so basis for W = {(1,0,0,0), (1,0,1,1), (1,-1,2,2), (0,-3,3,3)} ?
  2. jcsd
  3. Apr 13, 2010 #2


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    Homework Helper

    the transforms look ok, but the theorem assumes you know k the dimisenion of the T-cyclic subspace generated... how do you know it is 4?

    note that
    -(1,0,1,1) + (1, -1, 2, 2) = (0,-1,1,1)
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