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Linear algebra problem

  1. Oct 7, 2009 #1
    Let V be a vector space and let T: V [tex]\rightarrow[/tex] V be a linear transformation. Suppose that n and k are positive integers.

    (a) If w [tex]\in[/tex] V such that T[tex]^{k}[/tex](w)[tex]\neq[/tex]0 and T[tex]^{k+1}[/tex](w)=0, must {w, T(w),...,T[tex]^{k}[/tex](w)} be linearly independent?

    (b) Assuming that w [tex]\in[/tex] V such that T[tex]^{k}[/tex](w)[tex]\neq[/tex]0 and T[tex]^{k+1}[/tex](w)=0. Let W be the subspace of V spanned by {w, T(w),...,T[tex]^{k}[/tex](w)}. If v is a member of V such that T[tex]^{n}[/tex](v)[tex]\notin[/tex]W and T[tex]^{n+1}[/tex](v)[tex]\in[/tex]W, must {w, T(w),...,T[tex]^{k}[/tex](w),v,T(v),...,T[tex]^{n}[/tex](v)} be linearly independent? Explain.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Oct 7, 2009 #2
    (a) What is the definition of linear independence?
     
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