Linear algebra problem

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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.

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Homework Statement





Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
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(a) What is the definition of linear independence?
 

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