- #1
awef33
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Let V be a vector space and let T: V \rightarrow V be a linear transformation. Suppose that n and k are positive integers.
(a) If w \in V such that T^{k}(w)\neq0 and T^{k+1}(w)=0, must {w, T(w),...,T^{k}(w)} be linearly independent?
(b) Assuming that w \in V such that T^{k}(w)\neq0 and T^{k+1}(w)=0. Let W be the subspace of V spanned by {w, T(w),...,T^{k}(w)}. If v is a member of V such that T^{n}(v)\notinW and T^{n+1}(v)\inW, must {w, T(w),...,T^{k}(w),v,T(v),...,T^{n}(v)} be linearly independent? Explain.
(a) If w \in V such that T^{k}(w)\neq0 and T^{k+1}(w)=0, must {w, T(w),...,T^{k}(w)} be linearly independent?
(b) Assuming that w \in V such that T^{k}(w)\neq0 and T^{k+1}(w)=0. Let W be the subspace of V spanned by {w, T(w),...,T^{k}(w)}. If v is a member of V such that T^{n}(v)\notinW and T^{n+1}(v)\inW, must {w, T(w),...,T^{k}(w),v,T(v),...,T^{n}(v)} be linearly independent? Explain.
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