Are Elements Formed by Linear Transformations Linearly Independent?

[awef33]

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)$$\neq$$0 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)$$\neq$$0 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)$$\notin$$W and T$$^{n+1}$$(v)$$\in$$W, must {w, T(w),...,T$$^{k}$$(w),v,T(v),...,T$$^{n}$$(v)} be linearly independent? Explain.

 

[Donaldos]

(a) What is the definition of linear independence?