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Math Help - another T-invariant question

  1. #1
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    another T-invariant question

    Let T be a linear operator on a vector space V, and let v be a nonzero vector in V, and let W be the T-cyclic subspace of V generated by v.

    My question is, how do I show that W is T-invariant, and how do I show that any T-invariant subspace of V containing v also contains W?
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  2. #2
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    Quote Originally Posted by dannyboycurtis View Post
    Let T be a linear operator on a vector space V, and let v be a nonzero vector in V, and let W be the T-cyclic subspace of V generated by v.

    My question is, how do I show that W is T-invariant, and how do I show that any T-invariant subspace of V containing v also contains W?

    Let W=Span\,\{v,Tv,T^2v,...,T^kv\}\Longrightarrow\,\fo  rall\,w =\sum\limits_{i=0}^ka_iT^iv\in\,Span\{v,Tv,...,T^k  v\}\,,\,\,Tw= \sum\limits_{i=0}^ka_iT^{i+1}v=\sum\limits_{i=0}^{  k-1}a_iT^{i+1}v+a_kT^{k+1}v

    =\sum\limits_{i=0}^{k-1}a_iT^{i+1}v+a_k\left(\sum\limits_{i=0}^{k-1}b_iT^iv\right)\in\,Span\{v,Tv,...,T^kv\} , as T^{k+1}v\in Span\{v,Tv,...,T^kv\} per definition

    If W is T-invariant and contains v ==> it contains Tv ==> it contains T(Tv)=T^2(v), etc.

    Tonio
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