# another T-invariant question

• Nov 12th 2009, 12:16 PM
dannyboycurtis
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?
• Nov 12th 2009, 12:43 PM
tonio
Quote:

Originally Posted by dannyboycurtis
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