Cyclic sub-spaces, polynomials proof

Question

In a vector space $V$ of finite dimension over the field $F$, not zero $v\in V$ and $T:V\to V$, prove:

a) $Z(v,T)=\left<v,Tv,T^2v,...\right>$, (Span) is the intersection of the T-invariant subspaces containing $v$.

b) If $w\in Z(v,T)$ then there exist a polynomial $g(x)\in F[x]$ such that $w=g(x)v$ and $\deg g(x)<\deg m_v(x)$. Where $m_v(x)$ is the mynimal polynomial.

c) $u\in V$ not zero $Z(u,T)=Z(v,T)\iff p(x)u=v$, where $p(x)$ is a polynomial relative prime to $m_v(x)$

I don´t have a clue for any of these. Any answer or comment is appreciated.

Answer 1

Part a:

There are two things to prove here:

• Any ($T$-)invariant subspace that contains $v$ contains $Z(v,T)$
• $Z(v,T)$ is an invariant subspace containing $v$.

From there, we can conclude that $Z(v,T)$ is the intersection of all invariant subspaces containing $v$ (How)?

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Part b:

Write $w$ as a linear combination of $v,Tv,T^2v,\dots$.

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Part c:

Presumably you meant to write $p(T)u$. The idea here is similar to the idea for b.