Prove that $\{I, T, ..., T^k\}$ is linearly dependent (LD) if and only if $\forall v \colon \{v, T(v), ..., T^k(v)\}$ is LD.

Question

I want to solve this question, but my attempts failed :(

Assume that $T: V \rightarrow V$ is a linear transformation on vector space $V$.

Prove that $I, T, T^2, ..., T^k$ are LD $\iff$ $\forall v \in V$: $v, T(v), T^2(v), ..., T^k(v)$ are LD.

My Attemps:

1. Proof of $I, T, ..., T^k$ are LD $\Rightarrow$ $\forall v \in V$: $v, T(v), T^2(v), ..., T^k(v)$ are LD) is easy.

Assume that $I, T, ..., T^k$ are LD so exists $a_0, ..., a_k$ (where at least one of them isn't 0) and $a_0 I + ... + a_k T^k = 0 \Rightarrow a_0 I(v) + ... + a_k T^k(v) = 0 \Rightarrow a_0 v + ... + a_k T^k(v) = 0 \Rightarrow $ $v, T(v), ..., T^k(v)$ are LD.

2. There are some facts that can be proved easily such as:

   • if $\forall v \in V$: $v, T(v), ..., T^k(v)$ are LD for all $v$ exists $a_{v, 0}, ..., a_{v, k-1} \in F$ that $T^k(v) = a_{v, 0} v + ... + a_{v, k-1} T^{k-1}(v)$

but I can't solve this problem with them :(

Can you help me please?

Answer 1

The reciprocal can be proved by contradiction. Actually it is more easily stated as

$\{I,T,T^2,\ldots,T^k\}$ is linearly independent (LI) $\implies$ $\exists v\in V\colon\{v,Tv,T^2v,\ldots,T^kv\}$ is LI.

Put this way the proof might be more straightforward.