Show that the set {I, T, T2 , . . . } of powers of T is linearly independent in L(V ).

Question

Let V = $F^N$ be the vector space of all sequences {$a_n$}$_{n≥1} $ of elements of $F$.

Define T : V → V by T($a_1, a_2, a_3, . . .$) = ($a_2, a_3, a_4, . . .$).

Show that the set {$I, T, T^2 , . . . $} of powers of T is linearly independent in $L(V )$.

I've been told the following can be generalized into a proof, but I'm not sure where to start! Consider $ [aI +bT](0,1,0,0,0,0,....) = (0,0,0,0,.....) $

Then $(b,a,0,0,.....) = (0,0,0,0,....) $

Therefore $a=b=0.$

Answer 1

Assume $S=\sum_{i=0}^n c'_{n_i}T^{n_i}=0$ and not all $c_{n_i}$ is 0. Let $n=max(n_i)$. The subset sum can be written as $S=\sum_{i=0}^n c_iT^i=0$ with at least a $c_i\ne$0, and let k be the least index such that $c_k\ne 0$. Consider the sequence $a=\{a_i\}$: $a_i=0$ for $i>1$ and $a_1=1$. Let $b=\{b_i\}=Sa$. $b_{k+1}\ne 0$.

Answer 2

Let $x=(a_n) \in V$ such that $a_{n+1}=1$ and $a_j=0$ for $j \ne n+1$.

If $s_0,s_1,...,s_n \in F$, then we have

$[s_0+s_1T+...+s_nT^n](x)=(s_n,s_{n-1},....,s_0,0,0,...)$.

Can you proceed ?