1. ## T-invariant basis problem

Hello all, I am having trouble showing linear independence at the indicated step.

The setup:
Let $\displaystyle v\in V$ be a fixed vector in a finite-dimensional vector space $\displaystyle V$. Let $\displaystyle W=\{v,Tv,T^2v,...\}$.

Prove $\displaystyle W$ is $\displaystyle T$-invariant. (Easy)

Suppose that $\displaystyle \dim(W)=k$ and show that $\displaystyle B=\{v,Tv,T^2v,\ldots,T^{k-1}v\}$ is a basis for $\displaystyle W$.

It seems obvious that checking linearly independent is easier than span, but I am getting stuck trying to get all of the coefficients to be zero. Thanks to any and all help ahead of time!

2. Originally Posted by nqramjets
Hello all, I am having trouble showing linear independence at the indicated step.

The setup:
Let $\displaystyle v\in V$ be a fixed vector in a finite-dimensional vector space $\displaystyle V$. Let $\displaystyle W=\{v,Tv,T^2v,...\}$.

Prove $\displaystyle W$ is $\displaystyle T$-invariant. (Easy)

Suppose that $\displaystyle \dim(W)=k$ and show that $\displaystyle B=\{v,Tv,T^2v,\ldots,T^{k-1}v\}$ is a basis for $\displaystyle W$.

It seems obvious that checking linearly independent is easier than span, but I am getting stuck trying to get all of the coefficients to be zero. Thanks to any and all help ahead of time!

Use the lemma that says that if $\displaystyle \{v_1,v_2,\ldots,v_n\}$ is a lin. dep. set, then there's some vector lin. dep. on the PRECEEDING ones (i.e., the 2nd one on the 1st one, or the 3rd on the 1st and 3nd one., etc.)

So: if $\displaystyle \{v,Tv,\ldots\}$ is lin. dep. then there's some natural number r s.r. $\displaystyle T^rv$ is lin. dep. on $\displaystyle \{v,Tv,\ldots,T^{r-1}v\}\Longrightarrow T^rv=a_0v+a_1Tv+\ldots+a_{r-1}T^{r-1}v\,,\,\,a_i\in\mathbb{F}$

As $\displaystyle \{v,Tv,\ldots\}$ is a generating set for $\displaystyle W\,\,\,and\,\,\,\dim W=k$, it's clear from the above that $\displaystyle \{v,Tv,\ldots,T^nv\}$ cannot be lin. dep. for any $\displaystyle n<k-1$ (why?), so it must be that $\displaystyle \{v,Tv,\ldots,T^{k-1}v\}$ is lin. independent...and now end the argument.

Tonio