# linear operator

• Oct 12th 2008, 07:27 PM
maria_stoeva
linear operator
Can you please, help me with this:
Let T:V->V be a linear operator on the vector space over the field F. Let v is in V and let m be a positive integer for which v is not equal to 0, T(v) is not equal to 0, ...,T^(m-1)(v) is not equal to 0. Show that {v, T(v), ... , T^(m-1)(v)} is linearly independent set.
• Oct 12th 2008, 08:28 PM
ThePerfectHacker
Quote:

Originally Posted by maria_stoeva
Can you please, help me with this:
Let T:V->V be a linear operator on the vector space over the field F. Let v is in V and let m be a positive integer for which v is not equal to 0, T(v) is not equal to 0, ...,T^(m-1)(v) is not equal to 0. Show that {v, T(v), ... , T^(m-1)(v)} is linearly independent set.

I think you need the condition $\displaystyle T(\bold{v}) \not = \bold{v}$ because otherwise if $\displaystyle T$ is the identity operator than it satisfies the hypothesis but the conclusion fails.
• Oct 12th 2008, 09:36 PM
maria_stoeva
May be is a typo in the book, but I still don't get it
• Oct 13th 2008, 06:32 AM
ThePerfectHacker
Quote:

Originally Posted by maria_stoeva
May be is a typo in the book, but I still don't get it

And even with this assumption the problem as stated is still not true. Let $\displaystyle F = \mathbb{R}$ and $\displaystyle V = \mathbb{R}^2$. Define $\displaystyle T:V\to V$ to be a rotation operator by $\displaystyle \tfrac{\pi}{2}$. If your problem is true it would mean $\displaystyle \{ \bold{i}, T(\bold{i}), T^2 (\bold{i})\}$ is linearly independent. But it is not.
• Oct 13th 2008, 08:24 PM
maria_stoeva
Corrected Linear Operator problem
I asked the teacher and the correct settings are v is not equal to 0, T(v) is not equal to 0, ...,T^(m-1)(v) is not equal to 0 BUT T^m(v) IS equal to 0.
Would you help me?