# Nilpotent

• December 20th 2009, 04:04 AM
math.dj
Nilpotent
Let T:V-->V be a nilpotent linear map of index =Dim(V)=n.Then show that dim(Ker(T))=1.

We know that {v,Tv,...,T^n-1(v)} is linearly independent in V.now how do i show further..
• December 20th 2009, 04:14 AM
tonio
Quote:

Originally Posted by math.dj
Let T:V-->V be a nilpotent linear map of index =Dim(V)=n.Then show that dim(Ker(T))=1.

We know that {v,Tv,...,T^n-1(v)} is linearly independent in V.now how do i show further..

You mean: let be $0\neq v\in V$ s.t. $Tv,T^2v,\ldots,T^{n-1}v\neq 0$ , then $\{v,Tv,\ldots,T^{n-1}v\}$ are lin. ind. and thus a basis of $V$...well, you're done, since then $\{Tv,\dots,T^{n-1}v\}$ are lin. ind. AND contained in $Im(T)$ , so now just use the dimensions theorem...

Tonio