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..

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- Dec 20th 2009, 03:04 AMmath.djNilpotent
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.. - Dec 20th 2009, 03:14 AMtonio

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

Tonio