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 $\displaystyle 0\neq v\in V$ 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