Let T:R^3-->R^3 be a linear map such that T(e1)=(0,0,0) , T(e2)=e3 , T(e3)=(0,0,0).prove that T is nilpotent linear map and R^3 is not T-cyclic.
The matrix corresponding to that map is $\displaystyle \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 1 & 0\end{bmatrix}$. What are the definitions of "nilpotent" and "T-cyclic"?
Nilpotent: A linear map T:V--.V is said to be nilpotent if T^m=0, for some m in N.
T-Cyclic:Given a nilpotent linear map T:V-->V (dim(V)=n )then V is said to be t-Cyclic if there exist a v in V such that {v,T(v),...,Tn-1(v)} is a basis of V..