Answer to Question #215532 in Linear Algebra for Sujata

Question

If dim V

Answer 1

Let $e_1, e_2,\dots,e_n$ be a basis for V. Then there exist positive integers $k_1,k_2,\dots,k_n$ such that $T^{k_i}e_i=0$ for all $i=1,2,\dots,n$. Put $k=\max\{k_1,k_2,\dots,k_n\}$.

Consider any vector $v=a_1e_1+a_2e_2+\dots+a_ne_n\in V$.

$T^kv=a_1T^ke_1+a_2T^ke_2+\dots+a_nT^ke_n=$

$=a_1T^{k-k_1}(T^{k_1}e_1)+a_2T^{k-k_2}(T^{k_2}e_2)+\dots+a_nT^{k-k_n}(T^{k_n}e_n)=0$

Therefore, $T^k=0$, and that means that T is a nilpotent operator.