If $T$ is a linear operator on a finite dimensional vector space.prove $R(T^2)$ is subspace of $R(T)$, $R(T^3)$ subspace of $R(T^2)$ and so on

Question

As on question $N(T)$ is subspace of $N(T^2)$ and so on. I tried it well but couldn't get on conclusion.

Answer 1

Even without assuming finite dimensions, if $v\in R(T^{n+1})$, then $v=T^{n+1}w=T^n(Tw)$ for some $w$. Similarly, if $v\in N(T^n)$, then $T^{n+1}v=T(T^nv)=T(0)=0$.