Let $T:V \rightarrow V $ be a linear transformation and $R(T),N(T)$ denote range space and null space for $T$ respectively. $O$ is the zero operator.? The dimension of V is finite and even(to satisfy rank nullity theorem). Is it possible to show $T^2=O \implies R(T) = N(T)$.
I could show $R(T) \subset N(T)$. Whether other inclusion namely $N(T) \subset R(T)$ is possible for such $T$ in general?
Best Answer
Suppose $T^2 = 0$. Then:
$R(T) \subseteq N(T)$ (See here for a proof.)
It is possible for $R(T) = N(T)$. For example, consider $T: \mathbb{R}^2 \to \mathbb{R}^2, (x, y) \mapsto (y, 0)$.
It is not necessarily true that $N(T) \subseteq R(T)$: Consider the zero transformation on any non-zero vector space.