Q:

What can you say about the Linear transformation?

if $T:V \to V$ be a linear transformation and the composite $T(T)=0$,then $T$?
1) Kernel of $T$ is subset of image of $T$.
2)Image of $T$ is subset of kernel of $T$.
3) $T$ is a zero transformation.
4)$T$ is a non singular linear transformation.

A:

If $T (T)=0$, then you can conclude that $Im ( T)$ is a subset of $Ker (T)$.
To show: $Im(T) \subset Ker(T)$
Let $y \in Im (T)$ then there exists $x \in V$ such that $T(x)=y$,
now, $T(T)=0 \Rightarrow T(T(x))=0 \Rightarrow T(y)=0 \Rightarrow y\in Ker (T) .$
Thus,$Im(T) \subset Ker(T).$

