An example in my linear algebra textbook asks the following:
Let V be a vector space, and let T : V → V be linear. Prove that $T^2 = T_0$ if and only if $R(T)\subseteq N(T)$.
It gives the following answer:
Suppose $T_2$ = $T_0$. Then for all $u ∈ R(T)$, there exists some $v ∈ V$ such that $Tv = u.0 = T_2v = T(T(v)) = T(u)$, so $u ∈ N(T)$. Thus $R(T) ⊆ N(T)$. Suppose $R(T) ⊆ N(T)$. Then $∀v ∈ V , T(v) ∈ R(T)$ so $T(v) ∈ N(T)$. Thus $0 = T(T(v)) = T ^2v$. So $T^2 = T_0$.
Firstly, I don't understand what it means for the Rank to be contained in the Nullspace. How is that possible? Also, I don't how that would make the square of a transformation equal to the zero vector.
Can someone explain this proof in other terms?