Let $V$ be a vector space and $T$ a linear transformation from $V$ into $V$. Prove that the following two statements about $T$ are equivalent.
(a) The intersection of the range of $T$ and the null space of $T$ is the zero subspace of $V$.
(b) If $T(T\alpha) = 0$, then $T\alpha = 0$.
MY ATTEMPT
Let us assume (a) in order to prove (b). Since both the null space and the range are subsets of $V$, we have that $V = N\oplus R$. Thus, if $T\alpha$ belongs to the null space, that is to say, $T(T\alpha) = 0$, then $T\alpha = 0$, because it also belongs to the range of $T$ and their intersection is the zero subspace.
Let us assume (b) in order to prove (a). If $T(T\alpha) = 0$ implies that $T\alpha = 0$, it tells us that, if $T\alpha$ belongs to the null space, then its image equals the zero vector. In other words, if $T\alpha\in N\cap R$, then $T\alpha = 0$, which means that $V = N\oplus R$.
Am I reasoning correctly?