Proof of Theorem RSLT

(⇒) By Definition RLT, we know that $\rng{T}\subseteq V$. To establish the reverse inclusion, assume $\vect{v}\in V$. Then since $T$ is surjective (Definition SLT), there exists a vector $\vect{u}\in U$ so that $\lt{T}{\vect{u}}=\vect{v}$. However, the existence of $\vect{u}$ gains $\vect{v}$ membership in $\rng{T}$, so $V\subseteq\rng{T}$. Thus, $\rng{T}=V$.

(⇐) To establish that $T$ is surjective, choose $\vect{v}\in V$. Since we are assuming that $\rng{T}=V$, $\vect{v}\in\rng{T}$. This says there is a vector $\vect{u}\in U$ so that $\lt{T}{\vect{u}}=\vect{v}$, i.e. $T$ is surjective.