Proof of Theorem CSLTS

That the composition is a linear transformation was established in Theorem CLTLT, so we need only establish that the composition is surjective. Applying Definition SLT, choose $\vect{w}\in W$.

Because $S$ is surjective, there must be a vector $\vect{v}\in V$, such that $\lt{S}{\vect{v}}=\vect{w}$. With the existence of $\vect{v}$ established, that $T$ is surjective guarantees a vector $\vect{u}\in U$ such that $\lt{T}{\vect{u}}=\vect{v}$. Now, \begin{align*} \lt{\left(\compose{S}{T}\right)}{\vect{u}}&=\lt{S}{\lt{T}{\vect{u}}}&&\knowl{./knowls/definition.LTC.knowl}{\text{Definition LTC}}\\ &=\lt{S}{\vect{v}}&&\text{Definition of $\vect{u}$}\\ &=\vect{w}&&\text{Definition of $\vect{v}$} \end{align*}

This establishes that any element of the codomain ($\vect{w}$) can be created by evaluating $\compose{S}{T}$ with the right input ($\vect{u}$). Thus, by Definition SLT, $\compose{S}{T}$ is surjective.