Since $U$ and $V$ are isomorphic, there is at least one isomorphism between them (Definition IVS), say $\ltdefn{T}{U}{V}$. As such, $T$ is an invertible linear transformation.
For $\alpha\in\complexes$ define the linear transformation $\ltdefn{S}{V}{V}$ by $\lt{S}{\vect{v}}=\alpha\vect{v}$. Convince yourself that when $\alpha\neq 0$, $S$ is an invertible linear transformation (Definition IVLT). Then the composition, $\ltdefn{\compose{S}{T}}{U}{V}$, is an invertible linear transformation by Theorem CIVLT. Convince yourself that each nonzero value of $\alpha$ gives rise to a different function for $\compose{S}{T}$, which will require the selection of a nonzero vector from $U$. Then we have constructed infinitely many isomorphisms from $U$ to $V$.