Exercise 16

Let $f$ be a correspondence between vector spaces $V$ and $W$ (that is, a map that is one-to-one and onto). Show that the spaces $V$ and $W$ are isomorphic via $f$ if and only if there are bases $B\subset V$ and $D\subset W$ such that corresponding vectors have the same coordinates: ${\rm Rep}_{B}(\vec{v})={\rm Rep}_{D}(f(\vec{v}))$.