  Our main theorem is that the pullback of an associated noncommutative vector
bundle induced by an equivariant map of quantum principal bundles is a
noncommutative vector bundle associated via the same finite-dimensional
representation of the structural quantum group. On the level of $K_{0}$-groups,
we realize the induced map by the pullback of explicit matrix idempotents. We
also show how to extend our result to the case when the quantum-group
representation is infinite dimensional, and then apply it to the
Ehresmann-Schauenburg quantum groupoid. Finally, using noncommutative Milnor's
join construction, we define quantum quaternionic projective spaces together
with noncommutative tautological quaternionic line bundles and their duals. As
a key application of the main theorem, we show that these bundles are stably
non-trivial as noncommutative complex vector bundles.
