  We construct explicit BPS and non-BPS solutions of the U(2k) Yang-Mills
equations on the noncommutative space R^{2n}_\theta x S^2 with finite energy
and topological charge. By twisting with a Dirac multi-monopole bundle over
S^2, we reduce the Donaldson-Uhlenbeck-Yau equations on R^{2n}_\theta x S^2 to
vortex-type equations for a pair of U(k) gauge fields and a bi-fundamental
scalar field on R^{2n}_\theta. In the SO(3)-invariant case the vortices on
R^{2n}_\theta determine multi-instantons on R^{2n}_\theta x S^2. We show that
these solutions give natural physical realizations of Bott periodicity and
vector bundle modification in topological K-homology, and can be interpreted as
a blowing-up of D0-branes on R^{2n}_\theta into spherical D2-branes on
R^{2n}_\theta x S^2. In the generic case with broken rotational symmetry, we
argue that the D0-brane charges on R^{2n}_\theta x S^2 provide a physical
interpretation of the Adams operations in K-theory.
