Let T be a compact torus and (M,ω) a Hamiltonian T-space. We give a new proof of the K-theoretic analogue of the Kirwan surjectivity theorem in symplectic geometry (see Harada-Landweber, 2007) by using the equivariant version of the Kirwan map introduced in Goldin, 2002. We compute the kernel of this equivariant Kirwan map, and hence give a computation of the kernel of the Kirwan map. As an application, we find the presentation of the kernel of the Kirwan map for the T-equivariant K-theory of flag varieties G/T where G is a compact, connected and simply-connected Lie group. In the last section, we find explicit formulae for the K-theory of weight varieties.