In contrast to regular ordinary differential equations, the problem of accurately setting initial conditions just emerges in the context of differential-algebraic equations where the dynamic degree of freedom of the system is smaller than the absolute dimension of the described process, and the actual lower-dimensional configuration space of the system is deeply implicit. For linear higher-index differential-algebraic equations, we develop an appropriate numerical method based on properties of canonical subspaces and on the so-called geometric reduction. Furthermore, taking into account the fact that higher-index differential-algebraic equations lead to ill-posed problems in naturally given norms, we modify this approach to serve as transfer conditions from one time-window to the next in a time stepping procedure with window-wise overdetermined least-squares collocation. This results in the first fully numerical time-stepping procedures for general linear higher-index initial-value problems.