  Accurate initial conditions have the task of precisely capturing and fixing
the free integration constants of the flow considered. This is trivial for
regular ordinary differential equations, but a complex problem for
differential-algebraic equations (DAEs) because, for the latter, these free
constants are hidden in the flow. We deal with linear time-varying DAEs and
obtain an accurate initial condition by means of applying both a reduction
technique and a projector based analysis. The highlighting of two canonical
subspaces plays a special role. In order to be able to apply different DAE
concepts simultaneously, we first show that the very different looking rank
conditions on which the regularity notions of the different concepts
(elimination of unknowns, reduction, dissection, strangeness, and tractability)
are based are de facto consistent. This allows an understanding of regularity
independent of the methods.
