  We develop an algebraic version of Cartan method of equivalence or an analog
of Tanaka prolongation for the (extrinsic) geometry of curves of flags of a
vector space $W$ with respect to the action of a subgroup $G$ of the $GL(W)$.
Under some natural assumptions on the subgroup $G$ and on the flags, one can
pass from the filtered objects to the corresponding graded objects and describe
the construction of canonical bundles of moving frames for these curves in the
language of pure Linear Algebra. The scope of applicability of the theory
includes geometry of natural classes of curves of flags with respect to
reductive linear groups or their parabolic subgroups. As simplest examples,
this includes the projective and affine geometry of curves. The case of
classical groups is considered in more detail.
