We consider a finite set E of points in the n-dimensional affine space
and two sets of objects that are generated by the set E: the system Σ
of n-dimensional simplices with vertices in E and the system Γ of
chambers. The incidence matrix A=∥aσ,γ​∥,
σ∈Σ, γ∈Γ, induces the notion of linear
independence among simplices (and among chambers). We present an algorithm of
construction of bases of simplices (and bases of chambers). For the case n=2
such an algorithm was described in the author's paper {\em Combinatorial bases
in systems of simplices and chambers} (Discrete Mathematics 157 (1996) 15--37).
However, the case of n-dimensional space required a different technique. It
is also proved that the constructed bases of simplices are geometrical