Fundamental Domains , Lattice Density , and Minkowski Theorems

Now that we know that every lattice admits a basis, a next fundamental question is: given linearly independent vectors b1, . . . , bn ∈ L how can we tell if they form a basis of L? As we have seen previously, not every set of n linearly vectors in Zn is a basis of Zn. One possible answer is given in the following lemma. It says that the basic parallelepiped generated by the vectors should not contain any lattice points, except the origin. As an example, notice that the basic parallelepiped shown in Figure 1(c) contains the lattice point (1, 0) whereas those in Figures 1(a) and 1(b) do not contain any nonzero lattice points.