MathDB

Let

n, m \in \mathbb{N}

;

a_1,\ldots, a_m \in \mathbb{Z}^n

. Show that nonnegative integer linear combinations of these vectors give exactly the whole

\mathbb{Z}^n

lattice, if

m \ge n

and the following two statements are satisfied:
[*] The vectors do not fall into the half-space of

\mathbb{R}^n

containing the origin (i.e. they do not fall on the same side of an

n-1

dimensional subspace), [*] the largest common divisor (not pairwise, but together) of

n \times n

minor determinants of the matrix

(a_1,\ldots, a_m)

(which is of size

m \times n

and the

i

-th column is

a_i

as a column vector) is

1

1