Skip to main content
2 of 2
slight inaccuracy corrected
Richard Stanley
  • 50.8k
  • 14
  • 155
  • 279

You can compute the Smith normal form $B$ of the matrix with rows $a_1,\dots,a_n$. The rows generate $\mathbb{Z}^k$ if and only if the main diagonal entries of $B$ are $1$. (SNF over $\mathbb{Z}$ is determined only up to multiplication of the diagonal entries by $\pm 1$, and conventionally they are chosen to be nonnegative.)

Richard Stanley
  • 50.8k
  • 14
  • 155
  • 279