8
votes
2answers
449 views

Elementary proof that if $A$ is a matrix map from $\mathbb{Z}^m$ to $\mathbb Z^n$, then the map is surjective iff the gcd of maximal minors is $1$

I am trying to find an elementary proof that if $\phi$ is a linear map from $\mathbb{Z}^n\rightarrow \mathbb{Z}^m$ represented by an $m \times n$ matrix $A$, then the map is surjective iff the gcd ...
8
votes
2answers
232 views

Show that the matrix $A$ with integer entries is injective on the reals to $\mathbb{R}^m$ iff it is injective on the integer lattice.

Show that the $m \times n$ matrix $A$ with integer entries is an injective linear map from $\mathbb{R}^n$ to $\mathbb{R}^m$ iff it is injective as a linear map from $\mathbb{Z}^n$ to $\mathbb{Z}^m$. ...