We have a careful look at getting a good basis of an integral linear space through a specific algorithm which is essentially that of Hermite normal form. Usually in linear algebra courses this is framed in terms of matrices, but here we are taking more of an mset point of view, but the idea is still very much the same. This algorithm is an integral version of Gaussian elimination or row reduction, familiar to undergraduates around the world.https://www.youtube.com/watch?v=ipJYgeMVOng
Perhaps surprisingly, this integral version closely parallels Euclid's algorithm for finding the greatest common divisor of two (or possibly) more natural numbers. We will give an official definition of Hermite normal form for an mset of vexels.