Basis of the module
A student who is taking linear algebra asked me the following problem.
If we consider the field restricted to , and create a “vector space” on . How do we know if “spans” ?
Formally, what can we say about and if for every , there exist , such that .
We can generalize it and put it in terms of modules, as is only a ring but not a field.
is a basis for the module iff the matrix formed by the vectors is a unimodular matrix.
If , then are not linearly independent. If , then the parallelepiped formed by has volume . If there is any integer point not on the corners of the parallelepiped, then that point can’t be written as linear combination of . Notice that it must contain some lattice points not on the corners of the parallelepiped. One can see why by consider a large box that contain volume of such parallelepiped, but contain at least lattice points.
This shows if is not unimodular, then can’t be a basis.
Alternative proof: is not unimodular then contain a non-integer entry. This shows there exist a , such that the solution to contain a non-integer entry. (proposed by Thao Do)
implies it has a inverse over , thus for any always has a solution.