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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.

Theorem1

is a basis for the module iff the matrix formed by the vectors is a unimodular matrix.

Proof

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.

Posted by Chao Xu on .
Tags: math.