Lattice Points - Lattices in General Vector-spaces

Lattices in General Vector-spaces

Whilst we normally consider lattices in this concept can be generalized to any finite dimensional vector space over any field. This can be done as follows:

Let K be a field, let V be an n-dimensional K-vector space, let be a K-basis for V and let R be a ring contained within K. Then the R lattice in V generated by B is given by:

Different bases B will in general generate different lattices. However, if the transition matrix T between the bases is in - the general linear group of R (in simple terms this means that all the entries of T are in R and all the entries of are in R - which is equivalent to saying that the determinant of T is in - the unit group of elements in R with multiplicative inverses) then the lattices generated by these bases will be isomorphic since T induces an isomorphism between the two lattices.

Important cases of such lattices occur in number theory with K a p-adic field and R the p-adic integers.

For a vector space which is also an inner product space, the dual lattice can be concretely described by the set:

or equivalently as,