Lattice (group)
Lattices in general vector spaces

While we normally consider Z {\displaystyle \mathbb {Z} } lattices in R n {\displaystyle \mathbb {R} ^{n}} 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 B = { v 1, …, v n } {\displaystyle B=\{\mathbf {v} _{1},\ldots,\mathbf {v} _{n}\}} be a K - basis for V and let R be a ring contained within K. Then the R lattice L {\displaystyle {\mathcal {L}}} in V generated by B is given by:

In general, different bases B will generate different lattices. However, if the transition matrix T between the bases is in G L n (R) {\displaystyle \mathrm {GL} _{n}(R)} - 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 T − 1 {\displaystyle T^{-1}} are in R - which is equivalent to saying that the determinant of T is in R ∗ {\displaystyle R^{*}} - 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
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