Editing Dual lattice (section)

==Properties==

We list some elementary properties of the dual lattice:
* If <math display = "inline"> B = [b_1, \ldots, b_n] </math> is a matrix giving a basis for the lattice <math display = "inline">  L </math>, then  <math display = "inline"> z \in \text{span}(L) </math> satisfies <math display = "inline"> z \in L^* \iff b^T_i z \in \mathbb{Z}, i = 1, \ldots, n \iff B^T z \in \mathbb{Z}^n</math>.
* If <math display = "inline"> B </math> is a matrix giving a basis for the lattice <math display = "inline">  L </math>, then <math display = "inline"> B (B^T B)^{-1} </math> gives a basis for the dual lattice. If <math display = "inline">  L </math> is full rank <math display = "inline">  B^{-T} </math> gives a basis for the dual lattice: <math display = "inline"> z \in L^* \iff B^T z \in \mathbb{Z}^n \iff z \in B^{-T} \mathbb{Z}^n </math>.
* The previous fact shows that <math display = "inline"> (L^*)^* = L </math>. This equality holds under the usual identifications of a vector space with its double dual, or in the setting where the inner product has identified <math display = "inline"> \mathbb{R}^n </math> with its dual.
* Fix two lattices  <math display = "inline"> L,M </math>. Then <math display = "inline"> L \subseteq M </math> if and only if <math display = "inline"> L^* \supseteq M^* </math>.
* The determinant of a lattice is the reciprocal of the determinant of its dual: <math display = "inline"> \text{det}(L^*) = \frac{1}{\text{det}(L)} </math>
* If <math display = "inline"> q </math> is a nonzero scalar, then <math display = "inline"> (qL)^* = \frac{1}{q} L^* </math>.
* If <math display = "inline"> R </math> is a rotation matrix, then <math display = "inline"> (RL)^* = R L^* </math>.
* A lattice <math display="inline"> L </math> is said to be integral if <math display = "inline">  x \cdot y \in \mathbb{Z} </math> for all <math display="inline"> x,y \in L </math>. Assume that the lattice  <math display="inline"> L </math> is full rank. Under the identification of Euclidean space with its dual, we have that <math display="inline"> L \subseteq L^* </math> for integral lattices <math display="inline"> L </math>. Recall that, if  <math display="inline"> L' \subseteq L </math> and  <math display="inline"> |L/L'| <\infty </math>, then  <math display="inline"> \text{det}(L') = \text{det}(L) | L/L'|  </math>. From this it follows that for an integral lattice,  <math display="inline"> \text{det}(L)^2 = | L^* / L| </math>.
* An integral lattice is said to be ''unimodular'' if  <math display="inline"> L = L^*</math>, which, by the above, is equivalent to <math display="inline"> \text{det}(L) = 1. </math>