lattice in $\mathbb{R}^n$LatticeInMathbbRn2003-08-18 14:53:322003-08-28 11:39:54Definitionlattice in $\mathbb{R}^n$latticegrid% this is the default PlanetMath preamble. as your knowledge
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\newcommand{\Nats}{\mathbb{N}}
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\newcommand{\Rats}{\mathbb{Q}}\begin{defn}
A lattice in ${\Reals}^n$ is an $n$-dimensional additive free group over $\Ints$ which generates ${\Reals}^n$ over $\Reals$.
\end{defn}
{\bf Example}: The following is an example of a lattice $\mathcal{L}\subset {\Reals}^2$, generated by $w_1=(1,2),w_2=(4,1)$.
$$\mathcal{L}=\{ \alpha w_1 +\beta w_2 \mid \alpha,\beta \in \Ints \} $$
\quad
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\includegraphics[scale=0.75]{lattice}
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