handouts/Advanced/Lattices/parts/0 intro.tex

\definition{}
The \textit{integer lattice} $\mathbb{Z}^n \subset \mathbb{R}^n$ is the set of points with integer coordinates. We call each point in the lattice a \textit{lattice point}.

\problem{}
Draw $\mathbb{Z}^2$.

\vfill


\definition{}
We say a set of vectors $\{v_1, v_2, ..., v_n\}$ \textit{generates} $\mathbb{Z}^n$ if every lattice point can be written uniquely as
$$
	a_1v_1 + a_2v_2 + ... + a_nv_n
$$
for integer coefficients $a_i$.

\problem{}
Which of the following generate $\mathbb{Z}^3$?
\begin{itemize}
	\item $\{ (1,2), (2,1) \}$
	\item $\{ (1,0), (0,2) \}$
	\item $\{ (1,1), (1,0), (0,1) \}$
\end{itemize}

\vfill

\problem{}
Find a set of vectors that generates $\mathbb{Z}^2$.

\vfill

\problem{}
Find a set of vectors that generates $\mathbb{Z}^n$.



\vfill
\pagebreak

\definition{}
A \textit{fundamental region} of a lattice is the parallelepiped spanned by a generating set. The exact shape of this region depends on the generating set we use.

\vfill

\problem{}
Draw two fundamental regions of $\mathbb{Z}^2$ using two different generating sets. Verify that their volumes are the same.

\vfill
\pagebreak