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.

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

\vfill


\definition{}
We say a set of vectors $\{v_1, v_2, ..., v_k\}$ \textit{generates} $\mathbb{Z}^n$ if every lattice point can be written uniquely as
$$
	a_1v_1 + a_2v_2 + ... + a_kv_k
$$
for integer coefficients $a_i$. \par
It is fairly easy to show that $k$ must be at least $n$.

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

\begin{solution}
	Only the last.
\end{solution}

\vfill

\problem{}
Find a set of two vectors that generates $\mathbb{Z}^2$. \\
Don't say $\{ (0, 1), (1, 0) \}$, that's too easy.

\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.

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

\vfill
\pagebreak