Added lattice handout

Advanced/Lattices/parts/0 intro.tex

@@ -0,0 +1,49 @@
+\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 coeficcients $a_i$.
+
+\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
+
+\problem{}
+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 reions of $\mathbb{Z}^2$ using two different generating sets. Verify that their volumes are the same.
+
+\vfill
+\pagebreak