Edits

Advanced/Lattices/parts/0 intro.tex

@@ -15,17 +15,22 @@ $$
 for integer coefficients $a_i$.
 
 \problem{}
-Which of the following generate $\mathbb{Z}^3$?
+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 vectors that generates $\mathbb{Z}^2$.
+Find a set of vectors that generates $\mathbb{Z}^2$. \\
+$\{ (0, 1), (1, 0) \} doesn't count.$
 
 \vfill
 
@@ -40,8 +45,6 @@ Find a set of vectors that generates $\mathbb{Z}^n$.
 \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.