55 lines
1.3 KiB
TeX
55 lines
1.3 KiB
TeX
\definition{}
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The \textit{integer lattice} $\mathbb{Z}^n$ is the set of points with integer coordinates in $n$ dimensions. \par
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For example, $\mathbb{Z}^3$ is the set of points $(a, b, c)$ where $a$, $b$, and $c$ are integers.
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\problem{}
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Draw $\mathbb{Z}^2$.
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\vfill
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\definition{}
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We say a set of vectors $\{v_1, v_2, ..., v_k\}$ \textit{generates} $\mathbb{Z}^n$
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if every lattice point can be written as
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$$
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a_1v_1 + a_2v_2 + ... + a_kv_k
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$$
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for integer coefficients $a_i$. \par
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\textbf{Bonus:} show that $k$ must be at least $n$.
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\problem{}
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Which of the following generate $\mathbb{Z}^2$?
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\begin{itemize}
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\item $\{ (1,2), (2,1) \}$
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\item $\{ (1,0), (0,2) \}$
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\item $\{ (1,1), (1,0), (0,1) \}$
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\end{itemize}
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\begin{solution}
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Only the last.
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\end{solution}
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\vfill
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\problem{}
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Find a set of two vectors other than $\{ (0, 1), (1, 0) \}$ that generates $\mathbb{Z}^2$. \\
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\vfill
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\problem{}
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Find a set of vectors that generates $\mathbb{Z}^n$.
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\vfill
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\pagebreak
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\definition{}
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Say we have a generating set of a lattice. \par
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The \textit{fundamental region} of this set is the $n$-dimensional parallelogram spanned by its members. \par
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\problem{}
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Draw two fundamental regions of $\mathbb{Z}^2$ using two different generating sets. Verify that their volumes are the same.
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\vfill
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\pagebreak |