49 lines
1.1 KiB
TeX
49 lines
1.1 KiB
TeX
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\definition{}
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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}.
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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 $\{v_1, v_2, ..., v_n\}$ \textit{generates} $\mathbb{Z}^n$ if every lattice point can be written uniquely as
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$$
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a_1v_1 + a_2v_2 + ... a_nv_n
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$$
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for integer coeficcients $a_i$.
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\problem{}
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Which of the following generate $\mathbb{Z}^3$?
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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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\vfill
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\problem{}
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Find a set of vectors 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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\problem{}
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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.
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\vfill
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\problem{}
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Draw two fundamental reions 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
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