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2023-05-15 10:33:03 -07:00
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13
Advanced/Lattices/parts/0 intro.tex
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@ -1,5 +1,5 @@
\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}.
The \textit{integer lattice} $\mathbb{Z}^n \subset \mathbb{R}^n$ is the set of points with integer coordinates.
\problem{}
Draw $\mathbb{Z}^2$.
@ -8,11 +8,12 @@ Draw $\mathbb{Z}^2$.
\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
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
$$
a_1v_1 + a_2v_2 + ... + a_nv_n
a_1v_1 + a_2v_2 + ... + a_kv_k
$$
for integer coefficients $a_i$.
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 $\mathbb{Z}^2$?
@ -29,8 +30,8 @@ Which of the following generate $\mathbb{Z}^2$?
\vfill
\problem{}
Find a set of vectors that generates $\mathbb{Z}^2$. \\
$\{ (0, 1), (1, 0) \} doesn't count.$
Find a set of two vectors that generates $\mathbb{Z}^2$. \\
Don't say $\{ (0, 1), (1, 0) \}$, that's too easy.
\vfill