- What are Lattice Points?
Which points in x-y planes are Lattice Points?
Is (m,n) a lattice point where m,n are any integers?
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user3481652
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2 Answers
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No, that's not accurate. The points $(m,n)\in\Bbb Z^2$ are a lattice, but they are not the only lattice in $\Bbb R^2$, consider the sets:
$$\{(a,b\sqrt 2): a,b\in\Bbb Z\},\quad \left\{\left(a+{b\over 2}, b{\sqrt{3}\over 2}\right): a,b\in\Bbb Z\right\}\tag{$*$}$$
These are also a lattices.
Generally a lattice in $\Bbb R^2$ is a $\Bbb Z$ module of rank $2$ which contains a basis for $\Bbb R^2$.
As Cameron notes, this just means that you have integer combinations of two $\Bbb R$-linearly independent vectors from $\Bbb R^2$ (it's important that they be linearly independent over $\Bbb R$ and not something like $\Bbb Q$)
Adam Hughes
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3Or in perhaps friendlier terms: given two linearly independent vectors in $\mathbb{R}^2$, a lattice is all integer linear combinations of these vectors. – Cameron Williams Jul 12 '14 at 22:09
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@CameronWilliams This is the best answer here. – K.defaoite May 11 '20 at 11:24
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That is correct. The term "lattice points" usually refers to the points with integer coordinates.
DavidButlerUofA
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1About your $6x+8y=25$: Your book is probably talking about the integer lattice, the points with integer coordinates. If $x$ and $y$ are integers, then $6x+8y$ is even, so cannot be $25$. The line does not pass through any points $(x,y)$ of the integer lattice. – André Nicolas Jul 12 '14 at 23:11