Lattice Points in x-y plane
$begingroup$
- 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?
plane-curves
asked Jul 12 '14 at 21:54
user3481652user3481652
386
$endgroup$
add a comment |
$begingroup$
- 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?
plane-curves
asked Jul 12 '14 at 21:54
user3481652user3481652
386
$endgroup$
add a comment |
$begingroup$
- 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?
plane-curves
asked Jul 12 '14 at 21:54
user3481652user3481652
386
$endgroup$
- 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?
plane-curves
plane-curves
asked Jul 12 '14 at 21:54
user3481652user3481652
386
asked Jul 12 '14 at 21:54
user3481652user3481652
386
asked Jul 12 '14 at 21:54
user3481652user3481652
386
asked Jul 12 '14 at 21:54
user3481652user3481652
386
asked Jul 12 '14 at 21:54
user3481652user3481652
386
386
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
No, that's not accurate. The points $(m,n)inBbb Z^2$ are a lattice, but they are not the only lattice in $Bbb R^2$, consider the sets:
$${(a,bsqrt 2): a,binBbb Z},quad left{left(a+{bover 2}, b{sqrt{3}over 2}right): a,binBbb Zright}tag{$*$}$$
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 $Bbb Q$)
answered Jul 12 '14 at 22:06
Adam HughesAdam Hughes
32.3k83770
$endgroup$
2
$begingroup$
Or in perhaps friendlier terms: given two linearly independent vectors in $mathbb{R}^2$, a lattice is all integer linear combinations of these vectors.
$endgroup$
– Cameron Williams
Jul 12 '14 at 22:09
add a comment |
$begingroup$
That is correct. The term "lattice points" usually refers to the points with integer coordinates.
answered Jul 12 '14 at 21:57
DavidButlerUofADavidButlerUofA
2,672821
$endgroup$
$begingroup$
okay thank you for the help.But can i know why?
$endgroup$
– user3481652
Jul 12 '14 at 21:58
$begingroup$
does 6x+8y=25 pass through any lattice point?
$endgroup$
– user3481652
Jul 12 '14 at 22:00
$begingroup$
@user3481652 why? because that's the definition
$endgroup$
– leonbloy
Jul 12 '14 at 22:18
1
$begingroup$
About 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.
$endgroup$
– André Nicolas
Jul 12 '14 at 23:11
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
No, that's not accurate. The points $(m,n)inBbb Z^2$ are a lattice, but they are not the only lattice in $Bbb R^2$, consider the sets:
$${(a,bsqrt 2): a,binBbb Z},quad left{left(a+{bover 2}, b{sqrt{3}over 2}right): a,binBbb Zright}tag{$*$}$$
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 $Bbb Q$)
answered Jul 12 '14 at 22:06
Adam HughesAdam Hughes
32.3k83770
$endgroup$
2
$begingroup$
Or in perhaps friendlier terms: given two linearly independent vectors in $mathbb{R}^2$, a lattice is all integer linear combinations of these vectors.
$endgroup$
– Cameron Williams
Jul 12 '14 at 22:09
add a comment |
$begingroup$
No, that's not accurate. The points $(m,n)inBbb Z^2$ are a lattice, but they are not the only lattice in $Bbb R^2$, consider the sets:
$${(a,bsqrt 2): a,binBbb Z},quad left{left(a+{bover 2}, b{sqrt{3}over 2}right): a,binBbb Zright}tag{$*$}$$
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 $Bbb Q$)
answered Jul 12 '14 at 22:06
Adam HughesAdam Hughes
32.3k83770
$endgroup$
2
$begingroup$
Or in perhaps friendlier terms: given two linearly independent vectors in $mathbb{R}^2$, a lattice is all integer linear combinations of these vectors.
$endgroup$
– Cameron Williams
Jul 12 '14 at 22:09
add a comment |
$begingroup$
No, that's not accurate. The points $(m,n)inBbb Z^2$ are a lattice, but they are not the only lattice in $Bbb R^2$, consider the sets:
$${(a,bsqrt 2): a,binBbb Z},quad left{left(a+{bover 2}, b{sqrt{3}over 2}right): a,binBbb Zright}tag{$*$}$$
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 $Bbb Q$)
answered Jul 12 '14 at 22:06
Adam HughesAdam Hughes
32.3k83770
$endgroup$
No, that's not accurate. The points $(m,n)inBbb Z^2$ are a lattice, but they are not the only lattice in $Bbb R^2$, consider the sets:
$${(a,bsqrt 2): a,binBbb Z},quad left{left(a+{bover 2}, b{sqrt{3}over 2}right): a,binBbb Zright}tag{$*$}$$
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 $Bbb Q$)
answered Jul 12 '14 at 22:06
Adam HughesAdam Hughes
32.3k83770
edited Jul 12 '14 at 22:16
answered Jul 12 '14 at 22:06
Adam HughesAdam Hughes
32.3k83770
answered Jul 12 '14 at 22:06
Adam HughesAdam Hughes
32.3k83770
answered Jul 12 '14 at 22:06
Adam HughesAdam Hughes
32.3k83770
32.3k83770
2
$begingroup$
Or in perhaps friendlier terms: given two linearly independent vectors in $mathbb{R}^2$, a lattice is all integer linear combinations of these vectors.
$endgroup$
– Cameron Williams
Jul 12 '14 at 22:09
add a comment |
2
$begingroup$
Or in perhaps friendlier terms: given two linearly independent vectors in $mathbb{R}^2$, a lattice is all integer linear combinations of these vectors.
$endgroup$
– Cameron Williams
Jul 12 '14 at 22:09
2
2
$begingroup$
Or in perhaps friendlier terms: given two linearly independent vectors in $mathbb{R}^2$, a lattice is all integer linear combinations of these vectors.
$endgroup$
– Cameron Williams
Jul 12 '14 at 22:09
$begingroup$
Or in perhaps friendlier terms: given two linearly independent vectors in $mathbb{R}^2$, a lattice is all integer linear combinations of these vectors.
$endgroup$
– Cameron Williams
Jul 12 '14 at 22:09
add a comment |
$begingroup$
That is correct. The term "lattice points" usually refers to the points with integer coordinates.
answered Jul 12 '14 at 21:57
DavidButlerUofADavidButlerUofA
2,672821
$endgroup$
$begingroup$
okay thank you for the help.But can i know why?
$endgroup$
– user3481652
Jul 12 '14 at 21:58
$begingroup$
does 6x+8y=25 pass through any lattice point?
$endgroup$
– user3481652
Jul 12 '14 at 22:00
$begingroup$
@user3481652 why? because that's the definition
$endgroup$
– leonbloy
Jul 12 '14 at 22:18
1
$begingroup$
About 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.
$endgroup$
– André Nicolas
Jul 12 '14 at 23:11
add a comment |
$begingroup$
That is correct. The term "lattice points" usually refers to the points with integer coordinates.
answered Jul 12 '14 at 21:57
DavidButlerUofADavidButlerUofA
2,672821
$endgroup$
$begingroup$
okay thank you for the help.But can i know why?
$endgroup$
– user3481652
Jul 12 '14 at 21:58
$begingroup$
does 6x+8y=25 pass through any lattice point?
$endgroup$
– user3481652
Jul 12 '14 at 22:00
$begingroup$
@user3481652 why? because that's the definition
$endgroup$
– leonbloy
Jul 12 '14 at 22:18
1
$begingroup$
About 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.
$endgroup$
– André Nicolas
Jul 12 '14 at 23:11
add a comment |
$begingroup$
That is correct. The term "lattice points" usually refers to the points with integer coordinates.
answered Jul 12 '14 at 21:57
DavidButlerUofADavidButlerUofA
2,672821
$endgroup$
That is correct. The term "lattice points" usually refers to the points with integer coordinates.
answered Jul 12 '14 at 21:57
DavidButlerUofADavidButlerUofA
2,672821
edited Jul 13 '14 at 10:35
answered Jul 12 '14 at 21:57
DavidButlerUofADavidButlerUofA
2,672821
answered Jul 12 '14 at 21:57
DavidButlerUofADavidButlerUofA
2,672821
answered Jul 12 '14 at 21:57
DavidButlerUofADavidButlerUofA
2,672821
2,672821
$begingroup$
okay thank you for the help.But can i know why?
$endgroup$
– user3481652
Jul 12 '14 at 21:58
$begingroup$
does 6x+8y=25 pass through any lattice point?
$endgroup$
– user3481652
Jul 12 '14 at 22:00
$begingroup$
@user3481652 why? because that's the definition
$endgroup$
– leonbloy
Jul 12 '14 at 22:18
1
$begingroup$
About 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.
$endgroup$
– André Nicolas
Jul 12 '14 at 23:11
add a comment |
$begingroup$
okay thank you for the help.But can i know why?
$endgroup$
– user3481652
Jul 12 '14 at 21:58
$begingroup$
does 6x+8y=25 pass through any lattice point?
$endgroup$
– user3481652
Jul 12 '14 at 22:00
$begingroup$
@user3481652 why? because that's the definition
$endgroup$
– leonbloy
Jul 12 '14 at 22:18
1
$begingroup$
About 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.
$endgroup$
– André Nicolas
Jul 12 '14 at 23:11
$begingroup$
okay thank you for the help.But can i know why?
$endgroup$
– user3481652
Jul 12 '14 at 21:58
$begingroup$
okay thank you for the help.But can i know why?
$endgroup$
– user3481652
Jul 12 '14 at 21:58
$begingroup$
does 6x+8y=25 pass through any lattice point?
$endgroup$
– user3481652
Jul 12 '14 at 22:00
$begingroup$
does 6x+8y=25 pass through any lattice point?
$endgroup$
– user3481652
Jul 12 '14 at 22:00
$begingroup$
@user3481652 why? because that's the definition
$endgroup$
– leonbloy
Jul 12 '14 at 22:18
$begingroup$
@user3481652 why? because that's the definition
$endgroup$
– leonbloy
Jul 12 '14 at 22:18
1
1
$begingroup$
About 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.
$endgroup$
– André Nicolas
Jul 12 '14 at 23:11
$begingroup$
About 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.
$endgroup$
– André Nicolas
Jul 12 '14 at 23:11
add a comment |
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