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Real-world applications of fields, rings and groups in linear algebra.
Do groups, rings and fields have practical applications in CS? If so, what are some?Are there real world applications of finite group theory?Good undergraduate level book on Cyclotomic fieldsAlgebraic Intuition for Homological Algebra and Applications to More Elementary AlgebraText recommendation for introduction to linear algebraWhen is a field a nontrivial field of fractions?“Real”-life applications of algebraic geometryCharacters of a finite groupHow does Dummit and Foote's abstract algebra text compare to others?Algebra Text RecommendationsGood algebra book to cover these topics?What is “Field with One Element”?
$begingroup$
Real-world applications of fields, rings and groups in linear algebra.
A friend of mine asked me where one could use the definitions of rings, groups, fields etc. I was very embarrassed of the fact that I could only mention cyber security - nothing more (I'm studying IT). That's why I would like to get some more detailed suggestions. I would really appreciate that.
Thank you in advance!
linear-algebra abstract-algebra group-theory field-theory applications
edited Apr 27 at 15:21
Bernard
125k743119
asked Apr 27 at 15:15
DoesbaddelDoesbaddel
36613
$endgroup$
|
show 2 more comments
$begingroup$
Real-world applications of fields, rings and groups in linear algebra.
A friend of mine asked me where one could use the definitions of rings, groups, fields etc. I was very embarrassed of the fact that I could only mention cyber security - nothing more (I'm studying IT). That's why I would like to get some more detailed suggestions. I would really appreciate that.
Thank you in advance!
linear-algebra abstract-algebra group-theory field-theory applications
edited Apr 27 at 15:21
Bernard
125k743119
asked Apr 27 at 15:15
DoesbaddelDoesbaddel
36613
$endgroup$
4
$begingroup$
There is a lot of linear-algebraic group theory in quantum mechanics. See this wiki page for instance
$endgroup$
– Omnomnomnom
Apr 27 at 15:19
3
$begingroup$
I'm no expert, but I'm lead to believe that plenty of the tests in chemistry to detect various molecules work by examining symmetry groups.
$endgroup$
– Theo Bendit
Apr 27 at 15:23
2
$begingroup$
Counting theorem is a good one-see also math.stackexchange.com/questions/324253/…
$endgroup$
– John_dydx
Apr 27 at 15:28
1
$begingroup$
Wallpaper classification? Crystallography?
$endgroup$
– gidds
Apr 27 at 16:38
6
$begingroup$
What is this "real-world' people keep talking about? It sounds like a horrible, mathless place.
$endgroup$
– anomaly
Apr 27 at 17:12
|
show 2 more comments
$begingroup$
Real-world applications of fields, rings and groups in linear algebra.
A friend of mine asked me where one could use the definitions of rings, groups, fields etc. I was very embarrassed of the fact that I could only mention cyber security - nothing more (I'm studying IT). That's why I would like to get some more detailed suggestions. I would really appreciate that.
Thank you in advance!
linear-algebra abstract-algebra group-theory field-theory applications
edited Apr 27 at 15:21
Bernard
125k743119
asked Apr 27 at 15:15
DoesbaddelDoesbaddel
36613
$endgroup$
Real-world applications of fields, rings and groups in linear algebra.
A friend of mine asked me where one could use the definitions of rings, groups, fields etc. I was very embarrassed of the fact that I could only mention cyber security - nothing more (I'm studying IT). That's why I would like to get some more detailed suggestions. I would really appreciate that.
Thank you in advance!
linear-algebra abstract-algebra group-theory field-theory applications
linear-algebra abstract-algebra group-theory field-theory applications
edited Apr 27 at 15:21
Bernard
125k743119
asked Apr 27 at 15:15
DoesbaddelDoesbaddel
36613
edited Apr 27 at 15:21
Bernard
125k743119
asked Apr 27 at 15:15
DoesbaddelDoesbaddel
36613
edited Apr 27 at 15:21
Bernard
125k743119
edited Apr 27 at 15:21
Bernard
125k743119
edited Apr 27 at 15:21
Bernard
125k743119
125k743119
asked Apr 27 at 15:15
DoesbaddelDoesbaddel
36613
asked Apr 27 at 15:15
DoesbaddelDoesbaddel
36613
asked Apr 27 at 15:15
DoesbaddelDoesbaddel
36613
36613
4
$begingroup$
There is a lot of linear-algebraic group theory in quantum mechanics. See this wiki page for instance
$endgroup$
– Omnomnomnom
Apr 27 at 15:19
3
$begingroup$
I'm no expert, but I'm lead to believe that plenty of the tests in chemistry to detect various molecules work by examining symmetry groups.
$endgroup$
– Theo Bendit
Apr 27 at 15:23
2
$begingroup$
Counting theorem is a good one-see also math.stackexchange.com/questions/324253/…
$endgroup$
– John_dydx
Apr 27 at 15:28
1
$begingroup$
Wallpaper classification? Crystallography?
$endgroup$
– gidds
Apr 27 at 16:38
6
$begingroup$
What is this "real-world' people keep talking about? It sounds like a horrible, mathless place.
$endgroup$
– anomaly
Apr 27 at 17:12
|
show 2 more comments
4
$begingroup$
There is a lot of linear-algebraic group theory in quantum mechanics. See this wiki page for instance
$endgroup$
– Omnomnomnom
Apr 27 at 15:19
3
$begingroup$
I'm no expert, but I'm lead to believe that plenty of the tests in chemistry to detect various molecules work by examining symmetry groups.
$endgroup$
– Theo Bendit
Apr 27 at 15:23
2
$begingroup$
Counting theorem is a good one-see also math.stackexchange.com/questions/324253/…
$endgroup$
– John_dydx
Apr 27 at 15:28
1
$begingroup$
Wallpaper classification? Crystallography?
$endgroup$
– gidds
Apr 27 at 16:38
6
$begingroup$
What is this "real-world' people keep talking about? It sounds like a horrible, mathless place.
$endgroup$
– anomaly
Apr 27 at 17:12
4
4
$begingroup$
There is a lot of linear-algebraic group theory in quantum mechanics. See this wiki page for instance
$endgroup$
– Omnomnomnom
Apr 27 at 15:19
$begingroup$
There is a lot of linear-algebraic group theory in quantum mechanics. See this wiki page for instance
$endgroup$
– Omnomnomnom
Apr 27 at 15:19
3
3
$begingroup$
I'm no expert, but I'm lead to believe that plenty of the tests in chemistry to detect various molecules work by examining symmetry groups.
$endgroup$
– Theo Bendit
Apr 27 at 15:23
$begingroup$
I'm no expert, but I'm lead to believe that plenty of the tests in chemistry to detect various molecules work by examining symmetry groups.
$endgroup$
– Theo Bendit
Apr 27 at 15:23
2
2
$begingroup$
Counting theorem is a good one-see also math.stackexchange.com/questions/324253/…
$endgroup$
– John_dydx
Apr 27 at 15:28
$begingroup$
Counting theorem is a good one-see also math.stackexchange.com/questions/324253/…
$endgroup$
– John_dydx
Apr 27 at 15:28
1
1
$begingroup$
Wallpaper classification? Crystallography?
$endgroup$
– gidds
Apr 27 at 16:38
$begingroup$
Wallpaper classification? Crystallography?
$endgroup$
– gidds
Apr 27 at 16:38
6
6
$begingroup$
What is this "real-world' people keep talking about? It sounds like a horrible, mathless place.
$endgroup$
– anomaly
Apr 27 at 17:12
$begingroup$
What is this "real-world' people keep talking about? It sounds like a horrible, mathless place.
$endgroup$
– anomaly
Apr 27 at 17:12
|
show 2 more comments
2 Answers
2
active
oldest
votes
$begingroup$
Well, I'd consider the brand new book from Gilbert Strang: ''Linear Algebra and Learning from Data'', Cambridge Univ. Press, 2018. His learning from data culminates into the construction of deep neural networks.
Another application besides cryptography is coding theory, where finite fields are used to define linear codes. There is also a decent generalization to linear codes over the ring of integers modulo 4, which give rises by the Gray map to nonlinear binary codes which are better than any linear code with the same block length.
answered Apr 27 at 15:23
WuestenfuxWuestenfux
5,8991513
$endgroup$
add a comment |
$begingroup$
There's the GraphBLAS project [0] which is for graph algorithms expressed as linear algebra (see also [1]). It turns out that many such algorithms can be done this way if you change the semiring over which the matrices live. See for example this TOMS pre-print [2].
[0] http://graphblas.org
[1] Graph Algorithms in the Language of Linear Algebra, Kepner&Gilbert(ed), https://epubs.siam.org/doi/book/10.1137/1.9780898719918
[2] http://faculty.cse.tamu.edu/davis/GraphBLAS_files/toms_graphblas.pdf
answered Apr 27 at 16:01
JCKJCK
111
$endgroup$
add a comment |
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2 Answers
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active
oldest
votes
2 Answers
2
active
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active
oldest
votes
$begingroup$
Well, I'd consider the brand new book from Gilbert Strang: ''Linear Algebra and Learning from Data'', Cambridge Univ. Press, 2018. His learning from data culminates into the construction of deep neural networks.
Another application besides cryptography is coding theory, where finite fields are used to define linear codes. There is also a decent generalization to linear codes over the ring of integers modulo 4, which give rises by the Gray map to nonlinear binary codes which are better than any linear code with the same block length.
answered Apr 27 at 15:23
WuestenfuxWuestenfux
5,8991513
$endgroup$
add a comment |
$begingroup$
Well, I'd consider the brand new book from Gilbert Strang: ''Linear Algebra and Learning from Data'', Cambridge Univ. Press, 2018. His learning from data culminates into the construction of deep neural networks.
Another application besides cryptography is coding theory, where finite fields are used to define linear codes. There is also a decent generalization to linear codes over the ring of integers modulo 4, which give rises by the Gray map to nonlinear binary codes which are better than any linear code with the same block length.
answered Apr 27 at 15:23
WuestenfuxWuestenfux
5,8991513
$endgroup$
add a comment |
$begingroup$
Well, I'd consider the brand new book from Gilbert Strang: ''Linear Algebra and Learning from Data'', Cambridge Univ. Press, 2018. His learning from data culminates into the construction of deep neural networks.
Another application besides cryptography is coding theory, where finite fields are used to define linear codes. There is also a decent generalization to linear codes over the ring of integers modulo 4, which give rises by the Gray map to nonlinear binary codes which are better than any linear code with the same block length.
answered Apr 27 at 15:23
WuestenfuxWuestenfux
5,8991513
$endgroup$
Well, I'd consider the brand new book from Gilbert Strang: ''Linear Algebra and Learning from Data'', Cambridge Univ. Press, 2018. His learning from data culminates into the construction of deep neural networks.
Another application besides cryptography is coding theory, where finite fields are used to define linear codes. There is also a decent generalization to linear codes over the ring of integers modulo 4, which give rises by the Gray map to nonlinear binary codes which are better than any linear code with the same block length.
answered Apr 27 at 15:23
WuestenfuxWuestenfux
5,8991513
answered Apr 27 at 15:23
WuestenfuxWuestenfux
5,8991513
answered Apr 27 at 15:23
WuestenfuxWuestenfux
5,8991513
answered Apr 27 at 15:23
WuestenfuxWuestenfux
5,8991513
5,8991513
add a comment |
add a comment |
$begingroup$
There's the GraphBLAS project [0] which is for graph algorithms expressed as linear algebra (see also [1]). It turns out that many such algorithms can be done this way if you change the semiring over which the matrices live. See for example this TOMS pre-print [2].
[0] http://graphblas.org
[1] Graph Algorithms in the Language of Linear Algebra, Kepner&Gilbert(ed), https://epubs.siam.org/doi/book/10.1137/1.9780898719918
[2] http://faculty.cse.tamu.edu/davis/GraphBLAS_files/toms_graphblas.pdf
answered Apr 27 at 16:01
JCKJCK
111
$endgroup$
add a comment |
$begingroup$
There's the GraphBLAS project [0] which is for graph algorithms expressed as linear algebra (see also [1]). It turns out that many such algorithms can be done this way if you change the semiring over which the matrices live. See for example this TOMS pre-print [2].
[0] http://graphblas.org
[1] Graph Algorithms in the Language of Linear Algebra, Kepner&Gilbert(ed), https://epubs.siam.org/doi/book/10.1137/1.9780898719918
[2] http://faculty.cse.tamu.edu/davis/GraphBLAS_files/toms_graphblas.pdf
answered Apr 27 at 16:01
JCKJCK
111
$endgroup$
add a comment |
$begingroup$
There's the GraphBLAS project [0] which is for graph algorithms expressed as linear algebra (see also [1]). It turns out that many such algorithms can be done this way if you change the semiring over which the matrices live. See for example this TOMS pre-print [2].
[0] http://graphblas.org
[1] Graph Algorithms in the Language of Linear Algebra, Kepner&Gilbert(ed), https://epubs.siam.org/doi/book/10.1137/1.9780898719918
[2] http://faculty.cse.tamu.edu/davis/GraphBLAS_files/toms_graphblas.pdf
answered Apr 27 at 16:01
JCKJCK
111
$endgroup$
There's the GraphBLAS project [0] which is for graph algorithms expressed as linear algebra (see also [1]). It turns out that many such algorithms can be done this way if you change the semiring over which the matrices live. See for example this TOMS pre-print [2].
[0] http://graphblas.org
[1] Graph Algorithms in the Language of Linear Algebra, Kepner&Gilbert(ed), https://epubs.siam.org/doi/book/10.1137/1.9780898719918
[2] http://faculty.cse.tamu.edu/davis/GraphBLAS_files/toms_graphblas.pdf
answered Apr 27 at 16:01
JCKJCK
111
answered Apr 27 at 16:01
JCKJCK
111
answered Apr 27 at 16:01
JCKJCK
111
answered Apr 27 at 16:01
JCKJCK
111
111
add a comment |
add a comment |
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4
$begingroup$
There is a lot of linear-algebraic group theory in quantum mechanics. See this wiki page for instance
$endgroup$
– Omnomnomnom
Apr 27 at 15:19
3
$begingroup$
I'm no expert, but I'm lead to believe that plenty of the tests in chemistry to detect various molecules work by examining symmetry groups.
$endgroup$
– Theo Bendit
Apr 27 at 15:23
2
$begingroup$
Counting theorem is a good one-see also math.stackexchange.com/questions/324253/…
$endgroup$
– John_dydx
Apr 27 at 15:28
1
$begingroup$
Wallpaper classification? Crystallography?
$endgroup$
– gidds
Apr 27 at 16:38
6
$begingroup$
What is this "real-world' people keep talking about? It sounds like a horrible, mathless place.
$endgroup$
– anomaly
Apr 27 at 17:12