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!

  • 4
    $\begingroup$ There is a lot of linear-algebraic group theory in quantum mechanics. See this wiki page for instance $\endgroup$ – Ben Grossmann Apr 27 '19 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 '19 at 15:23
  • 2
    $\begingroup$ Counting theorem is a good one-see also math.stackexchange.com/questions/324253/… $\endgroup$ – John_dydx Apr 27 '19 at 15:28
  • 1
    $\begingroup$ Wallpaper classification? Crystallography? $\endgroup$ – gidds Apr 27 '19 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 '19 at 17:12

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.

| cite | improve this answer | |

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

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.