# Real-world applications of fields, rings and groups in linear algebra.

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!

• There is a lot of linear-algebraic group theory in quantum mechanics. See this wiki page for instance – Omnomnomnom Apr 27 at 15:19
• 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. – Theo Bendit Apr 27 at 15:23
• Counting theorem is a good one-see also math.stackexchange.com/questions/324253/… – John_dydx Apr 27 at 15:28
• – gidds Apr 27 at 16:38
• What is this "real-world' people keep talking about? It sounds like a horrible, mathless place. – anomaly Apr 27 at 17:12

## 2 Answers

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.

There's the GraphBLAS project  which is for graph algorithms expressed as linear algebra (see also ). 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 .

 http://graphblas.org  Graph Algorithms in the Language of Linear Algebra, Kepner&Gilbert(ed), https://epubs.siam.org/doi/book/10.1137/1.9780898719918  http://faculty.cse.tamu.edu/davis/GraphBLAS_files/toms_graphblas.pdf