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Can anybody recommend a linear algebra course based in finite fields? A lot of theorems like SVD and eigenvectors seem like they'd carry over, but they are alw
by thomasahle 10y ago
Can anybody recommend a linear algebra course based in finite fields?
A lot of theorems like SVD and eigenvectors seem like they'd carry over, but they are always proven over complex numbers.
- BlackFly 10y agoI believe one of the first theorems you go about proving in linear algebra is that given a vector space of a given dimension you can find a function that preserves the linear structure to any other vector space of the same dimension. After proving that they just proceed to work with complex numbers since every other (finite dimensional) vector space is the same in that sense.
- stablemap 10y agoI think you want a text like Hoffman-Kunze which gives the basic definitions and theorems over an arbitrary ground field. At some point the field starts to matter -- giving the real plane a quarter turn has no real eigenvalues, for example; and I don't see a good way to transport the statement of SVD to the setting of finite fields.
- walrus 10y agoOne possible approach is to take a class/read a book on coding theory. Algebraic block codes rely heavily on linear algebra over finite fields.
- thomasahle 10y agoThat is probably a good idea, but I'll always be wondering about the linear algebra tools they didn't use. Were they just not applicable? Or do they not work over finite fields?
- davidwihl 10y agoThe first third of Harvard's Math 23a includes finite fields in Linear Algebra. It is available as an online class. https://www.extension.harvard.edu/academics/courses/linear-algebra-real-analysis-i/15176 https://www.extension.harvard.edu/academics/courses/linear-a... For example, see https://youtu.be/c194lY79jOM https://youtu.be/c194lY79jOM minute 16:30 The course makes no specifics for finite fields but does illustrate that linear algebra works for any field, finite or not.