Fun with polynomials and linear algebra; or, slight abstract nonsense(guille.site)

Fun with polynomials and linear algebra; or, slight abstract nonsense

https://guille.site/posts/abstract-nonsense/

7 comments

To the author's credit, he has this in the first line, ie, that the article was not intended for others to read and enjoy.

> This is mostly a bunch of notes to myself

As Bessis has described in his book [1], it is extremely difficult to understand math someone else has written. The words and symbols dont convey imagery or ideas that the author has in their mind. I was surprised to read in that book that this applied to mathematicians just as it applies to you and I.

Coming back to this article, I wish it were written in the spirit of the essence of linear algebra [2] - conveying the essences in images and pictures instead of words. I am curious to hear from others if they feel this way or is it just me.

[1] Mathematica: A Secret World of Intuition and Curiosity

[2] Essence of linear algebra (3Blue1Brown, youtube)

seanhunter2 mesi fa

There are lots of people who write math in a way that is very easy for others (of an appropriate level of experience let’s say) to understand. I also didn’t find this particularly hard to follow, although some of it is I think a little fast and loose. eg

   > In general, given two finite-dimensional vector spaces U and W, then U ≃ W exactly when dim(U)=dim(W).

Is that really true? I don’t think it is. Specifically surely at least they have to be vector spaces either over the same field or over fields which are themselves isomorphic. I’m thinking say U is a vector space over R and W is a vector space over Q. Dim(U) = Dim(W)=1 but U and W are not isomorphic because there exists no bijection between the reals and the rationals.

LolWolf2 mesi fa

yes, definitely some of it is (purposefully) fast and loose, though (ideally!) mostly unambiguous with reasonable assumptions

I think that part should've been "vector subspaces" rather than vector spaces since that is how U and W are defined in the paragraph prior.

I'll add this as a note, thanks!

seanhunter2 mesi fa

It’s a cool article. I love linear algebra, particularly in settings like the polynomials.

LolWolf2 mesi fa

ha, thank you! it's very fun to write these

hopefully you also enjoy the next one which imo makes a fun connection between the linear algebraic CRT and the fourier transform :)

Chinjut2 mesi fa

I don't want everything to be images and pictures. Often, I enjoy words for communicating math.

LolWolf2 mesi fa

Fair on all accounts! Surely, this could be made way more lively if I were in front of a blackboard waving my hands and drawing images, but alas, the medium is what it is :)

Thanks for reading though!