5 ms·

I don't want to doubt the potential usefulness of the linked article to certain audiences, but I wanted to point out the following: It's important to acknowled

I don't want to doubt the potential usefulness of the linked article to certain audiences, but I wanted to point out the following: It's important to acknowledge that intuition from rotations etc. are crutches, and not substitutes for deeply understanding linear-algebraic constructions and their formal properties. Linear algebra is not really about the arithmetic of multiplying arrays of numbers - it's about the nice algebraic things that happen when you're working with things that come with "linear" operations. The linear-algebraic things that permeate all of mathematics aren't "rotations matrices" and such, but rather "universal constructions" like kernels and cokernels, products and tensor products etc. We even have abstractions that precisely formalize these nice properties of the category of modules/vector spaces, such as abelian categories and linear functors. Any introductory reference on these will provide you with an abundance of examples where things with these formal properties naturally arise.

dbranes 9y ago
A related remark: there is another (rather silly) way to "geometrize linear algebra". It goes something like whenever you see a ring R, think of it as the sheaf of functions (the structure sheaf) on some space. Whenever you see a module over R, think of it as the space of sections in some vector bundle over that space. Then anything you say about modules through this admits a sheaf-theoretic version, which is often easier to picture if you're geometrically inclined.
- joppy 9y ago
  In the case where the base ring R is a field (all of typical linear algebra), then the space associated to it is a point. So a vector bundle over this point is just a vector space, so we didn't really gain any extra geometry here - if you want a way of picturing linear algebra geometrically, you still need some other idea.
  - dbranes 9y ago
    Good point!
deleted 9y ago
[deleted]
heavenlyblue 9y ago
Would you suggest a set of intermediary-level books on the topic? For example: what would be beneficial to read if I understand that matrices are systems of linear equations, but I do not clearly understand why a cross product exists only for 3 dimensions?

dbranes 9y ago

A related remark: there is another (rather silly) way to "geometrize linear algebra". It goes something like whenever you see a ring R, think of it as the sheaf of functions (the structure sheaf) on some space. Whenever you see a module over R, think of it as the space of sections in some vector bundle over that space. Then anything you say about modules through this admits a sheaf-theoretic version, which is often easier to picture if you're geometrically inclined.

joppy 9y ago
In the case where the base ring R is a field (all of typical linear algebra), then the space associated to it is a point. So a vector bundle over this point is just a vector space, so we didn't really gain any extra geometry here - if you want a way of picturing linear algebra geometrically, you still need some other idea.
- dbranes 9y ago
  Good point!

deleted 9y ago

[deleted]

heavenlyblue 9y ago

Would you suggest a set of intermediary-level books on the topic? For example: what would be beneficial to read if I understand that matrices are systems of linear equations, but I do not clearly understand why a cross product exists only for 3 dimensions?