[gizmo686]

https://thenumb.at/Functions-are-Vectors/#proofs

It's not a particularly interesting proof, but the author does prove that real valued functions are vectors. The bulk of the article is less about proofs, and more about showing how the above result is useful.

[ttoinou]

Vectors in the way he talks about in the beginning. With indices (and then extending to "In higher dimensions, vectors start to look more like functions!"). Of course if you use the general meaning of every word, vectors are functions and functions are vectors, and this article shouldn't then have anything interesting to talk about.

  how the above result is useful

It doesn't seem useful at all to me, the examples in the article are not that interesting. On the contrary it is more confusing than anything to apply linear algebra to real valued functions.

[Tainnor]

I think you are confused about the analogies at the beginning at the article (although there is nothing technically wrong about them).

Here's the definition of a vector space which agrees with the one everyone in mathematics use: https://thenumb.at/Functions-are-Vectors/#vector-spaces

From this it's fairly easy to prove (and done in the article) that the set of all functions R->R is a vector space.

[ttoinou]

I understand that. The crux of the article is supposedly being able to apply what we learned with easy small vector space to infinite function vector space : we can't. But he's taking the example of polynomials and analytical functions which are MUCH smaller spaces than functions in general.

[Tainnor]

Yes we can. Several people have tried explaining this to you. Maybe you should sit down and work through the material.

[ttoinou]

Maybe you could envision that I learned maths a similar way than you and that's why I have problem with this material. It's trying to portray complicated space as simple, they're not