4 ms·

As evidenced by the confusion of at least one commenter, I do not think it is a good didactic way to introduce vectors by how they can be written in a particula

As evidenced by the confusion of at least one commenter, I do not think it is a good didactic way to introduce vectors by how they can be written in a particular basis. It is just unhelpful in many ways. It fixates on one particular basis and it results in a vector space with few applications and it can not explain many of the most important function vector spaces, which are of course the L^p spaces. In most function vector spaces you encounter in mathematics, you can not say what the value of a function at a point is. They are not defined that way. The right didactic way, in my experience, is introducing vector spaces first. Vectors are elements of vector spaces, not because they can be written in any particular basis, but because they fulfill the formal definition. And because they fullfil the formal definition they can be written in a basis.

lanza 11mo ago
> It fixates on one particular basis and it results in a vector space with few applications and it can not explain many of the most important function vector spaces, which are of course the L^p spaces. Except just about all relevant applications that exist in computer science and physics where fixating on a representation is the standard.
- constantcrying 11mo ago
  Most relevant applications use L^2 spaces which can not be defined point wise. If you want to talk about applications, then this representation is especially bad. Since the intuition it gives is just straight up false.
  - defmacr0 11mo ago
    Fwiw, my favourite textbook in communication theory (Lapidoth, A Foundation in Digital Communication) explicitly calls out this issue of working with equivalence classes of signals and chooses to derive most theorems using the tools available when working in ℒ_2 (square-integrable functions) and ℒ_1 space
- rnhmjoj 11mo ago
  In physics it is common to work explicitily with the components in a base (see tensors in relativity or representation theory), but it's also very important to understand how your quantities transform between different basis. It's a trade-off.
arjie 11mo ago
Haha, this works if you already know what a vector space is. But I think pedagogy needs to provide motivating examples. I'll quote one section of a text by Poincaré (translated by an LLM since most here do not speak French). > We are in a geometry class. The teacher dictates: “A circle is the locus of points in the plane that are at the same distance from an interior point called the center.” The good student writes this sentence in his notebook; the bad student draws little stick figures in it; but neither one has understood. So the teacher takes the chalk and draws a circle on the board. “Ah!” think the students, “why didn’t he say right away: a circle is a round shape — we would have understood.” > No doubt, it is the teacher who is right. The students’ definition would have been worthless, since it could not have served for any demonstration, and above all because it would not have given them the salutary habit of analyzing their conceptions. But they should be shown that they do not understand what they think they understand, and led to recognize the crudeness of their primitive notion, to desire on their own that it be refined and improved. The learning comes from making the mistake and being corrected, not from being taught the definition, I think. Anyway, it's from Science and Method, Book 2 https://fr.wikisource.org/wiki/Science_et_m%C3%A9thode/Livre_II,_%C2%A7_II https://fr.wikisource.org/wiki/Science_et_m%C3%A9thode/Livre... There's more to the section that talks about the subject. I just find this particular paragraph amusingly germane.
- constantcrying 11mo ago
  I have nothing against starting out with motivating examples, obviously they are needed for understanding. But they should motivate the definition of a vector space. Not the definition of vectors as mappings of indices. Functions are actually a great motivating example for the definition of a vector space, precisely because they are first look nothing like what student think of as a vector.
  - arjie 11mo ago
    Thinking about this specific case, I think you are right. The manner of describing actually confuses the concept more than if it never tried to introduce the index-mapping.
- 11mo ago
lvncelot 11mo ago
Completely agree. In uni, I (re)-learned about vectors in linear algebra, and for a good chunk of the course, we didn't write anything in "standard vector notation". We learned about vector axioms first, and then vectors were treated as "anything that satisfies the vector axioms". (When doing more practical examples, we just used the reals instead of something like R^3, but the entire time it was clear that for any proof, anything that can be added and multiplied in the way that the vector axioms describe would fit.) I think adopting this structuralist view really helps with a lot of mathematical studies.
phkx 11mo ago
Now I’m thinking that I have missed the point of the article. I didn’t read it as an introduction to vector spaces, but rather that the introduction served as to give an intuition how functions may be viewed as vectors (going back to the article, it’s even in the section heading). I found the next parts well written and to the point, leading along the steps to show that indeed the requirements for a Hilbert space are met by L^2 (even though those requirements are only spelled out in the end). I’m not actively working with mathematics any more, but I didn’t notice any major corner cutting. It’s not text book rigorous but lays out the idea in an easy to follow way. I took something away from it - or not, depending on whether I missed some inconsistency.
griffzhowl 11mo ago
> In most function vector spaces you encounter in mathematics, you can not say what the value of a function at a point is. Could you spell out what you mean by that? Functions are all defined on their domains (by definition) Are you referring to the L^p spaces being really equivalence classes of functions agreeing almost everywhere?
- rnhmjoj 11mo ago
  Yes, the L^p spaces are not vector spaces of functions, but essentially equivalent classes of functions that give the same result in an Lebesgue integral. For these reason, common operations on functions, like evaluating at a point or taking a derivative are undefined. If you care about these you need something more restrictive, for example to study differential equations you can work in Sobolev spaces, where the continuity requirement allows you to identify an equivalent class with a well-defined function.
  - griffzhowl 11mo ago
    Thanks for the clarification
anuramat 11mo ago
reminded me of "tensor is a bunch of numbers that transform in a certain way"; this should be illegal to teach, especially in physics
afiori 11mo ago
> In most function vector spaces you encounter in mathematics, you can not say what the value of a function at a point is. They are not defined that way. That is because they are not vector spaces of function but a quotient of one

lanza 11mo ago

> It fixates on one particular basis and it results in a vector space with few applications and it can not explain many of the most important function vector spaces, which are of course the L^p spaces. Except just about all relevant applications that exist in computer science and physics where fixating on a representation is the standard.

constantcrying 11mo ago
Most relevant applications use L^2 spaces which can not be defined point wise. If you want to talk about applications, then this representation is especially bad. Since the intuition it gives is just straight up false.
- defmacr0 11mo ago
  Fwiw, my favourite textbook in communication theory (Lapidoth, A Foundation in Digital Communication) explicitly calls out this issue of working with equivalence classes of signals and chooses to derive most theorems using the tools available when working in ℒ_2 (square-integrable functions) and ℒ_1 space
rnhmjoj 11mo ago
In physics it is common to work explicitily with the components in a base (see tensors in relativity or representation theory), but it's also very important to understand how your quantities transform between different basis. It's a trade-off.

arjie 11mo ago

Haha, this works if you already know what a vector space is. But I think pedagogy needs to provide motivating examples. I'll quote one section of a text by Poincaré (translated by an LLM since most here do not speak French). > We are in a geometry class. The teacher dictates: “A circle is the locus of points in the plane that are at the same distance from an interior point called the center.” The good student writes this sentence in his notebook; the bad student draws little stick figures in it; but neither one has understood. So the teacher takes the chalk and draws a circle on the board. “Ah!” think the students, “why didn’t he say right away: a circle is a round shape — we would have understood.” > No doubt, it is the teacher who is right. The students’ definition would have been worthless, since it could not have served for any demonstration, and above all because it would not have given them the salutary habit of analyzing their conceptions. But they should be shown that they do not understand what they think they understand, and led to recognize the crudeness of their primitive notion, to desire on their own that it be refined and improved. The learning comes from making the mistake and being corrected, not from being taught the definition, I think. Anyway, it's from Science and Method, Book 2 https://fr.wikisource.org/wiki/Science_et_m%C3%A9thode/Livre_II,_%C2%A7_II https://fr.wikisource.org/wiki/Science_et_m%C3%A9thode/Livre... There's more to the section that talks about the subject. I just find this particular paragraph amusingly germane.

constantcrying 11mo ago
I have nothing against starting out with motivating examples, obviously they are needed for understanding. But they should motivate the definition of a vector space. Not the definition of vectors as mappings of indices. Functions are actually a great motivating example for the definition of a vector space, precisely because they are first look nothing like what student think of as a vector.
- arjie 11mo ago
  Thinking about this specific case, I think you are right. The manner of describing actually confuses the concept more than if it never tried to introduce the index-mapping.
11mo ago

lvncelot 11mo ago

Completely agree. In uni, I (re)-learned about vectors in linear algebra, and for a good chunk of the course, we didn't write anything in "standard vector notation". We learned about vector axioms first, and then vectors were treated as "anything that satisfies the vector axioms". (When doing more practical examples, we just used the reals instead of something like R^3, but the entire time it was clear that for any proof, anything that can be added and multiplied in the way that the vector axioms describe would fit.) I think adopting this structuralist view really helps with a lot of mathematical studies.

phkx 11mo ago

Now I’m thinking that I have missed the point of the article. I didn’t read it as an introduction to vector spaces, but rather that the introduction served as to give an intuition how functions may be viewed as vectors (going back to the article, it’s even in the section heading). I found the next parts well written and to the point, leading along the steps to show that indeed the requirements for a Hilbert space are met by L^2 (even though those requirements are only spelled out in the end). I’m not actively working with mathematics any more, but I didn’t notice any major corner cutting. It’s not text book rigorous but lays out the idea in an easy to follow way. I took something away from it - or not, depending on whether I missed some inconsistency.

griffzhowl 11mo ago

> In most function vector spaces you encounter in mathematics, you can not say what the value of a function at a point is. Could you spell out what you mean by that? Functions are all defined on their domains (by definition) Are you referring to the L^p spaces being really equivalence classes of functions agreeing almost everywhere?

rnhmjoj 11mo ago
Yes, the L^p spaces are not vector spaces of functions, but essentially equivalent classes of functions that give the same result in an Lebesgue integral. For these reason, common operations on functions, like evaluating at a point or taking a derivative are undefined. If you care about these you need something more restrictive, for example to study differential equations you can work in Sobolev spaces, where the continuity requirement allows you to identify an equivalent class with a well-defined function.
- griffzhowl 11mo ago
  Thanks for the clarification

anuramat 11mo ago

reminded me of "tensor is a bunch of numbers that transform in a certain way"; this should be illegal to teach, especially in physics

afiori 11mo ago

> In most function vector spaces you encounter in mathematics, you can not say what the value of a function at a point is. They are not defined that way. That is because they are not vector spaces of function but a quotient of one