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.
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.
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.
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.
It's trivial to provide motivating examples for vector spaces, and there's no reason you can't do so while explaining what they actually are, which is also very simple for anyone who understands the basic concepts of set, function, associativity and commutativity. The notion of a basis falls out very quickly and allows you to talk about lists of numbers as much as you like without ever implying any particular basis is special.
I hesitate to call anything pedagogically "wrong" as people think and learn in different ways, but I think the coyness some teachers display about the vector space concept hampers and delays a lot of students' understanding.
Edit: Actually, I think the "start with 'concrete' lists of numbers and move to 'abstract' vector spaces" approach is misguided as it is based on the idea that the vector space is an abstraction of the lists of numbers, which I think is wrong.
The vector space and the lists of numbers are two equivalent, related abstractions of some underlying thing, eg. movements in Euclidean space, investment portfolios, pixel colours, etc. The difference is that one of the abstractions is more useful for performing numerical calculations and one better expresses the mathematical structure and properties of the entities under consideration. They're not different levels of abstraction but different abstractions with different uses.
I'd be inclined to introduce the one best suited to understanding first, or at least alongside the one used for computations. Otherwise students are just memorising algorithms without understanding, which isn't what maths education should be about, IMO. (The properties of those algorithms can of course be proved without the vector space concept, but such proofs are opaque and magical, often using determinants which are introduced with no better justification than that they allow these things to be proved.)
For many students, it is not so simple to grasp the concept of an abstract vector space. They could be taking linear algebra as college freshmen, without having seen any formal algebraic structures before. Many are unfamiliar with the formal notion of a set (and certainly have not seen the actual axioms of a set before). Most linear algebra students are not actually math majors; they are typically studying engineering, computer science, or some other physical science. Examples of abstract vector spaces are most often function spaces of some form (for example, polynomials of at most a given degree). These examples are not so motivating for non-math students.
The main reason why people care about linear algebra is that it lets you solve linear systems of equations (and perform related operations, such as projections). A linear system of equations has an immediate correspondence with a matrix of coefficients, a right-hand side vector, and a solution vector. For this reason, it is very natural to first talk about matrices and vectors (they can be used to represent concretely a linear system of equations), and then introduce the concept of vector space in cases where the abstract view can be clarifying or help with understanding.
From my perspective, the "right" way to teach linear algebra depends on the mathematical maturity of the students. If they are honors math majors, they can easily handle the definition of an abstract vector space right away. If they have less mathematical maturity, the abstract viewpoint isn't helpful for them (at least not without first familiarizing themselves with the more concrete concepts). Think about it this way: we don't teach school children about natural numbers and arithmetic by first listing the Peano axioms.
I think at least in the UK the lack of "mathematical maturity" among early undergraduates is partly the result of this very coyness about mathematical concepts. Enormous time at A-Level is spent rote learning algorithms, and very little on grasping the basic concepts of mathematics, so it's hardly surprising students turn up unprepared for such simple notions as "vector space".
I don't have first hand experience of the French system, but from what I understand the approach there is more along the lines I'm thinking of, and the relative over-representation of French graduates among my more mathematical colleagues suggests it may be rather effective in practice.
That's awful - just an awful way to teach. It's from more than a century ago when the point was to tame the children and turn them into good Prussian soldiers.
You don't have to start with anxiety, shame, and dominance - you can start with curiosity, a base of common understanding, and then experiment and problem solving.
What's awful is thinking that making mistakes is a form of shame, and that being corrected is a form of dominance. That view is something that is taught and acquired and I am very thankful that the people who taught me never made me feel that way. I make mistakes all the time and I never have to feel ashamed about it, nor do I feel that the people in my life who I've learned from hold some kind of position of power over me.
>It's from more than a century ago when the point was to tame the children and turn them into good Prussian soldiers.
If you judge by the outcome, that is probably the greatest education system of all time.
>You don't have to start with anxiety, shame, and dominance - you can start with curiosity, a base of common understanding, and then experiment and problem solving.
You can. The kids will learn nothing though.
School nowadays is a joke. An absolute waste of time. In a single semester of rigorous mathematics I learned more than in years in school. It is cruel to waste childrens time like that.
School needs to be authoritarian, rigorous and selective.
> 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.
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.
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
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.
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.
> 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
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.
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.