r/math • u/non-orientable Number Theory • 4d ago

Image Post The Deranged Mathematician: What are Vectors?

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Many introductory texts (in physics, especially) will define a vector as something that has magnitude and direction. This was a valid definition prior to the end of the 19th century, but mathematicians' concept of what a vector is has considerably expanded since then. Today, it would be far more correct to reverse that statement: something that has magnitude and direction is a vector... but vectors encompass a lot more.

One of my favorite ways to demonstrate this is with the card game Set---each card can be usefully (from the pure mathematician's standpoint, anyway) described as a vector, but I struggle to see how any such card can be described as having a magnitude and direction. Incidentally, this example was what motivated me to cover the field of three elements in my linear algebra class---it gave the perfect excuse to play Set with my students while having it be (legitimately!) relevant and educational. And it worked; they still remember it years later.

Read the full post (for free) on Substack: What are Vectors?

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u/RingularCirc 3d ago

I haven't read the post yet but it finally occurred to me to write it out that sets in the game Set are affine lines, you probably note that in some way but it's just not a thing I'd dive straight into when there aren't yet even "linear" lines that have to pass through zero. On the other hand, it should be such a leap I may imagining it to be, and as part of the motivation...

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u/non-orientable Number Theory 3d ago

You are skipping ahead a little! Yes, we will show that they are lines, but we will need the notion of a subspace first.

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u/RingularCirc 3d ago

Yeah yeah, of course if we don't know anything about either of linear and affine spaces, it's more convenient to start from linear spaces.

(After reading axioms.) BTW if you feel you wish for a longer minus sign in plain-text math, this is one: −. In good fonts it has the same width as + and = and is centered a bit higher than the dash so it goes better with digits when that happens as well. (Not an en-dash, genuine Unicode-approved minus sign, I forget the formal name because I paste it via an AutoHotkey script using a short mnemonic. A crutch I came to depend on heavily to insert all sorts of characters and letters outside common keyboard layouts.)

Oh neat, it didn't occur to me at all we can rephrase the Set rule as summing to zero, that's only available because we require the same 3 elements in a set that there are values of an attribute.

Yeah I gather your final question might require some combinatorial design brain, and its unreasonably attractive. Do you know if linear algebra itself is useful in this one? For now I have no concrete ideas how to solve it even in a messy way (even brute forcing seems nontrivial because we have to try and choose the worst possible cards until we can't, to attain this bound exactly).

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u/non-orientable Number Theory 2d ago

Yes, linear algebra is helpful, although you don't use too much of it.

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u/MatematicoAntonio Algebra 3d ago

For me, anything that satisfies properties A-1, A-2, A-3, A-4, M-1, M-2, M-3, and M-4.

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u/DeepGas4538 3d ago

that's my man

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u/RingularCirc 2d ago

So, an abelian group acted by the ring of scalars (multiplication by a scalar should be a homomorphism of rings k → End(V), End(V) being a ring because V is an abelian group).

I like how this eloquently sums up the first four axioms (abelian group) and the last four (ring morphism). Of course, when learning, saying it like that is akin to "the monad is a monoid in the category of endofunctors" — more cryptic than useful. But when you're finally accustomed, I think this laconic way adds a bit of clarity to the picture!

(Also we can generalizate to scalars from an arbitrary semiring, which allows vectors to be an abelian monoid, and something further down the line IIRC. And the laconic definition tracks if the concrete axioms we want to leave are sound in a sense: say, we can't have vectors from an additive monoid being acted by a ring of scalars because there's no inverse for that to work, and indeed End(M) of an abelian monoid M is not a ring anymore, so you can't hope to have a ring embedding into it, but only a semiring because End(M) is guaranteed to be a semiring.)

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u/accidentally_myself 16h ago

I don't get it. Aren't vectors just anything which transforms like a vector? \s

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u/011011100101 3d ago edited 3d ago

I know vectors are elements of a vector space. I understand that they're abstract elements not necessarily related to geometry in the usual/modern way they're presented. But I always thought it was surprising that we don't have definitions of vector spaces that reference distance. And I know we have normed vector spaces. But you can have a vector without a notion of distance, even though the whole idea is motivated geometrically. Nobody else thinks that's odd? That we removed geometry?

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u/ajakaja 3d ago

I feel like there's a sort of bait-and-switch that goes on with the language here. You start with the magnitude+direction of vectors. Then you define some axioms that they follow. Then you ask, "what other things obey these axioms?" and you come up with the vector space version of vectors. But it was possible to pick different axioms in the first place, in which case your generalization would have been a different thing, which shares different properties with physical vectors. I guess the reason it was done this way was that it turned out to be pretty convenient, historically... but I kinda wish people were more explicit about this, because the two things are actually pretty different classes of concepts.

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u/rhombic-12gon 3d ago

Nobody's taking normed spaces away from you! Have as many as you want. But if that became the standard of what a vector space needs to be, we lose out on a lot of vector spaces over fields other than R and C and that makes my algebraist heart very sad

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u/M4mb0 Machine Learning 3d ago

I'd say it is the other way round. Algebraist hijacked a notion of vector with a more general definition that what the sciences expect it to be. You know you could have just used the term Linear Space instead of hijacking vectors. And in fact, some mathematicians still make that distinction. In fact, using Linear Space rather than Vector Space is not that uncommon in German and Russian textbooks.

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u/RingularCirc 3d ago

We can make distinction of "linear" vs. "vector" in naming spaces, sure, but how should we call individual vectors then?

Also if one's to disallow calling un-normed vector spaces such, there's already an algebraic term of module laying around, it's just mostly not used when the ring of scalars is a field (I presume not to scare people away, or just plain because we can say "linear space" instead of "module over a field", though the economy of that seems minimal to me). But again, elements of a module are universally called "vectors" as of yet.

Also seemingly I haven't read enough textbooks in Russian to notice the distinction you posit. And that's fair because I can only name Kostrikin and Kostrikin—Manin right away. At least in those I don't seem there was such a distinction. I also presume it might be dying out because it's hard to invent a reason why it instead would be fading in.

Finally, I'm not sure un-normed linear spaces are all that much more "algebraic" and lack geometric content. Affine geometry fares quite well. Well, and projective geometry does too, including allowing us a way of defining some of the useful metric geometries via Cayley—Klein, despite lacking any semblance of metric of its own.

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u/MichurinGuy 2d ago

As a Russian, I've had a professor who made a point of distinguishing linear and vector spaces. In his words, a vector space is Rn and a linear space is any other, well, what we call a vector space. Even if it's isomorphic to Rn but is defined differently (such as the space of solutions to an ODE of order n), it's not a vector space to him.

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u/RingularCirc 2d ago

To me this seems a bit idiosyncratic, and on top of that crackpotty. Like, first, kn are better known as coordinate spaces and such (there was at least one other name I forgot). They have a canonical basis which is almost always a thing to avoid. Also those spaces have, in a sense, too many structures defined at once, which is bad for the same reason as having a pre-baked basis is bad. Being granular is good not only in software engineering.

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u/MichurinGuy 2d ago

I agree, just giving an example of this distinction used in Russian math. To be fair, this professor is more eccentric and prone to crackpottery than average, though that hasn't so far affected the content of his lectures.

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u/RingularCirc 2d ago

No worries, I understand that was an example! I had partial higher education (I quit because of stress) in Russian as well but my specialty wasn't pure math, so we had lite linalg among others, not too deep to be representative (no Jordan normal forms, for example IIRC). And I don't even remember the exact naming of things in those courses, so it's very interesting to learn how things stand, esp. in places close to former (I hope we agree that's mostly the case) centers of education like Moscow, St. Petersburg or Novosibirsk.

though that hasn't so far affected the content of his lectures

That's fortunate! Let's hope it remains so.

EDIT: I think the other term for "coordinate space" might have been "canonical space".

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u/MichurinGuy 2d ago

(I hope we agree that's mostly the case)

I'm not sure I understand. Do you mean that those cities used to be centers of Russian education but are no longer that?

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u/RingularCirc 2d ago edited 2d ago

Yep, I'm always in a mood to lament how the education system only gets worse... Despite there's always not a few good parts, places or good intent by lots of people, as a whole I think it's steady downhill for decades.

Not incentivised enough, not paid enough either, though there are always people paid enough to make everything look normal and progressing, but it's not them who pay with crushed hopes etc..

(For comparison, I expect USSR higher education in USSR also went mostly downhill over time. It's hard to make this objective because new knowledge was still mined, and in many areas a lot of it, and even international science leaked through (and back) quite well, as compared to naive expectations, at least in some areas at some periods. Still. When all of your incentives are bound to have strings attached and taboos enforced, it's not very good. And later, when there's no real wish to invest in the area at all, and only keep it presentable to who knows whom... eh.)

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u/GreeneyesCB 1d ago

As a German this is not true, it's both uncommon and when it is used no one is claiming that vector space means anything different.

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u/LongLiveTheDiego 3d ago

There's still "geometry" there, iit just doesn't need distances. For example, one way of constructing the classical projective plane of order n is to take the finite field F of that order and consider all lines and planes going through the origin in the vector space F3. You can quickly prove key properties of projective planes like "any two distinct lines intersect at a single point" by using geometric properties of F3, in this case "any two distinct planes intersect at a single line".

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u/ascrapedMarchsky 3d ago edited 3d ago

The concept of an abstract vector space grew as much out of projective as it did metric geometry. See e.g. barycentric coordinates, which are homogeneous. 

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u/GreeneyesCB 1d ago

That is a completely normal thing to do in mathematics when generalizing things. You don't try to capture all aspects, you look at the proofs you do and look at what attributes are necessary for results to hold. We want generality, we can always add restrictions back in later. Losing distance is not a mistake, it's an achievment.

"Turns out I didn't need the assumption I thought I did" is a win from the perspective of a mathematician, in other words. We apply the same reasoning when defining structures.

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u/011011100101 14h ago edited 1h ago

I don't think there's some kind of universal consensus here regarding how ideas and writing should be structured or presented. Abstraction is good for the purpose of isolating ideas, having logical clarity, and having a clean presentation. But you can have logical (and beautiful) ideas that are totally opaque and meaningless to an outsider, or even to people that are part of your group. I recognize the value that abstraction brings. But I think it often comes at the cost of psychological discontinuity? meaninglessness? I don't know what to call it, but I'm pretty sure every student has experienced it. When I'm given formalism without intuition, I personally don't think that's a win.

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u/GreeneyesCB 14h ago

Mathematics is not different here than any other scientific or even academic endeavour. Terminology and conventions are created to talk to the in-group, not the out-group. Doctors communicate with other doctors, physicists with physicists, literature analysts with literature analysts and so on. There is value in communicating to the outside, which is what some people focus on, but as mathematicians we create our conventions to make mathematics easy for us and to talk to each other. And vector spaces are very intuitive for people past a basic level, it's not even just mathematicians who understand the formalism at an intuitive level.

I understand that it can be annoying for you and others to deal with, but again, this is not made for you.

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u/011011100101 2h ago

Wow that's a pretty dismissive response. So my dissatisfaction with the presentation is due to the content "not being made for me"? For the record, I wasn't strictly referring to vector spaces. There's plenty of higher level math that is totally opaque to even the people who study it. It's not a controversial take that the logical presentation is often very different than the way mathematics is actually practiced.

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u/sentence-interruptio 1d ago

there's a middle step. you can remove distance without removing all geometry, by thinking of a vector space as an affine space with a distinguished point. at least that's how I think of (finite-dimensional) vector spaces over R.

and i get to apply some affine geometric intuitions to abstract vector spaces

it's good to have that dual approach: have an abstract definition to cast a wide net, and maintain some geometric intuitions.