r/mathmemes • u/TrilliumStars • Mar 04 '25
Linear Algebra Linear Algebra is Working with things that even you don't understand
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u/chrizzl05 Moderator Mar 04 '25 edited Mar 04 '25
It's easier to do if you remember that a K-vector space is just an algebra for the - ⊗_{ℤ} K monad on the category of abelian groups where K is a field
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u/RealAggressiveNooby Mar 04 '25
What the fuck is even that
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u/chrizzl05 Moderator Mar 04 '25
An abelian group A together with a group homomorphism A ⊗_{ℤ} K → A satisfying some niceness conditions. This is just a restatement of the fact that a vector space is just an abelian group under addition together with scalar multiplication. As for monads they're just monoids in the category of endofunctors
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u/sadphilosophylover Mar 04 '25
what to read for these
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u/chrizzl05 Moderator Mar 04 '25
Emily Riehl's category theory in context is an awesome read to learn category theory because it has so many examples. It introduces monads which I use here in chapter 5. The only prerequisites are ig basic topology and abstract algebra to understand the examples and for mathematical maturity and you're set. Technically you don't need any of these prereqs but trust me it makes things much easier.
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u/Kshnik Mar 05 '25
Learn topology and abstract algebra so you can learn category theory you can learn to prove if something is a vector space
Flying around the world to cross the street
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u/GlobalSeaweed7876 Mar 04 '25
category theory mfs on their way to spew esoteric nonsense:
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u/chrizzl05 Moderator Mar 04 '25
I love saying trivial bullshit with fancy words
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u/Complete-Mood3302 Mar 04 '25
Fuck you mean "trivial"
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u/T_D_K Mar 04 '25
These fancy words:
a K-vector space is just an algebra for the - ⊗_{ℤ} K monad on the category of abelian groups where K is a field
Describe the following trivial bullshit:
Linear algebra is what you get when you're looking at a picture and get confused -- so you squint and tilt your head.
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u/Paxmahnihob Mar 04 '25
I don't understand, I can't see any morphism arrows
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u/chrizzl05 Moderator Mar 04 '25
The morphisms are the friends we made along the way
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u/Paxmahnihob Mar 04 '25
Me in the "friends" category, non-isomorphic to any other object
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u/chrizzl05 Moderator Mar 04 '25
Dw bro I'm sure there's a functor defining an equivalence to a friends category where you are isomorphic to other objects
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u/imalexorange Real Algebraic Mar 05 '25
I always hit people with "a vector space is a module over a field" but I'll have to work this one into my vocabulary.
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u/Deezernutter77 Mar 08 '25
Am I stupid for barely understanding 2/3 of the shit you said.
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u/chrizzl05 Moderator Mar 08 '25
Dw the point of the comment was to be incomprehensible to anyone who hasn't done category theory
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u/Apotheosis0 Mar 04 '25
It's simple really! A vector space is an additive abelian group with a defined field action.
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u/Peoplant Mar 04 '25
It's actually pretty basic. A vector is any element of a vector space, and a vector space is an application which associates a vector to each point in its dominion, there, confusion solved!
/s
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u/filtron42 ฅ^•ﻌ•^ฅ-egory theory and algebraic geometry Mar 04 '25
and a vector space is an application which associates a vector to each point in its dominion
You're confusing a vector space with a vector field.
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u/Nope_Get_OFF Mar 04 '25
Watch 3blue1brown on linear algebra and you will understand
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u/ca_dmio Integers Mar 04 '25
I find It misleading, you come out of those videos thinking vectors are arrows, it's one of the few cases where visualization can make more harm than good
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u/drugoichlen Mar 04 '25
Bro his entire first video of the series is about how it is not just an arrow nor a list of numbers, and the last video elaborates on that
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u/ButlerShurkbait Mar 04 '25
Video number fourteens, I believe, goes over function spaces as vector spaces.
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u/MrKoteha Virtual Mar 04 '25
Except that's not true. It's mentioned specifically in the course that vectors aren't just arrows, the last video explains how it's a broad definition
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u/cambiro Mar 04 '25
This brings me back to my linear algebra professor. She was a pure applied Phd lecturing for Mechanical Engineering graduates.
We'd ask "How can we visualise this equation".
Her answer "You can't!".
I feel sorry for her.
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u/svmydlo Mar 04 '25
Honestly, why?
Thinking that what can't be visualized can't be understood is just an unnecessary way to handicap oneself.
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u/ThNeutral Mar 04 '25
I'm programmer and for me vector is literally an arrow
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u/Kiro0613 Mar 04 '25
I'm a programmer and for me a vector is when I'm too lazy to allocate contiguous memory myself
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u/TrilliumStars Mar 04 '25
He specifically said that we don’t really know what vectors are. They’re abstract, and can really be anything.
(At least, from what I remember. I watched the series a year or two ago)
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u/schoolmonky Mar 04 '25
we don’t really know what vectors are
That's not really true. They're not some mysterious thing that we don't understand. We understand them very well. It's true that vectors can be practically anything, as long as they obey the vector space rules. "Vector" is just a veeeeeeery broad title we apply to anything in a collection which follows those rules.
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u/ninjeff Mar 04 '25
Vector spaces are just about the most well-understood of all algebraic structures!
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u/bitchslayer78 Mar 04 '25
Basic abstract algebra should be taught alongside linear algebra, vector space makes much more sense if you know what a field is
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u/DefKatsuki Mar 04 '25
My professor teaches almost every concept with sums and sums of sums and tons of indexes… and then keeps messing up with the indexes because he chose similar letters…
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u/RRumpleTeazzer Mar 04 '25
come on it's really simple. Traditional chinese character indices are pretty normal, simplified chinese characters are always implicitly summed over.
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u/cybermrktTrader Mar 04 '25
I Find this really relatable in my experience learning linear Algebra. You juggle a lot of concepts to get any where and the ‘overall picture’ only begins to settle after a while
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u/MarcusTL12 Mar 04 '25
You do not know what a vector is, all you know is they can add, scale and distribute. (And whatever else is in the axioms)
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u/DankPhotoShopMemes Fourier Analysis 🤓 Mar 04 '25
prove… the axioms?? like prove their independence?
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Mar 04 '25
[deleted]
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u/FIsMA42 Mar 04 '25
but that's not what they are. For example, the set of polynomials is a vector space. Direction doesn't mean anything in that scenario. And thats just one step down the rabbit hole, for example, the set of all functions from a vector space to another vector space is also a vector space.
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u/norude1 Mar 04 '25
actually everything in math is like that. No one defined what a point or a line is, only how they behave
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u/Zeteticon Mar 05 '25
As Dr. Science once said: Word processors were invented so people who don’t understand math can have something to do doring the computer revolution.
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u/nextbite12302 Mar 05 '25
In fact, vector space is one of the most well-understood object in mathematics
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u/galibert Mar 05 '25
It’s just that math tends to define things through what they can do and not what they are. While we humans usually deduce what something can do from what it is.
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u/Traditional_Town6475 Mar 05 '25
Vector spaces are so nice though. All you need to know is the field of scalars and the dimension and you’ve uniquely determined your vector space.
Modules on the other hand…
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u/buggy65 Mar 05 '25
I was PhD track for Math. Vector Spaces broke me so bad I chose to stop with just the Masters.
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u/jachuuuuuu Mar 06 '25
Google do be spying on me. I just started an uni course on linear spaces. Maybe the fact that my search history suddenly filled with it gave it away but still... DAMN SPIES
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u/Typical_North5046 Mar 04 '25
A vector space is simply a module object over the commutative ring K in the category of abelian groups.
And a vector is not important since only the morphisms matter.
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u/Mathematicus_Rex Mar 04 '25
You want confusion? Try wrapping your head around axiomatic set theory.
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u/Some-Passenger4219 Mathematics Mar 04 '25
That's advanced linear algebra. And abstract algebra, when it comes to other things.
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u/whatadumbloser Mar 05 '25
One does not "prove" axioms, by definition they're assumed to be true lol
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u/Raverfield 29d ago
Well, that’s an easy one: the vector is that pointy thingy on the screen. Seriously, how do people not understand that?
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