Technically I think its mathematically correct to say "a set of functions with certain properties form a vector space" (For example the set of continuous functions on the real line form a vector space). For the purpose of Fourier transform intuition, let "function" just refer to the subset of functions that are a continuous mapping from the reals to the reals (although the definition of function is more broad than that).

I still remember the first time I understood this in a quantum mechanics class. It was mind blowing

Beautiful. I have a phd education in physics and never once thought of a function as a column vector with infinite components. The integral form of inner products makes a lot more intuitive sense now, I typically just thought of whatever the integrand was getting multiplied by as just as a "discrete" basis vector in some direction, but it was a column vector all along!

Vectors are functions.

I've never thought about Fourier transform like that. Thanks!

Mind Blowing!! Each time I dig into Linear Algebra, I open my eyes. I believe there will be more eye-opening moments.😮

Great video. I had good intuition on functions as vectors, but not in the sense of infinite (and uncountable!) dimensional vectors. Thanks for the video!

"take the dot product with itself and square root the result" just made me realize that thats exactly the same as Pythagoras theorem

Wonderful video please keep up.

Oh man this is great. That's an incredibly beautiful connection I probably would not have noticed on my own, thanks!

It's interesting how the space of all linear transformations between two subspaces is a vector space which means all the linear transformations are 'vectors'. which is like functions being vectors as well.

Great video with beautiful animations. Keep up the good work!

Amazed, Thanks. This is a fairly new intuition for me.

they are also points in the space C, L^2, in the schwartz class.

Unfortunately, the transition from R^n to infinite-dimensional function spaces isn't exactly trivial

Valen, this was fantastic - the dot product is the foundation for all signal processing (continuous or discrete - I'm following you to hopefully see more

I l❤ve it! It has helped me to see more clear why functions are vectors! Thank you to much!

Nice one! Looking forward to more videos! :)

vectors are lists of numbers, functions are (infinite) lists of numbers. integrals are dot products.

Linear algebra was a gift from God

this is mindblowing bro, excellent video intuition makes math so much more beautiful

Neat! Waiting for more videos!

Fourier transform as a change of basis (or reference system). Is there any other way to see it?

I'd encountered the concept of "functions as vectors" before but not the "functions as infinitely long column vectors." That tickled my brain

Thank you, looking forward to more content!

...and vectors are tensors of rank-1, so spinors are da freaks of the manifold.

Concise and well put, bravo

Awww man i feel like you didn't do the fourrier transform justice. It's sooo mindblowing when you start thinking about a projection of a function onto rotational space. It would have been more mindblowing if you didn't spoiler the fourrier transform before actually projecting it. But anyways, happy to see that it's not being gatekept

Great video 🎉

This is something you learn in physics by intuition because no one annotates their maths properly. And then multiplies a function by a matrix without explaining it

This is omg. You gave me a lot of happiness. So beautiful. Bro make more videos

This is very cool thx, in computer science we use this, we have a finite set of points of a given function say image or audio and apply a transformation like Furier and to do that we use linear algebra and we can use optimizations known for líneal algebra algorithms, like the FFT.

the vector space of functions is probably my favorite abstract vector space. and then start doing geometric algebra on it and it becomes even more fun

very cool, thank you!

Its funny because everything in math is a f*ing vector if you add the correct additive and multiplicative composition rules

I have studied for a while about image/video compression and it relies a lot on DCT (Discrete Cossine Transform). I actually never understood it very well, and treated this as pure magic. This video just made everything click inside my head. Great job.

Integratable functions map every point in each dimension to a novel point in another and a jacobian can be found that maps the change 😊

Mind blowing! Can you make a video that goes more in depth about the topic?

This is arguably the starting point for functional analysis…

That's really cool. What's a good book you'd suggest to study this subject a bit more?

this is an amazing video

Well done.

It's easier to visualize higher dimensions than you think. If you've ever used a tool like Blender, a 3D model is built up of vertices which are vectors. These vectors can have more than three dimensions. It's in fact common for vertices to have five dimensions; x, y, z, u, v. How do you work with this? Simple, each vertex is viewed from two separate perspectives, one 3D showing x, y and z and one 2D showing u and v.

I think you should have used Fourier series instead of Fourier transform, because it's not really correct mathematically to think about the Fourier transform as a dot product with some outher function in the same space.

Functions mapping to a vector space are vectors. An arbitrary function would not be a vector.

Really interesting!! Very clear and appealing visuals and a great explanation!! Please keep up this phenomenal work, you got a new loyal subscriber ❤

Wow

Good video! Did you use manim for this video?

I have a terrible problem. Any time I see a video about complex numbers I immediately wonder how it applies to quaternions and other hyper complex numbers. And anything dealing with any of the complex or hyper complex numbers or vectors makes me wonder how that translates to multi vectors and geometric algebra. So I understand how to use the geometric product to multiply vectors into multi vectors but I’ve only seen it applied to finite numbers of dimensions, (unless the most universal geometric algebra is invoke.) But I’ve never considered using functions with continuous domains as vectors. Are there vector spaces using function based vectors that can do something like Clifford algebras? Considering that Clifford Algebras can be boiled down to vectors with orthogonal basis vectors that square to either 1, -1, or (occasionally) 0, I’d be interested in how that works with continuous domain functions. (I don’t think the function itself necessarily has to be continuous unless it has to be manipulated with calculus.) I guess the weirdest part is that a n dimensional vector space used to create a geometric algebra creates a new vector space with 2^n dimensions. So would the function vectors create a multi vector space with the power set of the domain as its domain? The more I think about it the more unlikely it looks for a continuous domain function based geometric algebra. Maybe if I were a mathematician I’d have a clearer idea of whether or not they’d be possible or actually interesting. What does the power set of a real interval look like anyway? The only thing that I’m kinda sure about is that there are more elements in the power set of [0, 1] than in the numbers in [0, 1]. I’m not even sure how to build a function on such a domain. I guess I’ll stop my rambling here.

Great job and beautiful Animation, can u tell me what the logiciel that u use to make this animation 🙏🏽

Cooooool!

The pidgeon holing of everything through linear Algebra must be stopped! Still cool video for pointing out the identities

Functions are infinite dimension vectors

Functions with compact sypport are functionalas on radon measures

Could you please elaborate on how the bottom integral at 3:46 is derived?

Everything seems to be a vector

Very interesting, but some technical details of this video could be improved: -some visuals go too fast, and we don't have the time to "digest" them. Take more time when you have a lot to read. -the voice over is difficult to understand due to a weak sound level in general. Some audio mastering would help. I know that the goal was to be "intuitive", but anyway, these details are to be considered.

I didn't understand a thing

I get your point however I believe the expression «functions are vectors» is not mathematically accurate. What you mean to show here is that «a set of functions can be a vector space» and a few of the consequences of that fact. Or alternatively, «a function is a vector IN a function vector space». Because for a «function to be a vector» you need a few assumptions beforehand, as you showed yourself here. So, the statement «functions are vectors» is false. Anyway thanks for the video.

2:09 There shouldn't be any t on the left side there, I think.

Enjoyed the video, I understood the maths presented but I now work as a teacher. If you want to present information to even a fairly wide (first year uni student and beyond audience) you need to take slower steps. Explain why (not necessarily with proof) where all of the steps come from, do not jump from functions as a vector space to Fourier transforms. Realistically, if you are trying to attract a reasonable audience, getting to Fourier transforms would be the end of a several episode series of videos. It is worth reminding yourself that the average person watching these videos is not only below the level of the average first year maths student but probably struggles to solve linear simultaneous equations. This is absolutely not a hate comment, I hope you continue to do what you're doing.

The voice of the narrator is too quiet but otherwise its a good video.

I guess your functions have to have some field as a codomain to view them as vectors, otherwise is see no way to define a dot product.

Well yeah vectors are a different form of a co-ordinate system

Numbers are sets

hm nice topic but i feel like it woukd have been very helpful for understanding if the video was 10 minutes longer and if you went inti more details on the explanarions. too many question(mark)s touched/opened but not answered by this video. if the focus is to teaxh and not to entertain, that is. ( i think there are enough entertainers already).

Functions arnt just vectors. Functions are values.

Is it just me, or is there something wrong with the notation he's using? Taking it at face value, he's repeatedly showing various forms of X = integral of X from minus infinity to plus infinity with respect to t. This is like saying X = f(X) or f'(X) = f(X) which can be true in special cases, but not generalizable.

volume is very low, almost can't hear you

voice wayyy to silent compared to the music - and too silent in general

Hmm.. you give an example of the components of a vector being a function and show them, plotted together, making a sine wave or whatever it was. But fundamentally the dimensionality of components is an expression of their distinctness. Just because the fourier transform or DCT _can_ express any output as a frequency function doesn't make it meaningful. Let's make a vector. Component 1 is the coal output of Europe. 2 is the average toenail size. 3 is the wattage of a ZX81. So we have a 3d vector. And now we're going to treat it as a function and express it as a frequency. That's pointless.

No, all functions are not vectors, this is not true. Go look up a vector space and figure out why.