Abstract vector spaces | Chapter 16, Essence of linear algebra

This is really the reason linear algebra is so powerful.
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Home page: www.3blue1brown.com/
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3b1b.co/support
Thanks to these viewers for their contributions to translations
Russian: e-p-h
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[@taraspokalchuk7256]

I think we should all stand up and applaud

[@vasishtapolisetty639]

I just did! Alone in my room. A special moment!

[@christianwireko5871]

me too!

[@atarzwell]

1 liek = 1 clap

[@bhavulgauri7832]

I think many of us did, in front of our computer screens. This guy has linearly transformed us into better selves.

[@john_atkins]

I can't explain how much more confidence I have to not just tackle linear algebra, but use it as an actual tool.

[@DrJx-hk4np]

And he understood perfectly all of those geometric interpretations without watching his own videos first. He is a genius

[@snowy0110]

Agree He is genius not only he understood it without the visualisations he gave us, he is genius because he managed to entangle the mess of formulas and give us the simple and intuitive way of thinking (the way it was actually invented) which, I presume, is really hard to find on any paper book (because it is just paper book, not a video). I clearly see now that the classic book from my university has really no intention to teach linear algebra, but merely serve as cryptic notes for the lectures (pretty crap lectures, I gotta say, as they just wasted tons on peoples time without a decent explanation what actually we were doing). 3b1b, Grant is a genius, I love him.

[@l.1244]

@@snowy0110 tbh. That's how it should be. It's about mass first. You'll get the intuition once you start to really work with the concepts in later semesters. But the beginning is just a excersice in keeping your head down and accepting it as it is.

[@jackismname]

I would say he is more of a genius for executing these animated masterpieces. A lot of mathematicians are exposed to these ideas and the visualizations are given on paper, but I don’t think anyone really knew there was such a thirst for mathematical animations on youtube that Numberphile scratched the surface of.

[@mastershooter64]

if he is a genius then think about the mathematicians who invented these concepts, and about the mathematicians, who understood these flawlessly and built upon these concepts to invent new concepts

[@guyguy1811]

@@l.1244 I think that a terrible opinion. I would more then happy to share why if you steal believe what you wrote.

[@RSA_Shock]

I think I speak for many people when I say you have had the single largest impact on my academic life and how I go about learning (whether it be math or anything else really). This is true education, from the bottom of my and many others hearts, Thank you for all that you do Grant.

[@johnzhang1139]

Ultra-quality visuals and ingenious interpretations of Math from someone who really understands it and the difficulties it presents to most of us. One marvellous educator he is!

[@ruwanibailey7875]

simp

[@mortiloniperpomontinson]

E

[@suvrat4251]

E

[@deltamico]

E 404

[@10monthspregnant]

There is a special place in heaven for teachers like you.

[@Granitepunch]

is heaven a subspace of the IR world :^)??

[@Nelekochara]

you wish him dead?

[@Granitepunch]

Nelekochara I’m just asking him what kind of space heaven is :(

[@scottburgert9538]

A vector space? :D

[@lowkeylyesmith8545]

@@scottburgert9538 imaginary space....

[@AniketSharmacodes]

Never felt at the end of a educational series like "When will be the next season airing, damn!"

[@akasakasvault7597]

welcome to no blue no browns

[@TigaToonsELTiagor]

Aaaaahhh

[@moomoosattack7063]

I've always loved math, but had a linear algebra professor that would print the pages of the textbook and read them word for word as the lecture. It almost killed the love I had for all the fancy and elegant proofs there are, but you just saved and revived that. I'm so happy to be alive in a time where amazing people like you can reach out to millions inspiring entire generations.

[@youcefyahiaoui1465]

I am 52 year old and I considered myself until now as not so bad in math especially in linear algebra. But watching your series (that I could not stop by the way until the end) is the best education in linear algebra bar none. Kudos to you, outstanding everything! You got my 200/100.

[@jackkennedy98]

You are making the world a better place.

[@zes7215]

no such thing as better or worse

[@tomersvirsky5807]

you know that he meant that he likes better the world that 3Blue1Brown is helping shape. No need to always be a smartass.

[@LittleLionRawr]

@@zes7215 There is. Unless you're looking from a pessimistic nihilistic point of view.

[@notnilc2107]

@@LittleLionRawr I think you mean objective rather than pessimistic/nihilistic. Pessimism and nihilisim stem from objectivity (e.g. the existence and continuation of life has no meaning because it was basically a happy set of coincidences that have no intrinsic/objective value) but they aren't fully representative of being objective. For example there might be an optimistic person who looks at the pointlessness of life as a reason to live, since if life is pointless then the opposite of life (or the complete absence of the mere notion of life) is equally pointless, which allows them to be objective even though they are technically being subjective (by continuing to choose to live). This person would also say that there are no such things as better or worse, but would also say that life has objective meaning based on the axiom that sentience is good, since to be able to comprehend the pointlessness of sentience you'd have to have sentience in the first place (it's kinda obvious that last point).

[@LittleLionRawr]

@@notnilc2107 Well, no, I meant what I meant. Rather, according to your point there are other points of view that regard better/worse as non-existent. In any case, indeed given certain initial assumptions/axioms on the value of something, better or worse exist. The overall point being, there are such things as better or worse. But not from a perspective that rejects intrinsic meaning to or value of anything. (for example indeed in an attempt to be as objective as possible.) From other perspectives where meaning and value exist, better and worse do exist. Another interesting perspective is that sometimes concepts exist because we say they exist. In a broader sense, because we can think of it, the abstract concepts of good and bad/better and worse definitely exist, even from an objective perspective, since subjective concepts do exist as subjective concepts within an objective point of view.

[@alexwang982]

Wow. Derivatives are freaking linear transformations. My whole math just changed

[@hektor6766]

Now think about what transformations integrals and partial derivatives are.

[@taeyeonlover]

I just realised FT is linear and looked it up to find DFT is "a change of basis"

[@mukeshkhatri3850]

I might be wrong, but does this make derivative linearly dependent?

[@MrMctastics]

I'm in the process of getting a degree in math. I like to say I have a degree in linear operators

[@MuffinsAPlenty]

Mukesh Khatri - The derivative is a linear transformation, not a set of vectors, so it doesn't make sense to call it "linearly dependent" or "linearly independent". But while using the wrong terminology, I think you have still noticed something. The derivative linear transformation has a nonzero kernel. In fact, it sends the basis element {1} to 0, so the kernel is one-dimensional. It collapses a "line" (all scalar multiples of 1, i.e., all constants) in the set of polynomials to the origin. This means that the derivative is not reversible (you can't undifferentiate) since it collapses part of space. Luckily, it only has a one-dimensional collapse, which is why any two antiderivatives of a function different by a constant.

[@re1konn]

"Abstractness is the price of generality." I'll remember that at least😅

[@justamanofculture12]

That's what a Smart Artificial intelligence will say lol

[@domenicskelcher6408]

I might have shed a few tears at the wondrousness of this video. This series inspiring like no other. You may not realize it but you are acting as the building block for an entire generation of mathematicians. People who will go on to discover and innovate are only here because of you. And even though you may never see this. Know from the bottom of my heart that I and this whole world wide community thank you for it. All the love, DS

[@zostube]

Amen

[@propoop6991]

calm down

[@fastlearner292]

@@propoop6991 You need to calm UP

[@propoop6991]

@@fastlearner292 no lol

[@illyon1092]

"The mathematician's answer is to just ignore the question". This is so beautiful.

[@bhatman88]

also the most mathematician thing I've ever heard, lol

[@leo_artean5749]

It's also the politician's answer, but I believe it's a mere coincidence. (Though it would explain Cedric Villani's change of career)

[@SuperVapourizer]

Wouldn't a mathematician rather answer as "A vector is an element of a vectorspace"?

[@chaddaifouche536]

@@SuperVapourizer That is the point : a vectorspace is not defined as "a space of vectors", it's defined as a space that has certain properties. In the end, a vector in a mathematician mind is not defined by what it is, but what it can do and how it does it. This is a general rule in advanced mathematics : you cease to wonder *what* things are, you start to ponder *how* things behave, to the point that, more and more, you don't distinguish between different sets, if they are isomorphic (their elements behave the same, for a certain value of "the same").

[@insomnia20422]

now I know why my professor ignores my questions about all sorts of stuff he isnt lazy or hates me, he just knows that no answer he could give would be complete and sufficient... atleast thats what I hope.

[@homo-sapiens-dubium]

If there were a nobel price for didactic proficiency in academia, you sir would surely get it instantly. - every student watching this series

[@vitrums]

Well said. I don't care if he invented this idea of visualizing EVERY verbally spoken concept within the videos or not. The more important thing is that besides a superb visual implementation he also comes up with an increadibly polished narrative itself.

[@dougr.2398]

Nobel Price? Lots of work!

[@whatno5090]

I've watched many online educators before. It is clear to me that none but Grant have the passion for truly ensuring absolute and intuitive understanding, and that is why is teaching is so efficient, effective, and important.

[@byronwatkins2565]

Nobel did not fund a prize for math. The best he can hope for is the Field's prize.

[@justamanofculture12]

Exactly. Not even students but industry professionals.

[@nolanfaught6974]

First-year math undergraduate: "yeah, vectors are just lists of numbers" Second-year math undergraduate: "okay, maybe polynomials can be considered vectors, but I'm not sure why that matters" Third-year math undergraduate: "Before we can talk about vectors I'm going to need you to define the vector space, scalar field and tell me if we're using an inner product or norm." Fourth-year math undergradute: "All is vector. Images? Vectors. Functions? Vectors. Tensors? Vectors."

[@odysseus231]

Graduate math student: anything that's not infinite-dimensional is boring and useless.

[@rasmusturkka480]

@@odysseus231 Math post-doc: anything that's not impossible to prove is trivial

[@josh1ng]

@@odysseus231 "Be not afraid"

[@noemibedic3945]

1 year maths student and we are somewhere between your 3rd and 4th year 😅

[@sarthaknikhal5540]

Thank you for this video series. Very few people on the Internet can teach the way you just did. This video series is a piece of art. Just like applying math is.

[@DekuStickGamer]

I wouldn't have believed you if you told me 3 months ago, I would be watching math videos for fun. I love this channel.

[@3blue1brown]

+DekuStickGamer This is what I love to hear, glad I could play a part in converting you!

[@zbzb-ic1sr]

3Blue1Brown Could you do a follow up on the series but with tensors? I haven't found any intuitive explanation online. Thanks!

[@Goku17yen]

go to andrew dotson, he has an ongoing playlist about them :D

[@ASLUHLUHCE]

SAME

[@oriabnu1]

dear sir , i phd student but your lectures clear my views ,please upload vedioes on SVD and predictive sparse decomposition i love your way of teaching

[@nathanisbored]

the derivative thing blew my mind

[@shell_jump]

Try to think of an eigenvector. You will be amused.

[@debasishraychawdhuri]

e^x is an eigenfunction of the derivative matrix with 1 as the eigenvalue. just thought about it. mind blown :D

[@abhishekcherath2323]

all n^x, with ln(n) as eigenvalues.

[@TheBreadCatt]

Wouldn't the inverse matrix of it then be the matrix representing integration?

[@Zartymil]

I don't think so. because integration doesn't give you one function. it gives you a infinity of functions: a entire family of functions that once derived they go back to the original integrand.

[@vagueanxiety42]

3b1b: "Axioms are an interface" me: "If it looks like a duck, swims like a duck, and quacks like a duck, then it probably is a duck."

[@cheshire1]

It _definitely_ is a duck. Because ducks are defined as things that look, swim and quack like this :)

[@justamanofculture12]

"🦆: Peace was never an option. "

[@emlun]

"If it looks like a duck, swims like a duck, and quacks like a duck, then my theorem about duck-looking, duck-swimming and duck-quacking things will apply to it too."

[@jasonmaher5202]

same energy as featherless + bipedal = this is a man

[@GBY13]

9:07 For those who are wondering why operator p ∝ d/dx in Quantum Mechanics (QM) is also called "Matrix". The wave function (or state vector) is basically a linear combination of basis functions (or eigenvectors). This video solves the puzzle in the textbook of QM perfectly.

[@NovaWarrior77]

Nice

[@grates3652]

Holy shit i get it

[@chizhang2765]

Is it just me or does anybody else here get all teary-eyed watching the final flashback? Boy, what a journey!

[@josephkao2718]

Yes, especially after (got stuck and) watched Cahper 9- 11 a few times

[@bobjones5869]

张弛 what final flashback?

[@shamimasgari1095]

honestly my eyes are about to burst tear rn. why did it finish :(

[@user-dl2fb1ks7y]

岛主宁也看这个呀

[@anantchopra1663]

I felt so emotional at the end as if a really great movie/documentary just ended and changed my life!

[@firephoenix5556]

"What's your favorite vector space?" "Umm... I like the pi creatures."

[@kumaahito3927]

Just be sure to define them so that they correspond to the checklist

[@sudoLife]

Thank you so much! I cannot imagine how much effort was put into making this series, but I can attest to just how game-changing it's been for me.

[@cobaltcloud64]

I've come to the point where I actually need to understand these things (I just started uni) and I can't believe how lucky I've been for finding this channel five years ago. The best way I can express myself now is... you've definitely accomplished your goal of animating math for me (and surely for lots of other people). As you said, this really feels like reading a good novel: seeing how these squiggly lines and pointy arrows are actually the same thing makes for a perfect plot twist. Thank you :>

[@112BALAGE112]

The word "interface" brought me total enlightenment. A vector can be thought of just like an interface in computer programming. The "IVector" interface definition contains all methods (axioms) that the classes (vector spaces) must implement. When dealing with instances of these classes (actual vectors), we can be sure that the eight rules apply to them and we don't need to know anything else.

[@StephenMeansMe]

Yeah, I think the Bourbaki group did some damage to the concept of "axiom" when they insisted that all mathematics *start* from axioms rather than decide on axioms after working through various examples.

[@Holobrine]

I take it you mean interfaces in Java. Other languages don't have them.

[@ElchiKing]

+Stephen Peterson But after you decided on axioms, you start with them, test and evaluate

[@BigPapaMitchell]

Not true. An interface is really just a type of class, any language with objects and multiple inheritance can technically use interfaces. In Java, there's some stuff handled so you don't make mistakes you would if you were using classes and multiple inheritance

[@dekippiesip]

These are not really fundamental axioms of math. I would rather regard them as properties some system of objects(call them vectors) must have in order to apply the general results in linneair algebra. Just as group theory provides a framework for all classes of objects that have certain properties and relations towards one another.

[@Adityarm.08]

Now this quote makes a lot more sense to me. "Mathematics requires a small dose, not of genius, but of an imaginative freedom which, in a larger dose, would be insanity. And if mathematicians tend to burn out early in their careers, it is probably because life has forced them to acquire too much common sense, thereby rendering them too sane to work. But by then they are sane enough to teach, so a use can still be found for them." - Angus K. Rodgers

[@hektor6766]

It also requires obsessive rigor to attain that freedom. More angst for Angus.

[@tomxiao]

This teaching videos are gorgeous. Learnt linear algebra 12 yrs ago, but these videos blow my mind. Awesome stuff.

[@user-pr9yx2lx1f]

I think it is this video what elevate this series from an outstanding introduction to a definitive masterpiece. The whole series has try to be friendly and intuitive with best effort meanwhile trying to be 'accurate' as well. But still, what we have focused on earlier are just special cases. It it so because special cases, namely, examples on the ground are easier to understand and that's why we use 'e.g.' everywhere. And this last chapter wraps up by giving bigger picture underlying making our knowledge 'complete'. It is Complete, not in sense of knowing every detail, but rather that we have a structural understanding of the whole stuff which describe their relations. We know where to dive in further when we need to and where to put the incoming information at appropriately to assimilate these new stuff to grow systematically.

[@xxxhomiexxx5]

I was going to drop my Linear Algebra course, but I decided against doing so after stumbling upon this series. I definitely feel prepared for my exam tomorrow and my final in May. Thank you so much!

[@3blue1brown]

Good for you! Best of luck :)

[@yerr234]

so, how'd you do?

[@xxxhomiexxx5]

@@yerr234 passed the class! Graduated from uni a year ago and now a software engineer. It worked out for me!

[@yerr234]

@@xxxhomiexxx5 glad to hear my friend

[@xxxhomiexxx5]

@@yerr234 Thanks for checking in!

[@Math_oma]

Furthermore, why does the exponential function appear everywhere in math? One reason is that it (and all scalar multiples) is an eigenfunction of the differential operator. Same deal for sines and cosines with the second-derivative operator (eigenvalue=-1).

[@3blue1brown]

+mdphdguy1 Boy, that question is worthy of a full series in its own right.

[@Math_oma]

+3Blue1Brown Indeed. That's a good chunk of differential equations right there. Once you've trained the eye, one can see that oftentimes we're just solving eigen-problems. For example, what's the motion of a spring with a mass attached? It's going to be an eigenfunction (with some initial conditions, of course) of the second-derivative operator (from Newton's 2nd law and Hooke's Law) with eigenvalue -k/m, where k is the spring constant and m is the mass. There's your sines and cosines.

[@DekuStickGamer]

+3Blue1Brown Yes please, I have no disagreement to that idea.

[@UMosNyu]

I don't think anyone would be dissatisfied with 3Blue1Brown producing anykind of video.

[@martinkuffer5643]

There is a theorem that says that the solution tu the differential equation X(n)=AX (where X(n) is the order n derivative of X, X is a vector of m functions and A is a matrix whith m rows and m columns) is a vector space of dimention n*m and its basis functions are X=p(t)*e**(kt) where p(t) are polinomials, k is a complex number and e**(a) is exp(a). Considering that the sine and cosine are sums of complex exponential functions, if A is a real matrix you get that the solution of the equation is a sum of p(t)*f(t) where p(t) are polinomials and f(t) is exp(kx), cos(kx) or sin(kx), whith k some real number. So, every time you take a linear expression for something that will end up in a differential equation (which happens a lot in physics), you will get something of this sort PS: sory for my english, it's not my mother language

[@jcolinmizia9161]

Would love to see an extension of this series expanding on Tensors. Still something I don't really understand.

[@benshapiro8506]

read Spivak's books and then you will understand

[@nurpechbeimspielen3139]

A tensor is something that transforms like a tensor!

[@korwi7373]

the thing about tensors watch it

[@highgroundproductions8590]

tensors are extreeeeeeemely difficult and confusing, took me 4 EVER to at least somewhat wrap my head around it. The shortest way to describe it I can think of is a higher-order higher-dimensional object "built" from vectors.

[@MsThekiller02]

"Everything is a matrix in its own way" -my linear algebra teacher at cs

[@sujandutta8981]

I'm a first year Computer Science student. I was doing the annoying calculations related to matrices, determinants , eigen vectors etc. for last few weeks and suddenly I found these great series of videos, now truly speaking I have fall in love with this chapter. THANK YOU for making these.

[@sadhlife]

you should be reaching third year now. hope you still like math (:

[@sakshidwivedi4754]

Are you studying in India?

[@ASLUHLUHCE]

I'm so grateful to have watched these before ever sitting in a linear algebra class

[@bjdthethreecoolmathfolk2528]

Me to I’m not in high school math and because of this pi creature I’m smarter then my family with mathematics

[@maxcarvalho3202]

Sat in a linear algebra class yet?

[@joshuamahadevan9550]

I watched the series 2 years ago when i was at high school. now that am in college and its been 2 years, i totaly was struggling and had to re watch it. LMAO😂

[@insomnia20422]

this series makes more sense if you already learned some linear algebra the hard way and try to make sense of it

[@sir4982]

Only problem is, that this is just the tip of the iceberg in my Linear Algebra class 1 and 2... But for the things the video talks about, it's mind blowing

[@perseusgeorgiadis7821]

Officially covered more in a day than Math HL AA could ever hope to cover in a month. The lessons you uploaded in this series helped me immensely with understanding this considerably hard concept so easily. I can now confidently say that I have developed a solid intuition around linear algebra

[@Devinchy02]

Well, 3B1B, I don't know if you'll see this comment in some time of your busy life, but I just wanted to say thank you for this series. I have recently discovered your channel because of my needs to better understand Calculus, Algebra, and other college subjects, and I ended up finding the greatest math youtuber ever. I have just ended finishing and studying this series of Algebra videos for my Linear Algebra journey, and now not only you have clarified my mind with all these sorts of topics (I was very frustrated with Algebra because of my lack of abstract thinking) and now is like being flashed by God's light. And most importantly, you have revived my interest in mathematics and my joy for science. I was thinking to drop out of university because of the education system that makes you memorize without understanding, but thanks to these videos now I see how beautiful maths can be, and how all these numbers and symbols describe the Universe in such a precise way is kind of terrifying... but astounding. Thank you for everything. Keep up the fantastic work.

[@Lotschi]

That‘s a nice comment? How are you doing now?

[@SirFloIII]

My LinAlg Prof once said in a lecture: "What is a vector? An element of an Vector Space."

[@niraalii6707]

Best gift i received from an almost stranger in my first 17 years of life. Love your work sir.

[@braigetori]

had i had this resource when i was in undergrad, the entire trajectory of my life would have been much different

[@tgg3249]

never too late.

[@Alexyxg]

This is, "not one of", the best ever Linear Algebra tutorials I have seen. Appreciate, sir!

[@MozartJunior22]

It's amazing how this makes everything add up. You can easily see how e^x is an eigenvector of the derivative transformation. While I was considering how you would go about calculating the determinant of an infinite matrix, I realized that it is just 0 because the first column is 0. Which makes sense since the derivative reduces a "dimension" from the polynomial.

[@riccardoorlando2262]

Which alsoexplains why it is not invertible, also known as the reason why integrating is such a mess.

[@zairaner1489]

If you ever wondered why there is the +c after integration, that is the reason

[@gaspardsagot628]

You can start in the vector space of polynomials of limited degree, noted Rn[X] where X is the formal Indeterminate

[@mjtsquared]

Pretty much every exponenrial function e^kx is an eigenfunction of the derivative operator. Although I’m not sure for sinusoidal functions (the second derivative of sin(x) is also an eigenfunction but not for the first derivative). So it seems like exponentials and sinusoids are closely related to one another, in fact the ultimate relationship between these two classes of functions is already given by Euler’s formula. It feels so good to see a connection between two things in math!

[@KabeloMoiloa]

I want to point out that there's actually a lot of depth to defining the invariants, i.e. geometric properties like the trace, determinant, set of eigenvalues, etc. of a linear transformation when you have infinitely many dimensions. For instance, with the determinant you have to multiply an infinite collection of numbers and you have to ask questions like: "When will this infinite product converge?"In finite dimensions you have a discrete set of eigenvalues, but for infinite dimensional transformations you can have a full continuum of eigenvalues as well. This deep interplay between linear algebra and real analysis is the subject of functional analysis.

[@joulesjams20]

I cried a little when it finished. Please can you do another series or just more videos your channel is absolutely amazing. Keep up the great work

[@brandonkayes5530]

me too man

[@scienceofeverything5996]

You're an inspiration. A true teacher. I've a million things I wish I could've discussed with you.

[@mastershooter64]

3b1b: what are vectors? big brain me: rank 1 tensors

[@biblebot3947]

Only if their components are contravariant

[@sofia.eris.bauhaus]

YOU JUST TURNED CALCULUS INTO LINEAR ALGEBRA. WHAT. :D great vector of videos. thanks so much for this. :3

[@ryanpiotrowski6801]

better binge watch than netflix

[@shivamsinghaswal303]

The most beautiful things my eyes have seen is matrix multiplication performing the function of a derivative. Great work sir. Love

[@brandonvalentin8083]

All my life I always thought I hated math. I took algebra twice in high school like "how tf is this applicable to anything?". Now at 26 Im watching MIT lectures on yt trying to teach myself derivatives n finance instruments like "slow down wtf is an eigenfunction?". Just finished this series to wrap my head around the basics and I wish I had to time machine to make my 16 year old self watch these videos. I thought I hated math but I never saw the beauty in it until I watched you bring it to life over these 15 vids. I just binged this math course like a netflix series. I don't normally drop comments ever but hats off to you bruh you just changed the way I see the world. May God Bless you

[@tennenrishin]

As an undergrad, it took a lot of effort and time for me to gain these intuitions. Now you're making it all so accessible! I'm a little angry and jealous of those who can learn it all so easily from your videos. Your videos are so good, that I still watched every single video in this series just to make sure I didn't miss some useful intuition, and indeed the cross-product video (2nd-part) revealed and closed a hole in my understanding that I had given up on as an undergrad. I wish you would do a similar series on some advanced topics, such as, for example: Projective Geometry Differential Forms Calculus of Variations Stochastic Calculus I know these topics would cater to a smaller audience, but you seem to have the perfect skills, tools and interests to cover at least the first two. Or perhaps you have already. I'd better go look at what else you've done. You're doing humanity an enormous service. Keep up the good work.

[@littlenarwhal3914]

I get your feeling of anger and jealousy, it's always like that when you put effort into things and then others just get up to your level pretty quickly because they learn it through a better platform...

[@E1phel]

I think that these videos are a good way to have the initial intuition of "what are we doing with these vector spaces". But, they don't give to the watcher all the expertness you acquired by experiencing situation like "in what concrete example of mathematics these concepts help me to have a complex result with a simple method".

[@adeelali8417]

@@E1phel Maths exams for me are generally computational, but watching these videos has made the concepts come to life if you will.

[@NomadUrpagi]

Damn man ive also felt so jealous but couldnt express it or admit it aloud. My knowledge has always been my precious jewel that i took pride in and was protective of when it got to understanding challenging stuff, as if saying, "fuck off ive spent years to understand it: im not gonna tell you the secret of it so you can grasp it in 10 minutes without putting in the work". Thank you for this honesty it made me feel more likely to share it.

[@blubblubber9460]

now I'm jealous that you came up with these intuitions all by yourself

[@s3cr3tpassword]

I am a physics Ph.D. student, and this series has helped a lot! I have been getting as many of my friends to watch this since it should help them too. Thanks for making these videos. What's next?

[@ICYSHADOW12]

Absolutely amazing! As a software engineer currently studying computer graphics programming in my free time, this video series has been immensely helpful. This episode was especially impactful for me. The concept of the "Axioms" of linear algebra being an "interface" for how to use all this theory was unexpectedly familiar! In C#, for example, there are "Interface" classes that define what functionality a given class that implements said interface must have. It was striking to see this same concept from a mathematical perspective!

[@PeterBernardin]

I feel like when I listen to your videos I sigh with the relief of understanding something in a way I never did before! Thank you!

[@NoobLord98]

Thank you so much for this entire series, this has seriously boosted my math grades, which went from a 5/10 for calculus to a 9.3/10 for linear algebra. Without this series that would've been impossible.

[@looksintolasers]

What I thought when you said "interface" public class Function implements AbstractVectorSpace {... }

[@birdboat5647]

this was honestly the lightbulb moment for me (with a programming background), all of the vector spaces must satisfy this contract...

[@timberlin4075]

@@birdboat5647 when he started talking about abstractness of vectors it also reminded me of abstract classes and how you dont have to worry about how the class is implemented when coding it

[@alphalekter4474]

Remember lambda functions? There's a whole branch of math that studied them, before programming was even a thing. I'd bet there's a connection between eigenvalues and those functions.

[@duartecastro623]

This is beautiful. Just binge watched the whole thing, and im out of words...this series is pure art. Thank you for sharing it with us

[@georhodiumgeo9827]

“If you watched and understood this video you have a solid foundation for the underlying concepts.” So if I watched it twice but still don’t understand then I should still be ok right. Maybe I’ll just watch it a bunch of times to be safe.

[@cadu7698]

He is an excellent teacher, but some concepts take quite some views to be truly apprehended .

[@johanottoson1373]

@@radwanalaghawani7053 Did you write this while drunk my dude?

[@HappyAndroid42]

"pause & ponder for a moment" That's what I've been doing! Thank you!

[@madacol]

I've seen this series 3 or 4 times through out the last 3 years. Everytime grasping something new. Terrific work!!!

[@zephir]

Haha same, watch probably my 3rd time watching the series after 2 years, I feel like I finally understand it all !

[@prasunkumar2106]

Felt like an ending to a thriller movie, with all dots connecting one by one! Thanks Grant:)

[@kleinesfilmroellchen]

May I just say that I sincerely appreciate the leading quotes in the series? Wise words, it does feel like no one _wants_ you to learn math properly.

[@BearsThatCare]

I'm about to move to university and want to go into mathematics as an academic. This series not only changed my perspective on algebra, what I've been learning for most of my life, but on mathematics in it's entirety. If you ever read this thank you.

[@ottografstrom9511]

Me and almost everyone in my class at Lunds University in Sweden are infinitely grateful to you for making this series. Your series make linear algebra not only much easier to understand, but also much more enjoyable. So thank you, Grant.

[@vijaysinghchauhan7079]

As a Math major student, I cherish the fact that there are people like you in the world that gives life to math and help those demotivated and motivated math majors to appreciate it more and get comfortable with it. Thank you!

[@ultramadscientist]

I rewatch this series every couple months just to refresh my intuition on the subject this whole thing was so fantastic. I hope you realize how influential you've been to mathematics education and that there's no way that you aren't helping absolutely brilliant mathematicians, physicists, and engineers overcome very difficult hurdles in their math education. I know it sounds like I'm exaggerating but the work you're doing is going to have multiplicative effects through the future and I really appreciate all you've done Grant

[@user-kh5tv9rb6y]

A whole semester of tedious classes condensed to a couple hours of fantastic videos and explanations. Thank you so much for this series!

[@sara-hc7wb]

I mean I agree that this is a really good video series that gives good intuitions about things but ... you cannot seriously argue that one could watch this series and then be able to nail a real test in linear algebra. it's too high level. and it does nothing to replace the tedious hours of just grinding problems until you get a feel for them. or even give the basic tools needed to actually calculate some of the things. videos like these are a support to help give understanding, but they can never replace hard work and doing problems. give your classes more credit.

[@ely_mine]

Totaly agree, time to get back to Gauss Jordan elimination !

[@raindropssonroses]

what you said is so true. I don't get comments of people when they say that a video made them understand EVERYTHING and now they're getting high scores just because of them. I mean, sure these videos are immensely fascinating but at the end of the day, it depends how many practice problems one has gone through in order to "ace" a test.

[@nickanselmo6353]

I completely disagree unless the only type of test you could be taking is one that forces you to simply regurgitate what you have been told. After taking a formal linear algebra class (in which I received an A on every test only because I did many practice problems) I could not apply the concepts to anything other than a problem I had seen in the textbook or on a homework assignment. To me, this means I really learned next to nothing other than how to plug some numbers into a formula that I had memorized. Throughout the following years I have stumbled across many other good explanations of what these operations really mean that have completely changed my understanding of linear algebra. This video series has managed to collect all of those explanations and then some in one place, and I can guarantee that these videos in parallel with a linear algebra class would greatly increase ones understanding of the subject, and definitely could result in someone going from poor grades to Acing the tests. Or maybe I just had a really crappy teacher when I took the class and a good professor would have explained these concepts better in the first place. All I know is that it would have been immensely helpful when I was taking the class and I have heard similar stories from many others who have taken a number of different linear algebra classes so it would seem the issue is not limited to just one professor at one university.

[@DieselBoulder]

@@sara-hc7wb yes and no. I'd say both are needed. With a deeper understanding (this channel) you can grasp the concepts and extrapolate much MUCH more easily than by just grinding through problem sets applying formulas you don't understand. If you know how the machine works, using it becomes easier. Not to say that you could ever get by without doing the calculations and practice problems but the goal of education isn't to be able to pass the tests, it's to then be able to apply the knowledge to a broader set of problems. The understanding that this channel adds, opens up a much wider field of application by adding the understanding to what it is you're doing. AND if you're stuck on an understanding of what you're doing or why you're doing it, but you grind through practice problems, you're not really learning, you're just putting in effort for little to no benefit. By understanding what I'm doing I"ve been able to bypass hours of practice problems by being able to visualize the problem. You don't need to memorize anything if you understand what you're doing.

[@deepakmecheri4668]

I can't tell you how great this series was. You're doing a great service.

[@seriousmax]

Hi.

[@user-mv4oh8yp1y]

Thank you for making this awesome series! Linear Algebra is a subject that made me felt pain before, but now all my confusions got clarified with intuitive "graphical proof" memorized in my brain. btw, I really like the quotes you given in the starting of each episode too!

[@zack_120]

10:25 - COOOOOOL ! The graphic presentation of everything on this channel is genius! It makes learning math visual and hence much easier to understand. Thank you soooo much for spending the time to benefit everyone of interest.

[@satanas1729]

Thank you very much for this series. It brought me a lot of intuitive understanding of a subject that appears a lot in my field (I'm graduating to become a telecommunications engineer) and a lot of things have "cleared up" in my head. I realize that it must have taken a lot of work, both in the animations and in trying to explain things in the most intuitive way possible, but know that the work you've put on did not go unnoticed. Thank you a lot for these videos, and know that, to me, your channel is the best maths channel in youtube

[@3blue1brown]

+Bruno Chaves Thanks so much, this means a lot to me.

[@ollpu]

It..it's...over? :(

[@zairaner1489]

No series rules forever my son

[@Gnenguin]

apart from infinite series

[@ollpu]

Sapphire Charm right

[@neerajtiwari5365]

ollpu don't make me sad 😯

[@mr.champion7304]

Don't forget mythbusters

[@jinyunghong]

I really appreciate your efforts in creating this whole series of linear algebra. You definitely help me broaden my perspective of mathematics. Thank you so much!

[@jinks908]

Grant, the only available words I can use to describe the level of unimaginable appreciation and gratitude as well as the level of insight that I have gained from your videos is, "There are no words to describe it." Your videos are invaluable and I'm quite certain I can speak for literally thousands of people when I say that you, sir, have changed our understanding of mathematics and the reality within which it exists. Thank you.

[@mattstokes3881]

Loved the series! Can you do topology next?

[@bjb2738]

+

[@bengski68]

-

[@taraspokalchuk7256]

+

[@SotirisSimos]

+

[@MuhammadSaad]

+

[@davidwright8432]

Beautifully clear and satisfying explanations! If all math were this well taught from grade school on, we'd be a nation of math-lovers, rather than haters. And who knows? - maybe people would write theorems rather than novels, and movies would involve characters who were functions. Or vectors. Or whatever!

[@user-vw2tk2ru2r]

Never commented on a video before but my god this was indescribable. I am in a proof-based linear algebra course in college right now and I did not understand why we defined vectors like this, or how functions (and now I know many things) could be compared to vectors. This was amazing, spectacular, beautiful, gorgeous, stunning and absolutely insightful. You sir, are a legend. I am now able to understand the beauty of linear algebra to a far greater depth. Thank you.

[@neutralzone7775]

Wow, this is so beautiful. I‘m really starting to fall in love with math. Thanks 3b1b

[@TanishqIsHere]

*Can we have two minutes of silence please.*

[@MisterYagibe]

This serie of videos is just amazing. Even with some basis in linear algebra, you give some amazing intuitions, and the video are clear even for a french speaker. By the way, if you want your videos to be subtitle in french or other languages for instance, I'm pretty sure some viewers (me for instance) will be happy to help you. And more importantly thank you and best of luck also ^^

[@3blue1brown]

+MisterYagibe If you want to add subtitles, I'd love that.

[@MrGustavier]

your videos are genius ! i'd like to do a similar series for french youtube community, can you tell me which software you're using for animation and editing ?

[@karolakkolo123]

He programs the animations in Python, he doesn't use any animation program

[@MrGustavier]

@Play It On Linux je travaille dessus en ce mmoment meme

[@charlesnyiha4641]

The idea of the derivative of a function as a linear transformation has completely blown my mind.

[@169harumaru7]

このような素晴らしい教材を提供してくださり、本当にありがとうございます。

[@hcblue]

This has been an excellent series. If you plan on doing more series in the future and assuming you've been doing these in your own time, I'd love to help support you on Patreon or a similar platform.

[@sebastiangarcia3270]

Thank you. Thank you thank you thank you! I'm struck with disbelief at how informative and intuitive and clearly explained the entire series was. I just...man. I don't even know. You're amazing. I can't say thank you enough!!

[@nicoguzmanp]

I ABSOLUTELY loved this series! Thank you so much for sharing the knowledge with the world in such an understandable and deep way at the same time. Your voice is great, the manner in which you explain and say everything is fascinating and the animations make the subject much more clear and easy to understand. I found the videos perfect for anyone who wants to get the fundamentals of the subject but also for those who want to dig in more deeply in details, I spent a week watching these videos with my pencil and notebook and I´m sure no college course would have been able to provide me such understanding. I already took Linear Algebra in college last year but I could wrap my head around it and this was JUST what I needed, I am already starting to watch the differential equations "tour guide" and looking forward to studying the whole Calculus series as well!!

[@jeremiahwilson6651]

I watched this series a year ago right before I took a linear algebra course at university and I still come back to watch it every so often. I find that every time I rewatch it I understand a concept better and take away something new that I hadn't thought of before.

[@RiotGearEpsilon]

I loved this whole series. Thank you for producing it. I'll tell all my friends about it.

[@BurakBagdatli]

Thank you for the hard work you've put in these videos.

[@freshdawg3070]

I just want to say thank you, there was a time when i thought i could never pass this exam about Linear Algebra, your videos gave me the understanding and i began to see the inner logic in this topic. Yesterday at 12 pm i received the message from my prof that i passed and its really incredible for me, when i think where i started. Ps.: Even i am not especially good in the english language, the visuals are absolutely brilliant.

[@magnusmuller5223]

I watched your series today to get some sort of recap. But this series has " transformed " my thinking. Thanks for all the Ahhhhh - moments!

[@AoSCow]

I'm in tears, thank you so much. This episode, and the series of course, was beautiful.

[@gregorymacchio4077]

I have been in love with math since I was a kid and this channel makes my free time worth it.

[@paulstahlhofen5017]

I’ve learned linear algebra several years ago and have applied it for various problems since then, but when I watched this playlist I really got a much better understanding for some of the key concepts. Providing a good intuition is, in my eyes, really the best thing teaching can do. Thanks a lot for these beautiful videos!

[@KaranTejwani-ft1bl]

No words to actually appreciate your content. I have understood the concepts so clearly man just because of you. It was a very exciting journey with this playlist enjoyed each and every concept, visual, animation and quote at the very start. Would really love to start another playlist of yours. Finally thanks again and again for such an amazing understand and way of teaching and that so for free.

[@NScott45]

chemical reactions form a vector space, where the stoichiometric coefficients are the vectors' coordinates. have fun

[@NomadUrpagi]

Damn i was just thinking " lets apply it to chemistry, how exactly it would look like". In that context, what would the eigenvectors mean in the physical world? Scaling the amount of one of the reactants will always bring scaling results/chemical process rates i guess?

[@Magnasium038]

The Wei & Prater method applies these concepts of vector spaces and eigenvectors to chemical reactions to plan the minimal number of chemical experiments you need to run in order to determine all the rate constants for a set of reactions. If you use species concentrations as the coordinates in your vector space, then eigenvectors are starting compositions which will move in a straight line in vector space towards equilibrium.

[@bhatman88]

omg www.cs.helsinki.fi/bioinformatiikka/mbi/courses/06-07/memo_07/Lecture5.pdf

[@EayuProuxm]

"Take a moment right now, to imagine yourself as a mathematician developing the theory of linear algebra" So imagine yourself as Herman Grassman?

[@OnamKingtheKing]

dai graass

[@Subcrafting]

Well, as many people, from what I can see in the comment section, I am deeply grateful for your work. Despite knowing your channel for a while, I only found this serie a couple of weeks ago (I've just finished it now), but it couldn't have been a better time though. I am a first year math student in Belgium, and I happen to attend a linear algebra class this semester. I don't have the words to describe how fascinating it was for me to travel between both my algebra cours and your video at the same time. It allow me to discover this subject with a unique perspective, and I'll never be thanksfull enough for the work you did and share. Having the exam in a couple of day, I feel as if I was able to grasp some deep understanding which would have been way harder to get without your help... Just, thank you !

[@Jack22866]

This series has been a continuous build up for this video and it's just beautiful, the way it opens your mind feal just like magic. Ty for you work 3b1b

[@shivamkaushik6637]

Whenever I watch your video, these words from Ratatouille come to my mind Not everyone can become a great artist, but a great artist can come from anywhere.

[@112BALAGE112]

Best video in this awesome series.

[@Victual88]

Thank you so much Grant, I know myself and millions of others are grateful for your work!

[@christosgeorgiou1656]

Thank you for all the work behind these videos. Your contribution to my understanding is priceless.