Hey, everyone! So I’m pretty excited about the next sequence of videos that I’m doing. It’ll be about It’ll be about linear algebra, which—as a lot of you know—is one of those subjects that’s required knowledge for just about any technical discipline, but it’s also—I’ve noticed—generally poorly understood by students taking it for the first time. A student might go through a class and learn how to compute lots of things, like matrix multiplication, or the determinant, or cross products—which use the determinant—or eigenvalues, but they might come out without really understanding why matrix multiplication is defined the way that it is, why the cross product has anything to do with the determinant, or what an eigenvalue really represents. Often times, students end up well-practiced in the numerical operations of matrices, but are only vaguely aware of the geometric intuitions underlying it all. But there’s a fundamental difference between understanding linear algebra on a numerical level and understanding it on a geometric level. Each has its place, but—roughly speaking—the geometric understanding is what lets you judge what tools to use to solve specific problems, feel why they work, and know how to interpret the results, and the numeric understanding is what lets you actually carry through the application of those tools. Now, if you learn linear algebra without getting a solid foundation in that geometric understanding, the problems can go unnoticed for a while, until you’ve gone deeper into whatever field you happen to pursue, whether that’s computer science, engineering, statistics, economics, or even math itself. Once you’re in a class, or a job for that matter, that assumes fluency with linear algebra, the way that your professors or your co-workers apply that field could seem like utter magic. They’ll very quickly know what the right tool to use is, and what the answer roughly looks like, in a way that would seem like computational wizardry if you assumed that they’re actually crunching all the numbers in their head. As an analogy, imagine that when you first learned about the sine function in trigonometry, you were shown this infinite polynomial. This, by the way, is how your calculator evaluates the sine function. For homework, you might be asked to practice computing approximations to the sine function, by plugging various numbers into the formula and cutting it off at a reasonable point. And, in fairness, let’s say you had a vague idea that this was supposed to be related to triangles, but exactly how had never really been clear, and was just not the focus of the course. Later on, if you took a physics course, where sines and cosines are thrown around left and right, and people are able to tell pretty immediately how to apply them, and roughly what the sine of a certain value is, it would be pretty intimidating, wouldn’t it? It would make it seem like the only people who are cut out for physics are those with computers for brains, and you would feel unduly slow or dumb for taking so long on each problem. It’s not that different with linear algebra, and luckily, just as with trigonometry, there are a handful of intuitions—visual intuitions—underlying much of the subject. And unlike the trig example, the connection between the computation and these visual intuitions is typically pretty straightforward. And when you digest these, and really understand the relationship between the geometry and the numbers, the details of the subject, as well as how it’s used in practice, start to feel a lot more reasonable. In fairness, most professors do make an effort to convey that geometric understanding; the sine example is a little extreme, but I do think that a lot of courses have students spending a disproportionate amount of time on the numerical side of things, especially given that in this day and age, we almost always get computers to handle that half, while in practice, humans worry about the conceptual half. So this brings me to the upcoming videos. The goal is to create a short, binge-watchable series animating those intuitions, from the basics of vectors, up through the core topics that make up the essence of linear algebra. I’ll put out one video per day for the next five days, then after that, put out a new chapter every one to two weeks. I think it should go without saying that you cannot learn a full subject with a short series of videos, and that’s just not the goal here, but what you can do, especially with this subject, is lay down all the right intuitions, so that the learning you do moving forward is as productive and fruitful as it can be. I also hope this can be a resource for educators whom are teaching courses that assume fluency with linear algebra, giving them a place to direct students whom need a quick brush-up. I’ll do what I can to keep things well-paced throughout, but it’s hard to simultaneously account for different people’s different backgrounds and levels of comfort, so I do encourage you to readily pause and ponder if you feel that it’s necessary. Actually, I’d give that same advice when watching any math video, even if it doesn’t feel too quick, since the thinking that you do in your own time is where all the learning really happens, don’t you think? So, with that as an introduction, I’ll see you in the next video. Captioned by Navjivan Pal

Reviewed by Johann Hemmer 07/08/16

More videos please on number theory and probability

Good starting to LA lol

I'm crying inside, I wish I would have had these videos during my first linear algebra class. My frustrations were always exactly what you described…my professor absolutely neglected the geometrics of linear algebra. I barely got anything out of that course. I feel validated in my frustrations now though. When all you do is crunch numbers, you lose sight of what it all really means!

I LOVED linear algebra. Majored 30+ years ago (pre computer era) and it was one of the most intuitive for me (much more than any advanced calculus). Number theory was my other favorite. Ended up eventually becoming an actuary for a few years then pursued medical school, so this was a nice reintroduction . Looking forward to the series.

I really appreciate your lectures about linear algebra. Thanks for your helpful lectures, I am really interest in linear algebra itself or related problems : )

Where have you been all my life!!

Acabo de encontrar oro puro.

Where can i find more info on extention of sine function and how to visualize it?

Ingenious. Harmonic.

Speaking of both mathematical and general counsel..

Wow man, well put. Hit home with the computer brain analogy for Physics. It's nice to see someone committed to addressing the fundamentals rather than rinse repeat computations.

That moment you realize you got an A in Linear Algebra and didn't learn Linear Algebra

Good feel for any concept is important to make a new concept.

Does any know about the piano music? Really liked it. Shazam is not able to pick it up too

This video series is better than most university courses

<3

Hi, massive thanks for your videos! Have you done any tutorial on Norms (Euclidean norm, max norm, etc)? I am struggling to understand them.

Does anyone know the piano piece 3b1b has in his videos? It's beautiful! 😀

The educational system is so utterly messed up. You wouldn't believe how much money (proportional to my income) I paid to go to college, and how much debt I have now because of it (proportional to my assets). This is so infuriating to me considering than because of financial reasons I had to quit it, so I didn't even get a degree out of it. With these series of videos that you make available FOR FREE, I have learn MORE and FASTER than the semester I spend on college learning the same topic.

Since high school I see that everyone who perform well academically, had done it through YouTube Videos. I don't wanna dismiss the academy. Universities are extremely valuable, but the lower half of them fall short of what you guys (educational YouTuber) do for the well of all.

I know this sounds kind of over the top, but I would consider this a public service and I am thankful for it. You all are playing an important roll, and I think that should be recognized. So, thanks you.

Do series on the trigonometry

Thanks a lot for motivating numerical learners ( or rather muggers) to grow up and understand the geometric intuitions…many many thanks….

You are great…..!!!!

You are straight up a freaking wizard. Keep doing what you're doing. You're doing such a great service to the world, and more people need to know about this incredible playlist resource @3Blue1Brown

This playlist gave me a 86/100 on a linear algebra test.

Congrats on a million views! Well deserved on such an amazing playlist.

Hi, how do you do your animations? Please answer.

I really found some very useful stuff like this which makes me understand the depth and language of Maths and how to motivate myself for a subject which I can take up in future. Thanks.

Hi , please can you make a series of videos on tensors? it is really difficult to imagine them geometrically.

i am a fan of your videos now

Amazing set of videos! These videos can deliver the "right" linear algebra concepts into your head in a few minutes instead of struggling for days on end using regular methods. Educational innovation of high order.

These videos are fantastic …..It really teaches the inner concepts of linear algebra. I wish every education system should teach like this.

Thank you very very very …….much!

~225 Upvote to Downvote ratio; One of the best I've ever seen.

legendary

Typo at 0:30, det(A) is ad-bc not ac-bc. 3b1b you are my hero thankyou for making these videos

I'm so dumb. I don't just pause video, I even slow down the playback speed. But the only one fact encouraging me is that Albert Einstein was stupid too. )))

Hlo Sir,please upload video lecture on Geometric Interpretation of the Eigenvalue Decomposition for Symmetric Matrices.

thank you for the dark background it really helped me watch this video comfortably in bed just before I went to sleep

😀 We humans worry about the conceptual half while computers do the number crunching, I like it.

Could you please tell me , What a Application that you use for Visualization ?

education in my country is 100 years after this…

Grant, you just introduced me to a new world of Math which i've never seen. You're a great server of the community. To top up with this, can you suggest me book or video series that get's into the details(more technical) of the Linear Algebra. I'm trying to complete my pre-requisites for Machine Learning

I just want to thank you for the amazing work you have been done, you are amazing and please don't give up

Just started a robotics course recently. It's all about rigid bodies rotation and coordinates transformations.

These series of videos helped me a lot. It all seems so clear and intuitive now. Thank you for your great work

please make a series of videos on probability and statistics. I am eagerly waiting for it..!!!

Would love to see a course on analytic geometry , if you can do that the same way as these videos, it will be an awesome presentation.

3Blue1Brown should be classified as medical channel on Youtube, because it legitimately saves lives.

I wish there was a pdf of these amazing series. That would be awesome!

Hi

Beautifully made video, this really got me interested in delving into a subject that daunted me. Keep up the brilliant work!

Thank you

I am going to show these videos to my kids. They haven't been conceived yet.

i searched so many site.. they all show application of linear algebra.. but your title = content of the video truly the essence. thanks for the video

Hi guys, I wish to learn linear algebra. Where do I start? I have 0 knowledge about it.

I wish I had Youtube in my school days 🙁

hello, would you consider doing game related math at all? I have looked at various resources which focus on linear algebra, vectors and trig for game programming. but also hear one needs solid understanding of algebra to calc as well. but when learning such things communicative factor, fractions and alike, i can not find the use of those formulas and stuff for game programming. any help or advice would be great? Also can not figure out what most formulas and equations can be used for, why are they needed?

I'm Creating a revolutionary new; collaborative animated video platform, which has very similar experience to YouTube, however when one user is watching a video, and notices an incorrection, they can edit and correct the visual / audio / narrative (With the possibility to cite the source information for the change). Would allow high quality works to be put together, and hyper linking of visual material inside videos.

Would love to hear from any interested parties, to help me make this a reality. I would really love for the community to have a say in the development for are looking for core team members to join me in building this (non-technical and technical, everyone is welcome).

Samuel Holt, https://www.linkedin.com/in/samuel-holt/ Email: [email protected]

Summarised in more detail in a short paper here: https://docs.google.com/document/d/1ZTWNKzkeDlKYnm-E0sykmgzhNvfRNiJzjzg2WU988ME/edit?usp=sharing (Comments are very welcome, and encouraged)

Summarised in a complete white paper, here: https://docs.google.com/document/d/1Ar3FsTltuQ–msNVgMHvHWHfOdDTInMxX5fbAN5QZSY/edit?usp=sharing (Again comments are encouraged)

Would absolutely love to hear from anyone who is interested, can provide advice, or wants to join the project and create the future with me.

Look forward to hearing from you 🙂 Samuel Holt, https://www.linkedin.com/in/samuel-holt/ [email protected]

How did you know that infinite polynomial is how calculators evaluate sine? How/where can I learn more about how calculators work and their electronics?

Thank you for your video in linear algebra helping me to secure my A+ 😉

To understand linear algebra you need 16 subjunctive logic cases, not 4 that are taught in high school. I began to understand this when I was in office hours listening to Elyndra and Ettica Tesfay and I noticed the professor was essentially accusing them of over-analyzing the material. You really need Discrete Mathematics and Set Theory before Linear Algebra. Ettica eventually let the faculty have it in an attempted lecture. She was completely correct and the faculty talentless by comparison. They expelled her.

Charlie Price says I'm right on that girl being mentally retarded.

I have linear algebra this semester… So i cried when i found this video.. THANK YOU!

I'm currently taking linear algebra after watching your series over the summer. Thanks to you, I am able to understand the motivation and geometric interpretation behind what my professor talks about in class, allowing me to easily follow along and gain insight I would not otherwise have been able to get. Your channel is truly one of the greatest resources for learning math, and you help inspire me to always seek the underlying insights in math, and to never be satisfied with unclear explanations.

I want to learn this but have no mathematical background, where should I start?

Just wanna say thank you, for putting hope into eyes of the algebra students. This is exactly what the majority of teachers don't understand or do not have time for. It's the questions "why?" and "for what reason?".

So you fill this gap and for that I truly thank you! <3

You started this video by proposing some problems with the actual understanding of it all and I felt baffled by how on point it was. University forced us to sit through so much of it without giving us any application it's just painful when you come to the realization that all you can do is jump through a hoop because you were obliged to.

It's honestly very sad that learning isn't about learning anymore; it's about being a number that succeeded or failed.

2:29 is quite disturbing. Is there anything more to be aware of?

Sir you are a math genius. I'm a high school student and was longing for a series like that…so elegant.

School teachers are just there to tell students how to perform stuff rather than letting them know why the stuff they talk about makes sense…Once again… sir you are great!

Thank you for this

Why not translate eigen? Inherent value doesnt sound too bad

This series is the bestttttt on internet😍

Does anyone know what 3d modeling tool is being used for these tutorials?

can I get a track ID for the song that starts out in the first seconds of the video? you have quite a good taste 3blue

ông nào dịch hay thế :)) mà gần thi mới tìm thấy video này damnnnnnnnnnnn

You are a genius.

I only realised the purpose of some linear math concepts when I got into image processing – Gaussian blurs … etc. Back in high school – and in college, the teachers didn't seem to care about Math as a problem solving tool, they treated it as a puzzle that one has to solve to get to the next level

@3Blue1Brown I remember this series starting to come out the summer before I took my Linear Algebra class and it was absolutely invaluable. You are truly wonderful, and run my favorite YouTube channel. Never stop making this stuff so clear!

Linear algebra is absolutely one of my favorite fields of mathematics and my favorite university course yet. I've been learning multi-linear algebra for about a month or so now in preparation for differential geometry and I love it even more.

Great video! Completely illustrates the problem associated with students getting disinterested in Math courses at Undergraduate or Post graduate level. Especially the part about students getting baffled by the 'wizardry' conducted by their professors when they display their intuition through problem solving but don't articulate the 'conceptualization' that's taking place in their brains. Thank you so much for making these videos.

I absolutely hated math when I was in school but this seems very interesting to me 🙂

Yeah procastrinating is great. I now have a reason to first learn english math vocabulary and then watch this basics series and then learn for my exam which is in 5 days. God, im fucked

The problem with modern education is that it leaves the conceptualization of Math onto the students, there are some students who will be able to manage to understand it on their own, but majority of the students do not have that type of thinking or imagination. Learning math should be about teaching the "where" and "why" of an equation or concept, not "how".

Diste ferovci

Blah blah blah 5 minutes without even describing what the hell linear algebra is

this channel is best for all mathematics lovers

love from nepal❤️

great lecture

I am a student right now and let me make a commitment, as soon as I start making money, I will support, financially, this channel and people like you. keep the Patreon alive. Thanks for providing high-quality information for free.

For people who wanna play around with vectors and 2D matrix transformations as they watch the videos I made a little website that could be of help https://las.nullp0tr.com. It's heavily inspired by the videos:) thanks a lot @3Blue1Brown for all your work!

Thank you very much for your work. I really enjoy watching your videos.

I am currently a student of mechatronics engineering.

Greetings from Mexico.

the first min includes my feelings for all math classes..

My comment might get lost between all others, but i'm still gonna write it: I've took math classes my whole life, always trying to understand the real essence of math; despite all my efforts, every time i asked a professor the reason of a formula or the way a problem went, their answer were always "because thats the way formula says", its kinda sad because now im 21 and after all my years at school i feel like i haven't really learnt nothing but pretty hard formulas, but that changed since i started watching your series its really amazing to see how math behaves behind all that complicated formulas. I really like all your videos, you are great, man!

The part about applying trigonometry to physics describes my experience in physics last semester, and I feel attacked.

We should create an Academy Award for teaching and give it to Grant.

hey, I have a math problem for you

…

-3x^2+1467x-86940=0

hehehe…

(has 2 answers, both equally funny)

What is generally more difficult: Calculus 3 or Linear Algebra?

the instructor is just too good 🙂

if I pass, you are the only reason I won't off myself

Good day sir. The is one of the best video collection I have seen so far on YouTube. I am a learner of Robotics and Automation. Is this something I can follow up, or do you have any suggestions for me. I know robotics involves transformation of frames. But is there any other thing that I am missing out?

Well there goes my next few hours

is it useful for Data science, as i am not getting any lucid video too understand the linear algebra, if YES, Kindly reply and if NO, then kindly suggest the good Video lectures on youtube as soon as poosible.

Thanks

I just finished college level calculus and I'm just now learning there is a definition for sine that I can computer by hand and get a rough estimate. I don't know how bad that is but it feels bad.

I count myself as one of the luckiest people to have stumbled on this video just before taking linear algebra in college