Essence of linear algebra preview

[@serock3]

I just got my grade in linear algebra, highest score. Couldn't have done it without these videos, they didn't just make the subject much easier, but also more intuitive and fun.

[@3blue1brown]

+serock3 Congrats!

[@serock3]

Thanks :)

[@avijeetjha8774]

3Blue1Brown would you plz tell what 3rd 4th or higher order derivatives mean mathematically like the 2nd derivative shows rate of change of slope then what does higher derivatives mean and specially what does nth derivative mean.

[@gopallaheri6209]

Serock .

[@sir_aken9706]

avijeet jha you should probably watch his Calculus series. The Nth derivative is slope at every single point on the n-1 derivative of a function.

[@VincentRubinetti]

This video illustrates the big problem with how we teach math in the United States. Teachers and professors leap right into equations and numbers, completely skipping any conceptualization of them such as illustrations or high level explanations. For example, I remember back to my algebra class in high school when we first learned about functions. The teacher started to explain how to graph functions, and how to write them using f(x) notation. We learned more advanced concepts such as factoring, completing the square, etc. Not once did the teacher say anything to the effect of "a function is a thing you put a number into and get a result out". They never explained how a graph is a visualization of how a specific function's result changes based on the input. They only described the mechanics of it, and "how to do it". It's no wonder when many kids get to calculus that they have no idea what's going on, because they still don't actually understand in their head what a function or graph really is. The unfortunate consequence is that so many people now think they're just bad at math, when really it was just never properly explained to them on a conceptual level, and the ones who did well in high school math were simply the ones who could make the conceptual leaps on their own.

[@dianac4480]

Vincent Rubinetti THIS.

[@ledlowcostanza5920]

It seems all the problems is the same of the math teaching in the world.what a pity.

[@learner_3673]

I can exactly relate to what you just said

[@chuckbecker4983]

Feel free to blame the United States if that's what it takes to get you through the day, but I can tell you that it's much worse in some other parts of the world. Particularly in much of Asia, the simple rule is "ruin your life as much as you need to, to get the grades you need to get into Berkeley" ... or (so unfortunately) die trying. Considering the level of achievement here, the United States in general is pretty humane in the way we teach and learn math and physics. The actual "big problem" in the United States is millions upon millions of entitled students who feel that the professor owes them a personalized explanation of everything, with no effort on the student's part expected or offered.

[@AverageJoe8686]

Chuck Becker: "...owes them a personalized explanation of everything, with no effort..." VERY WRONG! We ask the text books not have errors in the answers and proofs, so we may be efficient and excel. We ask the professors understand the subject they teach, instead of reading verbatim from the book full with errors. Your boss WILL expect this error-less standard from you, thus you MUST expect this standard from your teacher. No exceptions. This is not unreasonable.

[@jlpicard7]

I had an awful linear algebra instructor when I was working through my BS in engineering so this subject has been somewhat of an Achilles heal for me for many years. I could always do the math, but the geometric interpretations you provide here has really helped me develop the intuition that I have been striving for. Thanks for putting together this excellent series!

[@calaphos]

What a coincidence! Looking for videos to procrastinate from studying for my linear algebra exam I found this. Looks like I can combine these things now

[@KoruGo]

I'm literally doing the same thing lmao

[@mahxylim7983]

Same! it is hw for my though

[@ignacioarroyo3385]

Mine is next week. I have to read the proof of the uniqueness theorem of the Jordan form and I'm procrastinating it xd

[@alanr4845]

Im doing the opposite. I'm watching all the math that I might have to potentially do, while in Calc 2

[@tomasbeltran04050]

@@alanr4845 U doing Calc 2 before linear algebra? :0

[@KhalilEstell]

Thank you for taking on this topic. My friend who is brilliant at maths told me 3 years before I took linear algebra about how it work geometrically, and I thought the subject was amazing. Then when it came time to learn it in college, I got almost none of that. I felt that either I missed out or I was just not understanding it as well as he did. But this video series has given me hope to understand and see Linear Algebra like he does. Thank you. I can't wait!

[@3blue1brown]

You're not alone, and I think that's indicative of most students experiences. Hopefully this series helps.

[@lerubikscubetherubikscube2813]

+3Blue1Brown Thank you in advance for the videos. I'm looking forward to them because i have linear algebra next semester :). Hopefully this series of videos will give me good intuitions about the subject.

[@n8n988]

+3Blue1Brown Thank you for caring about us I love the way you think and your style. Do you have other sources for me to check out? i mean i only have your channel as a GOOD source to learn and would like to have more sources

[@kazimierz198]

You hit the nail on the head in this intro. I went through uni studying electrical engineering. In 1st year we did linear algebra and all we focused on was how to do various calculations. It was easy to memorize and I did very well but it wasn't until 3rd/4th year where linear algebra was applied to various topics like control systems that I realized how little I actually understood the topic. I understood the HOW but not the WHY or WHAT. Sadly this was over 10 years ago and I wish I had access to such wonderful resources like your videos. Keep up the great work!

[@iwainste]

These videos highlight where innovation in education can take us. Well done!!

[@fengwang8674]

WELL SAID!

[@tghhh19]

We need videos like this for middle school math

[@contrazzed3651]

3.5 years deep into a bachelor's in computer science and I haven't taken a linear algebra course since high school. I've already passed calc II, probability and stats, discrete mathematics, and some logic courses. This information has been taken for granted in countless situations and it has always been a fuzzy subject in mathematics for me. Honestly, the visualization of the graph skewing blew my mind. It never clicked like that for me before. And yet it seems like such a simple thing to cover. Thank you for this course. I should probably be paying you thousands of dollars for this information.

[@BondJamesTheGamer]

Just a small nitpick at 2:12, calculators and computers do not use Taylor polynomials to calculate trigonometric functions, but rather employ a method of using pre-calculated tables and values, called CORDIC, to find the answer

[@iersssel]

🤓

[@jdm89s13]

I appreciate you making these videos so much! I've always asked professors about the geometric interpretations of linear algebra, especially eigenvalues, and never once did I get even an attempt at an answer. You are a godsend, sir.

[@crclayton]

I really look forward to this series! My linear algebra class was all pure math, with proofs and matrices operations, and had to frantically catch up in my engineering courses. Your sin example was spot on, and really articulated some of the things I'd been feeling for a while!

[@PepePopo2011]

This channel is just amazing, it might not get as many views as a popular channel that shows "hot pranks kiss" crap. But it has definitely a higher quality, please keep making these amazing videos. They have helped me a lot and I'm looking forward to this new series of algebra videos!! Amazing work!

[@3blue1brown]

Thanks so much, I'm glad they could be of help.

[@하민박사]

KZbin has been created for this video series.

[@KarnKaul]

I am starting out as a game engineer, and everything you said is absolutely correct. Although my linear algebra is decent, I now realise that the Achilles Heel of my high-school-undergrad learning is the fact that I always chose only ONE interpretation to understand a concept: as soon as one clicked, I never bothered to comprehend the others, because in my head, I knew at least one way to approach it. This has come back to bite me, because now I need to do the opposite: figure out how to use those tools to solve real-world problems (rather than been given a tool and a made up problem). I know unit vectors, dot products, and parametric forms, yet it took me a really long time to figure out how to easily determine if a given ray intersects a given a plane, in an actual job interview. I realised it's because of all those holes I've left in my knowledge of mathematics. It's quite disheartening! Is it even possible to develop a solid foundation now? :(

[@doodelay]

Karn Kaul yes of course it is! all you have to do is start from the beginning and play with the numbers a bit. starting from the beginning sounds horrific but it won't be nearly as difficult to learn initially now that you have math knowledge already. you already understand the operations and rules, it's just a matter now of exposing yourself to different methods which will inevitably be related to those you're familiar with. do not stress.

[@tatoforever]

I always say that to anyone leaving holes in their maths. Start from the beginning. Will pay better in the longer.

[@nin10dorox]

I HAVE BEEN WAITING FOR SOMEONE TO EXPLAIN THIS!!! You are even more awesome now, man!

[@dhinas9444]

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!

[@Tomartyr]

The moment you realise that your maths teachers failed you in ways you don't fully understand yet.

[@saurabh75prakash]

@Tomartyr , it happened to them as well, much worse.

[@BlakeryMath]

Does the blame lie with your teachers and not with yourself?

[@vibodhj349]

@@BlakeryMath Both. The teacher didn't take efforts to understand the student's perspective and the student didn't take enough effort to understand what the teacher was saying because the student could not relate to what the teacher was saying and was plainly disinterested . This is my experience.

[@amicloud_yt]

it seems the older i get the more i realize most of the adults in my life failed me in ways i'll never be able to fully understand

[@777Skeptic]

To be fair, teachers have a chalkboard and students are busy writing notes as the teacher is speaking. This guy has animations, visuals, sound effects at his disposal, and people watching are completely focused on the video without feeling rushed to take notes and can always rewind/pause if they need. Plus, this is a review for most of us. Like watching a movie a 2nd time, we're picking up things we missed the 1st time.

[@samgdotson]

Okay the analogy to having an intuitive feel for trig functions blew my mind. I'm taking linear algebra right now, and the geometric intuition about certain concepts is absolutely being watered down. This series is incredible for that. Thank you!

[@shainmays247]

I wish I had seen this stuff 7 years ago, thanks so much for making this content. You're really causing positive change in thousands of people's educations.

[@WhattheHectogon]

@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 KZbin channel. Never stop making this stuff so clear!

[@car-keys]

Great timing! I am just about to start linear algebra in 2 weeks.

[@repker]

ditttttttoooo

[@silpheedTandy]

i never undersetood the point of eigenvalues and eigenvectors way back when i took linear algebra. you'll have a leg up on others becuase you'll actually understand why they're important, by the time you learn them in class :P

[@fobenga]

lol me too

[@heckler73]

silpheed, Do you understand them any better now? I still don't generate an intuitive picture in my head when I think of them.

[@silpheedTandy]

i have no understanding of them :P i'm hoping 1Brown3Blue's videos will help!

[@OskarElek]

It is almost creepy how well the video's narration fits my life... Being subscribed for only a few months, I just dicsovered this series and am very much looking forward to it!

[@fobenga]

This is the kind of video that I REALLY need in the moment. I started Linear Algebra in college this month. I'm graduating in mechanical engineering.

[@sarahszabo4323]

I'm terrifically excited for this. I'm reviewing linear algebra over the summer so I can tutor it next semester :D Kind of reminds me of when I was learning determinants, and I asked my professor what exactly a determinant was and why we were doing this strange sequence of numerical operations. He was totally stunned and didn't know exactly XD.

[@3blue1brown]

I think you'll really like chapter 5 then.

[@sarahszabo4323]

3Blue1Brown Sounds wonderful, thank you so much!

[@Peter-bg1ku]

Same here, the professors never understood the math at a fundamental level. Sad

[@sarthaksingh1761]

Thanks!

[@bruceedward3079]

The educational system is so dumb, colleges make students to compute numerical operations over and over again with out knowing the actual logic, good video

[@gamemakingkirb667]

Bro stated facts (albeit ones mentioned in the video) and got no likes in the last seven years 😭

[@Brawlingsilver]

This is exactly why im here

[@antoniovelazquez9869]

I'm incredibly happy about this! And I thank you so much! I had a good course on linear algebra about a year and a half ago as part of my Computer Engineering undergrad, but I had almost forgotten about much of it; so a couple of weeks ago I decided to take a book named "Coding the Matrix" which is a FANTASTIC book for someone who likes programming and learning cool applications of linear algebra in Computer Science. I highly recommend that book!

[@piyushdongre4283]

at 0:31, det should be ad-bc

[@patrisagar6548]

Maybe he's checking whether we are really concentrating or not

[@avory7938]

@@patrisagar6548 that's what a math teacher says when a students corrects a mistake he made 🤣🤣🤣🤣

[@bears7777777]

What's worse is that saying this will one day be racist. Unfortunately i'm not kidding in the slightest. Its already happening.

[@harshsharma03]

@@bears7777777 what?

[@abdulwasey3506]

@@bears7777777 how tho

[@kaiiiichen]

THANK YOU FOR THIS! I love how you talk about having a visual pedagogy or geometric understanding will make this easier. I have a TA in my class who just assumes I already understand the geometric understanding, so I'm grateful for your videos

[@zachb.4429]

I'm looking forward to it. You really have a gift for teaching, and I love your videos.

[@PieroSavastano]

Totally agree with your philosophy. Math is beatiful when shown visually, focusing on intuition. Many books and courses only deal with symbols and computation. Hope you have an high return with these videos, they are of great humanity :)

[@DardanUkaj]

Wow... god bless you. I'm in Linear Algebra right now and I'm a prospective math major and I feel the same frustrations that you outlined in this video. My final is in two weeks!!

[@SatyamKumar-zx7yu]

This channel is just great, they way you show mathematics is not taught in majority of institutions. Visualisation of mathematics is what makes it beautiful. For example, for fourier transform we never knew how to visualise it, it was just restricted to calculations. Then I saw your video, which changed the way I think of fourier series. Similarly, this linear algebra series is excellent

[@SirusDas]

I cant imagine Maths without this... and thats why i hated it the most as no one could tell me why is it what it is like, they thought i was a mad freak who questions everything, all they care was those digits they get in exam and all i cared was to value those digits i score and Maths was still a mystery to me until this videos show up. It's a heart full of thanks to all you great people who bothered to share this knowledge to the world for free, trust me they never knew what they have missed! Thanks!

[@maxv2.0]

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 KZbin 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 KZbinr) 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.

[@BenMunro]

I'm studying for a 4th attempt at a maths unit final exam in my science bachelor and this series is helping a lot. Without this sort of intuition, maths can be like playing a song from sheet music that you've never heard before. You can know technically if you're doing it right, but without hearing the original song, you can't feel what it's supposed to feel like.

[@compton8301]

How did it go?

[@BenMunro]

@@compton8301 lol i failed the 4th attempt and dropped out

[@compton8301]

@@BenMunro Haha, cool. At least you tried!

[@circe3976]

This is one of my favorite courses on KZbin. Thank you! It helped me during my exams, and now I'm revisiting it purely for enjoyment. 😊

[@CODandponies]

wait, there's a formula to calculate sin? i mean we had soh cah toa but beyond that it was a calculators, here i am 2 years into collage and just learning this exists? wow thanks.

[@2Cerealbox]

How do you think your calculator figures out sine? It's gotta come from somewhere. It's just generally not worth memorizing the formula since it's such a well-worn computation that even before calculators people look up the values in tables and books.

[@SGManiac1255]

i must have assumed that most calculators had a look up table of some kind and did minor calculations on the values to be more accurate. i had no clue there was a formula for Sin Cos and Tan.... (mind=blown)

[@BlueMelon555]

maclaurin polynomials

[@CODandponies]

+Ryan N ya I know I just wish they would have made me aware of it.

[@EmperorZelos]

Of course, that is how people calculated it before calculators existed, which they themselves uses these days.

[@3isr3g3n]

These videos are exceptionally good. Talking about geometric understanding of numerical principles and the intuition behind it truly is genious and could help many people to grasp abstract concepts by visualizing them in a more intuitive, mentally isomorphic way. Subscribed and liked.

[@VikasVJois]

Looking forward to the series. Love your explanations and animations. BTW, I don't think calculators use the taylor series for numerically calculating the transcendental functions. That would converge slowly and would require lot of computation for the number of digits. They use the CORDIC algorithm which is best suited for binary arithmetic.

[@dan131m]

CORDIC is a workaround for when you don't get multiplication "for free" from your processor. If you do have hardware multiplication, what you do is reduce the problem to a particular interval using trig (or whatever) identities, and then use the first few terms of the projection of your function onto a basis of orthogonal polynomials.

[@3blue1brown]

Well, I just learned something. Thanks for sharing!

[@nadranaj]

Thanks

[@hoarfyt]

So excited for this!

[@mezzanine510]

You have no idea how much these videos have helped - I never got to this stuff earlier in my life and now I needed to learn it for computer graphics. Eternally grateful!

[@EugeneTulushev]

Thanks for your work. The pacing, animations and insights of the videos are wonderful (all of them). I hope making them is also rewarding for you, and you will continue. (sorry for bad English, thanks again!)

[@PremKumar-ym3vh]

The introduction was the most relatable in my case. Not having a practical understanding of the linear algebra put brakes on my machine learning course. Thank you 3blue1Brown for making such videos.

[@katetranscribes]

I'm taking linear algebra next semester and this is absolutely perfect!

[@not_potaytoes_hobbit]

Kyle Silver same here haha how did it go?

[@VictorWaknin]

From time to time I always come back here, I must have suggested this video to more than 50 persons. Excelent series, excelent explanation. It's the best visualization I ever saw on the matter.

[@silpheedTandy]

super curious about the videos to come! i love your focus on exploration and discovery! one bit of feedback: i have seen in past videos, as well as this one, the "transformation animation" where you have two grids superimposed on top of one another, and then one grid gets rotated and stretched [taking along with it whatever curve was plotted on it]. i always found these a little bit confusing because my brain is somehow trying to make sense of the whole criss-crossing of the two grids, rather than just focussing on a few line segments that get transformed. maybe if you first showed the transformation of the curve without the second transforming grid? or maybe if you had highlighted line segments of the transforming grid so i'm supposed to know where to focus on the transforming grid? or even if the transforming grid was "cropped" so only a small segment of it [the place where the curve is transforming, too] shows, while most of the screen only shows the original, non-transformed grid?

[@3blue1brown]

Hmm, good suggestions. In the chapter 3 video introducing this concept, I try to make it clear by only introducing each part step-by-step. I'm curious to see how well you think I address your concern.

[@silpheedTandy]

ah, the step-by-step approach might do it. also, can i say, i'm happy to see you back (and we know why you were away for so long -- you were probably spending much time preparing for this series!). sometimes i play your videos without trying to focus too much, just because your voice is so soothing, so i look forward to all the hard work you're going to soon put up!

[@aliensoup2420]

Thanks very much for these videos. Exellent material! They totally demystify an otherwise perplexing subject. Over the years, I have delt with Linear algebra in the form of computer graphics and viewed numerous lectures and textbooks, yet up to now, never acquired a meaningful understanding of "why" any of it works, or how to apply it. Most other treatments of the subject suffer from TMI - too much information. They dwell on the formal mathematics, while the real value is obfuscated beneath theorems and terminology. Learning the mechanics is easy enough, but without the pure intuition that you have provided, the mechanics is just mimicry or a parlor trick. Looking forward to more great videos.

[@seriousmax]

Amazing! Looking forward.

[@jonran]

Can't thank you enough for going so in depth in all these videos. Just finished writing ten pages of notes and sitting for 4 hours trying to fully understand. I'm currently a senior in undergrad physics at UIUC, and I haven't taken linear algebra because it wasn't required but recommended. I feel dumb for not taking the class years ago; this made every concept click instantly, and you turned pages of formulas and work into something I can do in my head. Again, thank you.

[@wicowan]

been watching your videos for a while, always as a hobby, but I found myself back to *study* , exams coming ahahaha

[@EspinosaJ]

This is so awesome man! I start my Linear Algebra class on the 22nd of this month and you have me eager to get at it! Thanks again for the great content, your work is much appreciated.

[@mattcay]

At 1:49 a statement can be seen: 'B^-1*A*B will also have positive eigenvalues'. Is that true and if so, why? My reasoning is that if we call this B^-1*A*B matrix C than this relation shows that A is similar to C so it has the same eigenvalues. Why would they be necessarily positive, though?

[@3blue1brown]

+maciej53 In my mind, this was some out of context sentence in a lecture where A, whatever it is, had already been shown to have positive eigenvalues. Of course, it doesn't really matter, the point is that the sentence is a hard one to make sense out of without a visual.

[@mattcay]

+3Blue1Brown Oh, I see. Thanks. Great video series anyway. I'm lucky enough to be learning from a book that lets me build some intuition for linear algebra's concepts and yet I enjoy watching them a lot. They are great for some revision and really hammering the ideas deep down into your head. Also it's always very nice to find a new perspective (I've never thought about looking at a dot product as an R^n -> R^1 linear transformation). Cheers :)

[@gamma-bv6ty]

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.

[@swiftyedits4067]

the determinant is ad-bc bro 0:30

[@riteshdshetty]

I love your videos, the presentation style, the voice is just makes even the complex matter easy to understand. Thank you!

[@JM-us3fr]

Thank you so much! I always thought it was weird that I understood abstract algebra better than linear algebra. I'm already into Galois Theory (much of which can more easily be learned in terms of linear algebra), but there's just this big knowledge gap. However, in defense of linear algebra, I had a terrible teacher for linear.

[@0x90meansnop8]

Man, you are doing a great job explaining those things. The educational system didn't manage to teach us the half of that what we should have learned. I respect your work.

[@glitchgatsby4290]

"You can't learn a subject just from a short series of videos." My aspergers-having ass: "You're not my supervisor"

[@elijahizere]

Absolutely perfect timing +3Blue1Brown that I begin learning linear Algebra only just this week in my Engineering course and now you start this series? Wonderful.

[@christianjorquera]

This is a dream come true. Saludos desde chile! :D

[@АндрійКоваленко-и2й]

There are no words how to describe the feeling when you want to research something in math, search though KZbin and find series from 3Blue1Brown! You feel that you will learn math beautifully 🎵

[@mouhammadabomoharam]

Thanks alot for the efforts put to make such videos see the light. I am though a little bit curious about the way these video are made. I mean which tools or software are used to make them? I hope I can get an answer from you :) I've always wanted to make such wonderful animations, but didn't really find the right way !!

[@wenminwu5240]

Have you figure out the tools used to make such wonderful animations? If you have, please let me know, thanks!

[@forresterickson6225]

I was curious about this too. Seams like it might be a nice learning tool. Work through some numerical examples for transformations of progressively more complex effects. Notice that the animation to the transformed unit vectors is not linear. The coordinates swoop around a bit. I assume he paramaterized the transformed vectors and put a "different" speed vertical and horizontal on each i and j so that the swooping helps visualize the relation that you would miss if both i and j simply were animated in a straight path to the final destination.

[@simicsabi1998]

I read he has written all the animations in Python

[@bandar1606]

My advice to young students to take the benefit of these videos to have a deep understanding about linear algebra. Long time ago when I was an undergrad student (i.e. 2003), not only we had no ample visualized materials, but we couldn't access the Internet with a reasonable speed (i.e. dial-up connection and not available at our homes). We could solely determine eigenvalues and eigenvectors with no understanding about their meaning whatsoever . The worst came to the worst is that English is not our native language yet English is mandatory at our colleges. We were good at math because the math calculations doesn't require a strong linguistic background however, every thing we were good at has been ceased to exist. The reason was the fact that we didn't have an intuitive explanation about a lot of math materials that helps our memory to remember math concepts for a long time. We've wasted a lot of time on carrying out calculations which is supposed to be the task of the computer and forgotten to remember the main picture. My advice to students is that this time being an undergrad student will ever come back again. Learn materials and focus on the big picture and keep some notes for each course.

[@n8n988]

+3Blue1Brown Thank you for caring about us I love the way you think and your style. Do you have other sources for me to check out? i mean i only have your channel as a GOOD source to learn and would like to have more sources

[@3blue1brown]

Khan Academy is great to really practice the operations (plus, I'll be revamping the linear algebra there in the next couple weeks). A friend of mine going through MIT opencourseware linear algebra material and speaks very highly of it. Also, I cannot speak highly enough of just sitting down alone with a pencil and paper and exploring on your own.

[@adityaekbote8498]

yep starting with the open course tomorrow

[@cristian-bull]

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

[@zashtozaboga]

What's the background music? It's so satisfying!

[@gabrielepichierri]

It's so significant that you are discussing this standard and yet fundamental topic in Mathematics. I've studied math at university and linear algebra has always popped up in every course I've taken. It's probably the one thing that should be taught to students in high school with much more care, or even considered in middle school. I can't wait to brush up my knowledge, I'm super excited about your videos, I know they're going to be great!

[@MrPoutsesMple]

It seems to me like a big conspiracy of the math teachers I had in high school not to reveal me the fact that Linear Algebra is indeed geometry. Multidimensional geometry. And that geometry doesn't end with Euclid's plane geometry...Thankfully, I learned this on my own. By studying Physics instead, and working backwards in my free time. So a big thank you to my physics teachers, that taught me to love Physics and Mathematics, and a big FU to my maths teachers. That said, thanks for these series of videos, I have the feeling I'll enjoy them very much !

[@zombiedude347]

I'm in a linear algebra coarse right now. Pretty much everything we do is shown mathematically, basically in the format: set up vectors into augmented matrix and solve using row reduction. Minimal explanation goes on into what or why. And the hoemwork relies on us knowing definitions that were easy to miss during the lectures. These videos are helping a lot.

[@rodrigoappendino]

Can you subtitle the videos, please?

[@silpheedTandy]

i wonder -- are youtube viewers able to help with this? i've heard other youtube channels say "thank you"s for viewers who have subtitled their videos in other langauges? i know i often like subtitles in my videos, so i can appreciate the appetite for them from others.

[@silpheedTandy]

(although, for what it's worth, i'm finding the autogenerated English subtitles to be surprisingly accurate.)

[@danielcookman3971]

Yeah, viewers can help out with the subtitling, but it requires the creator to flip a switch on their end when they upload the video. I myself would be happy to subtitle the videos, if that's okay with +3Blue1Brown.

[@dragoncurveenthusiast]

he would need to open this option up. right now we can't. below the video, if you click on 'more>transcript' and then click on the drop down menu, it should have the option 'add subtitles/CC' at the moment this option doesn't exist.

[@rodrigoappendino]

silpheedTandy I don't think the auto subtitle is accurate. It depends.

[@Arkalius80]

I'm looking forward to this series. I love the idea of using geometric visualizations to help build intuitive understandings of how some of these calculations are built.

[@rachelnanshija251]

Is the software you use for visualizing linear transformations available for public use?

[@keithfernandez2546]

Rachel Nanshija It is believed that he uses Python to create all of his animations.

[@abhidevathi1382]

THANK YOU SO MUCH.. This came at the perfect time as I'm taking linalg next semester along with a bunch of other hard classes. Your videos are amazing and I know these will save my life. Bless your soul.

[@BarackObamah]

what the actual fuck am I doing here

[@nerd_entangled]

WOW, these videos are incredible. I'm taking linear algebra in college right now and these videos might have just made linear algebra one of my most favorite branches of math.

[@EmperorZelos]

I don't think you should call it "geometric understanding", it's algebra more than anything

[@KhalilEstell]

No its not.

[@EmperorZelos]

***** Actually it is, it is just a very special case of module theory, except you're using a field rather than a ring.

[@KhalilEstell]

***** I concede my defeat! You know more about this than I do.

[@EmperorZelos]

***** You surprised me in a pleasent way, I have had the misfortune of dealing arrogant pricks for far too long it seems. Well done that you can admit the limitations of your knowledge, well done! It's not the easiest :) If you want I'll gladly inform you more of what I mean however.

[@justlikedirt9634]

He's trying to teach to geometric side of it

[@Schlynn]

I've taken a couple different linear algebra classes. And while ive never found it particularly difficult I do believe you 100% that they go way to hard on the computations. I'm very very excited to watch these videos.

[@TheDavidlloydjones]

Boring reminder: when you throw a baseball the applicable law is Kepler's. Think ellipses, give or take air resistance. Earth is not flat, and there is no reason for teechers to tell kids that any parabolas are involved.

[@TheDavidlloydjones]

***** Excellent! What you wrote there is exactly what I wish every physics teacher in the world would say *first* before going on to tell all those lies about parabolas. The lesson that we sometimes have to use an incorrect, but malleable, approxiamation instead of the real thing is much more valuable than the lesson about parabolas. Nobody cares about parabolas and ellipses in t the real world, and the Willie Mays Algorithm, keep the angle constant, for catching a fly ball doesn't involve either one. The judicious use of a good approximation, on the other hand, is something every kid is going to need all their life. Let the teaching be real. Your good post above is a fine example of how it should be done! Cheers, -dlj.

[@TheDavidlloydjones]

"Although parabolas have nothing to do with projectile motion, we can use them as an easy hack for small problems. "If your projectile is a baseball, a parabola will approximate it roughly, and the error from using the wrong math will only be a couple of microns -- much less than the error from air resistance. "On the other hand, if you try to use parabolas when your projectile is a Minuteman III, Moscow is not worried about you. 'Course San Francisco might be in trouble when you aim for Moscow..." The point is not knowing that a cannonball dropped off the Leaning Tower of Pisa takes 4.3 seconds, or whatever it is. Thep point is getting the ideas right. In this case, the whole point of teaching math to kids is to tell them that math is a bunch of tools, all of them approximate hacks, and you choose the one that comes closest -- being careful about it because there are a lot of people who get it wrong. Whaddya think? -dlj.

[@TheDavidlloydjones]

You haven't answered my question a month later, so let me put it more directly. Above you say "One loses almost nothing in accuracy and gains a world of intuition and understanding from the vastly simpler equations. " How can you gain "a world of intuition" when your teacher is forcing you, over and over and over, to "intuit" something that is factually false? Far better than being brainwashed with falsehoods incorrectly called "intuitions," surely it's far better to learn "Sometimes we use convenient lies because they get us into the ballpark. (and in the ball park we put up with the air resistance to the furshlugginer ball because we breathe the stuff)." Doncha think? -dlj.

[@TheDavidlloydjones]

They haven't learned about Kepler, but they've learned about conic sections? Hunh? And no, I'm not suggesting an "everything... is a lie narrative." My idea is that we not tell them lies in the first place. -dlj.

[@TheDavidlloydjones]

OK, I guess we've pretty much come full circle -- although you seem to be getting a wee bit French-deconstructionist about things if you insist on calling the ordinary discourse of everyday life "necessary lies." Meet me at Deux Magots for Sunday morning coffee some time. My point is that things will go better if the "lie," since you insist, that you start with is as accurate as possible. "We're using parabolas because they're a good enough approximation" would seem to do the job, imho. Going along implicitly using parabolas as if they were the actual case is a dishonest lie. You've forced me to coin a phrase. :-) The Big Point, it seems to me, is that a lot of kids are turned off to science by the fact that it is taught as a The Truth, and they can see right away that this is horsefeathers. Cheers, -dlj.

[@skaai]

I already subscribed to your video and can't wait to see the entire series. I spend my spare time explaining linear algebra to junior high school students (at a very basic level, of course!) and part of why I do it is I absolutely SUCKED at it when I took it. I recall tutors and teachers doing their best to help me, but they were too smart: calculations they did in one step took me 5 or 6 steps, and to make matters worse, smart folks cursed with such knowledge could never understand this!. Long story short: now that I understand it, I want to distill it to these kids in a way that is accessible to them. I've done a good job so far, but I have been waiting for someone to make a visual description of what takes me tons of words to describe. It's true that there is no substitute for imagination, pencil and paper, but a good intuitive video can do wonders... me (and my students) thank you!

[@KindFire756]

I am not exaggerating. I feel this series of videos is great service to all the students studying mathematics and to the society as a whole. Awesome job. Sir, you blew my mind!

[@kaustabhchakraborty4721]

This series is highly valuable to me , being a student from india where you r taught every time to replace your heart by a text book and your brain by calculator.But these videos just helps become a human again. I just cannot Express how thankful I am to you Grant. From a student of Btech Computer science.

[@zakku1732]

I'm an econ student planning on taking a lot of statistics and analysis based classes (forecasting, econometrics, data analytics etc), and linear algebra is the first time in a long time I've felt very interested and invested in a math subject. I'm studying up on the basics of it for my future classes and this series is extremely helpful

[@PuglyWont]

Wow this is just what I've always wanted. Thank you so much for making this video series.

[@yuvneesh]

Algebra is like sheet music. The important thing isn't "Can you read music", it's "Can you hear it? Can you hear the music?"

[@theflaggeddragon9472]

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

[@dddtl]

I can't wait to watch the rest of these videos! I had to teach myself much of the computational side of linear algebra for P Chem without ever having taken a class on it, so I look forward to understanding more than just what to put into Maxima.

[@mahidharc4866]

Awesome! I've tried to watch the MIT Open Courseware lectures of the Linear Algebra (18.06) class by Gilbert Strang. They were absolutely fascinating, but were very dense to go through, and I sort of gave up after the 4 or 5 videos. Hopefully the upcoming videos will help me get back into those lectures, and work through the assignments. Thanks for putting out all your excellent videos! :)

[@Abhraneil_Bhattacharya]

Grant's etude is the most beautiful piece of music that my ears have heard yet, to define it in a straightforward way : When I listen to this piece of genius, I feel butterflies in my stomach like people feel when their crush likes them back :).

[@y__h]

Thank you. Thank you. Thank you. I'm currently having a lot of problems with Linear Algebra, and I see it like how you described in this video: Magic. I and possibly many people here will be more than thankful enough to watch your tutorial again and again.

[@judahb3ar]

These videos from 3B1B really represent to me what I hope is the future of education. Excellent visualisations and explanations which are accessible to anyone with internet connection. I cannot believe more teachers do not incorporate these Into their classes … these could easily be a universal standard in classrooms.

[@Rubikorigami]

That is just so well-timed! I'mma have to learn linear algebra this year and this is really relieving me : the whole matrices thing really scared me. Looking forward to the series!

[@ThruVision]

So grateful for these videos!

[@AaronRobeson]

Thanks for doing this. I'll be going into Linear Algebra in the fall, and while I've done the operations before I really have little understanding of why we do things the way we do. I found a simple abstract understanding of what derivative and integrals were and how they related really helped me with Calc. Here's hoping this will be of similar use.

[@brettwoodcock7997]

Halfway through my semester of linear algebra and I know what I’m doing, but have no idea what I’m actually doing. It’s currently 2 AM and I have to study for a quiz tomorrow, but I will be back!

[@elisauhura]

Your work is simply AWESOME ! I can't wait to watch everyone of the episodes!

[@TheStallion1319]

god in one sentence u said everything in my head and said out loud for years let's worry about the conceptual half and leave the calculation to the computer but everyone just want more complicated problems and tricks not related to the physical meaning and the concept it's understood as symbols and puzzles not as logical consequence of applicable ideas , thanks keep it up and keep spreading science and information .

[@chasestubblefield]

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

[@Cyber_Homestead]

I wish videos like this existed when I was in school. I'm a computer engineer, yet never did very well in algebra classes. In fact, I failed the same algebra class 2 times in college before I took it a third time and passed. I have no problem with geometry at all, and algebra actually makes sense when I program, so it makes me wonder why teachers always taught the boring stuff first with no basis for application. Thank you so much for these videos!