Linear transformations and matrices | Chapter 3, Essence of linear algebra

Quite possibly the most important idea for understanding linear algebra.
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Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld
Spanish: Juan Carlos Largo
Vietnamese: @ngvutuan2811
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[@nenemunsigiri]

All of 3Blue1Brown videos should be stored in case of a global apocalypse

[@brianweir112]

11/10 comment

[@jayocaine2946]

Global apocalypse? Any apocalypse is global

[@sythatsokmontrey8879]

Yeahhh

[@black_jack_meghav]

damn yes

[@shrayammitra6447]

@@jayocaine2946 No

[@brunoborgatti4880]

3Blue1Brown: "you can make high schoolers memorize this..." . . me, studying for my master degree in engineering: **looks away in shame**

[@AM-qx3bq]

You're not the only one, brother. You're not the only one.

[@zuen7762]

wait why are you learning about the basics of linear algebra when you're in your masters

[@brunoborgatti4880]

@@zuen7762 cause my linear algebra professor during bachelor was super bad, i got the maximum grade at his exam but still had no idea what everything really ment geometrically ^-^"

[@zuen7762]

@@brunoborgatti4880 Ah, explains, i was kinda curious because it doesn't seem like something you would do in a master's degree

[@sz8670]

@@brunoborgatti4880 Me too. I just got 100% on my linear algebra midterm, but I had no idea what the operations actually meant geometrically. Thus I went online, and I found this...wow

[@bioshazard]

I recently picked up a book far above my level and this series is EXACTLY what I needed to comprehend the first page. Thank you for all your amazing content.

[@yogitshankar6348]

thank you for the donation! I can't do much on my behalf

[@hexagon8899]

wait you can donate in comments?

[@bioshazard]

@@hexagon8899 iirc, there is a donate button below the video at least in desktop and your comment in the donation apparently goes into the video comments

[@bioshazard]

I see in mobile it's called "thanks" and has a dollar sign

[@hexagon8899]

@@bioshazard yeah ive seen that for ages but had no idea what it did

[@dabblingfrancis]

I've just finished the entire Essence of Linear Algebra series, a truly wonderful (re)introduction.

[@3blue1brown]

Thank you!

[@meaninglessjunk9594]

3blue1brown is at the forefront of a revolution that he started. he’s innovating a new way to teach math with exceptional visual representations. so talented with the animations and so skilled at explaining. not all professors/teachers can perform what 3blue1brown can, but i think that’s where education is headed. absolutely exemplary job.

[@ShomeAvi]

meaningless junk true man true..so true dammit..makes me wanna cry

[@Boog1137]

@kirwi kirwinson While thats true you ignore the value of video over a still image. Youre not going to get as much out of something like 3D vectors from looking at pictures rather than something this well made.

[@Boog1137]

@Vela S Youre not wrong but way to be a dick about it

[@Canonall]

@kirwi kirwinson true and I would add to that, that this is just a visual representation, it doesn't go into detail about axioms or proof, which is the most important thing. It's just a good supplement to visualize the concepts.

[@lukedimitriades9835]

Vela S people like you are what ruin education. Understanding something conceptually is way more important than the math. If this is how that helps someone, then it’s what helps. Fuck off with negativity else where

[@alexsere3061]

whos the monster that taught me matrices but forgot to tell me about their origin and beauty

[@alexsere3061]

it all makes so much bloody sense now

[@StephenMeansMe]

Right? For some reason their existence is mentioned in, like, late middle school or early high school, and yet nobody actually does the cool stuff with them!

[@98Amberke]

Well, in the UK, if you take Further Maths as one of your A-levels(High-school equivalent IIRC), you will do some matrix work in two modules, but nowhere near as well explained, or detailed as this is. He truly makes even the hard stuff look very intuitive.

[@gorgolyt]

You don't know who your own math teacher was?

[@Kaiju3301]

Alex Sere I didn't even get that. They just taught is how to multiply them together and add them. I had no idea what a matrix actually was supposed to be.

[@Jleonpolanco]

I took Calc 1 - 4 and linear algebra on my way to an engineering & physics degree in college. I got what I'll call a "forced A" in all classes. I did not understand any of it, but I studied enough to apply the concepts in a test. Only when I studied physics did I understand enough of the math to apply it correctly to physics. After watching this video along with "The essence of calculus" series, I've realized that I didnt know anything. These videos are gems, thanks a lot!

[@ronaldrussell2048]

I too finished graduate and undergraduate degrees from top ranked institutions with math through difEq. I obtained solid grades in physical chemistry including quantum mechanics and then graduate courses in biophysical chemistry. My thesis work was largely aided by the application of this rigorous coursework. But when I observe and pay careful attention to 3Blue1Brown presentations, I feel almost embarrassed at how little I really understood of the fundamental mathematical truths or facts. I'm taking online course for a greater understanding of deep learning and a little AI. As I review the math today with the help of gifted content developers and teachers as is 3Blue1Brown is here, everything is so much more meaningful, even strangely beautiful. I, too, have am so thankful.

[@debrachambers1304]

You had a Calc 4? Did you mean to type 3?

[@alextsvetkov3211]

@@debrachambers1304 diff eq likely

[@joelquiles5339]

Exactly the same here! Enjoying these.

[@Jleonpolanco]

@debrachambers1304 sorry you're right we didn't call it Calc 4 we called it diff eq.

[@deepjyotichakraborty8739]

Your videos are a gift to humanity, Grant. Words do not suffice to express how brilliantly you have brought focus on intuition behind mathematics, away from notations and terminologies. It is not hard to see that it is this intuition that helps one to think from first principles regarding a concept and further, derive the formulas. Thank you many times over and wish that you keep making videos like these!

[@3blue1brown]

Thanks so much!

[@Colonies_Dev]

@@3blue1brown hey, quick note..isnt this kinda the basis for gilbert strangs' lectures? the topic in video- since u said it was not commonly tought?

[@luisalbertotorrescruz422]

Finally, I've seen the matrix.

[@lowkeylyesmith8545]

Morpheus is proud.

[@jerekabi8480]

me too

[@jeffreynatividad1937]

And you didn't have to take a pill either.

[@DeViLTh0rn]

@@lowkeylyesmith8545 LMAO

[@MatteoBucci95]

You are joking but after this video I felt like a sense of euphoria that is difficult to describe :)

[@AnanyaMuddukrishna]

You made me cry again. These are tears that flow when some obstacle in the mind is removed forever. Thank you, great teacher.

[@slayerq3]

If this made you cry, then that is weird.

[@djlinux64]

slayerq3, this is because you do not understand what true suffering feels like. many around the world still live in total shit conditions and have to endure them while also attempting to study topics such as mathematics without great teachers. thus their struggle is compounded.

[@mikeyking3670]

ABC123 it’s pretty racist that you just assumed he lived in awful conditions 😂

[@kaustubha7371]

Especially for Indians where parents teachers and the whole society believes in rote learning. As a proud Indian it makes me cry too😅

[@venceremosallende422]

The Beautyfulness of math brings me sometimes near to tears, especially when I see how maths get's pulled through the mud in school. I am so sorry for the many times I cursed this universal wonder because I was totally wrong connected with it due to school. Sorry me dear ♥️

[@kkh6174]

I should've watched this when I was a sophomore. This 11-minute visualization is much more worthy than my last 6 years. Awesome.

[@Jammoud]

My exact same thoughts hahaha.

[@brandonwang6338]

When the alleged best math teacher in my school gets EASILY bested by a random talented dude online LOL. This channel is the best discovery on KZbin I've made so far. Your explanations and visuals combined simply beats any textbooks out there. Keep up the great work my man!

[@tomasnedzinskas7768]

To be fair, the pool of math teachers at a given school never really match up with Grant Sanderson (and his small behind-the-scenes team) :D. I also love this content - it's one part of the unique access our generation (I'm assuming you're a teen like I am) has, for free, to learning from people that are simultaneously highly expert on their subject and brilliant at teaching. We live in kick-ass times.

[@adityasanthosh702]

You are mistaken if you think he is a random dude online. He is from Stanford and has been doing these videos since years and has gained huge following all over the math community. He popularised the python visualisation package manim which he himself has coded

[@huyxiun2085]

I am astonished but the level of stupidity this comment could reach, WITHOUT being deleted asap out of shame by its author, AND liked by 69 fools. Man, don't get me wrong, I'm glad you appreciate 3Blue1Brown channel and content. But keep that state of mind and you'll never grasp anything out of it, nor school, nor life itself. You just illustrated that you understand NOTHING about statistics... Fine, most people don't. However, you should realize how it lower the quality of your praise toward a math channel. It's high time you learn that every reaction is a couple. And within, the hierarchy is secondary in the quality of the reaction. Learning and understanding is not about a teacher who is good or bad. It's about how good, and adequate to each other, both the teacher and the student are. So that bested math teacher from your school... Trust me, they are doing a lot to adapt to you guys... if they every sound dumb, now you know who they get that from. Meanwhile did you ever try to adapt to them?

[@tomasbeltran04050]

@@adityasanthosh702 I mean only to help by saying ðe following: it's "for years", not "since year". We use "for" wið periods of time and "since" for specific moments in time

[@mridulagrawal2370]

That random dude online teaches "computational mathematics" at MIT, idk if the expertise level gets above that lol

[@securedigit]

My life has been linearly transformed by 'shear' joy of watching this video!

[@dojinho]

Nicely put! ;-)

[@odyseuszkoskiniotis6266]

Noice

[@rajkiran5490]

very true... I am starting to like Maths again..

[@abhishekarora4007]

good one

[@eliya3424]

😆

[@diahidvegi8536]

I can already see next generation kids growing up learning math this way and then asking us, "Seriously, what made it so hard for you to get this, Mom??"

[@dasav6724]

That sounds amazing

[@francescocostanzo8225]

@Fluffybrute because math

[@legacies9041]

This is how it has been taught at MIT and UC Berkeley for eons

[@hema5748]

I have a felling that we were so young to understand all of these and math was always a problem for all of us, instead of having these in 6th grade this should have been in 10th for our brain to understand and process I mean how could a 11 year old process all of these

[@mayasudhakar9595]

Absolutely..Its going to happen for sure😂

[@krishtundwal7794]

i think a really really helpful thing in the future videos would be to add some practice questions for the people who really are learning from this playlist and not just watching it, i feel that the way to learn something like this is to get your hands dirty and make mistakes, and so a set of questions attached to each video would really boost the understanding and learning process. and i absolutely love this series, thank you so much!

[@eliad6543]

This!

[@peterlohnes1]

yes i noticed that too...a few examples and how they are transformed would be good

[@kavitownsend9919]

I think the issue with this is it takes away from what he is trying to get across which is a deep conceptual understanding of what is going on as opposed to learning the methods and techniques to solving specific rleated problems. I imagine the target audience for these videos are those studying this content already but haven't fully grasped why they are doing what they are doing.

[@TheProblembaer2]

When somone truly understands their subject they are really capable of explaining it in easy terms. For me, this is one of the signs of true academic mastery when you are able to explain things really understandable and to the point. Thank you 3Blue1Brown for making the lifes of many students so much easier. Additionally I want to add, this is the way I always whished to understand math, not as a applications of rules, but understanding why that rules exist. These videos do a tremendous job of really intuitively explaining those rules.

[@KanjoosLahookvinhaakvinhookvin]

3b1br has to be one of the only animators out there who can say "you've gotta admit, watching [my animation] is pretty beautiful" in the middle of it and have no one disagree or annoyed.

[@brent2004]

This is because he is admiring the beauty of the linear transformation itself, as his animation is merely a graphical representation of said transformation

[@MrMarc1991]

And the crazy thing is that he programs them with python.

[@W___G]

Or we say "this is how animations are done, and how it should be used in the first place". It comes from math, now used to teach math, which is essentially, itself.

[@abbasdehghan8709]

For all my teachers, f__k you b__; This is Math, not those shits you are teaching at school.

[@tppiotrowski]

This video finally gave me the intuition to explain why multiplying a vector by the identity matrix produces the same vector. It is because the identity matrix is a linear transformation which does not move i-hat and j-hat from their original positions. Great!

[@JiananLi]

Exactly what I was thinking!

[@Warby579]

Same!

[@meaninglessjunk9594]

mind=blown. thanks for this comment

[@TanKSeang]

yeah, thank you too!!

[@shabir301]

I also knew at that time when they thought what transformation is...

[@takeiteasy8847]

After having already taken a basic course in linear algebra it finally all comes together. 1000 thanks your videos rock!

[@edwardwong654]

Not being the sharpest shed in the tool, I like to watch these videos, take breaks and watch them again, to really appreciate the beauty of math and Grant's passionate ability to explain it.

[@youceflabiad5700]

WTF, why would they teach me to multiply matices without telling me the point of it?? it just makes sense now and i instantly memorised it against my will, thank you

[@eliya3424]

IKR??!!!!

[@abhishekrbhat8919]

We need to find the source which started teaching it wrong

[@randycalifornia]

@@abhishekrbhat8919 bourbaki

[@abhishekrbhat8919]

@@randycalifornia Are they the bad guys?

[@randycalifornia]

@@abhishekrbhat8919 Yes Neo, if you want to see how deep is the rabbit hole then google "On teaching Mathematics" by V.I. Arnold muahahaha

[@lennertjansen4299]

when Grant said "Then, you can make high-schoolers memorize it, and hide the most crucial part that makes it intuitive." i felt that

[@lilsunflower9655]

Lennert Jansen seriously!!!! This is the piece of the puzzle I was missing! Linear transformations are hard to conceptualize

[@63ch31]

It's like every single math professor watched this video, but most of them didn't get the sarcasm

[@dioptre]

finally i got to understand the fkin multiplication

[@georgiavanfossen3109]

I keep coming back to this video. I watched it once in my linear algebra class, but over and over again in new classes I find this information makes everything else click. Reference frame transformations in orbital mechanics, stiffness matrices in FEM, this is where I go when I don't understand something! Thank you!

[@jax1831]

I absolutely loved how smooth the transition from explaining transformations numerically to introducing matrices was. I didn't remember or hadn't ever realized that such concepts were so intuitive.

[@andrewknapton7665]

Oh. My. Word. This was brilliant! I have a PhD in physics and I've never understood matrices. I've learnt how to manipulate them, but I've never understood how they relate to anything (which is a huge failing in my education, by the way!). In one fell swoop you've not only explained what the numbers in the matrix actually represent, but also why you multiply columns by rows in a way that actually makes sense!

[@nurahmedomar]

I feel you, I am a current Ph.D. student in Electrical Engineering, I know how to manipulate matrices but never understand the meaning of it. After watching this video, it makes so much sense.

[@fishfeelpain7764]

That gives me hope, I'm in the middle of my studies and feel like an impostor because I get good grades even though I don't really understand the fundamentals. Most of my professors are just useless, it's so sad. They often have a really bad accent and/or monotone voice, bad handwriting and no creativity when it comes to explaining complex topics, they just read from the text book. So nice to hear you made it to a PhD. Just discovered these videos and they give me so much hope.

[@clementyau7192]

@@fishfeelpain7764 reminder that profs are worse versions of teachers. theyre there to research and we pray to god that they are decent enough at being able to explain their knowledge to other people

[@fishfeelpain7764]

​@@clementyau7192 Yeah, and I really don't get why this system hasn't changed yet. Just let researchers do their thing in the lab, and let spirited people (who wouldn't necessarily need a PhD) explain linear algebra etc. to undergrads.

[@pangipang1144]

I also have a Ph.D. I got an A+ in my undergrad linear algebra course, but I did everything mechanically. This gives me a whole new understanding of linear algebra.

[@BangMaster96]

I always hated matrices, i thought it was the devil's concept of representing numbers in square blocks. But now, seeing as to how beautiful they are, i realized that it's not so bad after all, and i can actually understand all of this. We need some serious questioning about our current method of education and teaching kids about Mathematics and Sciences. I mean come on, all it takes is one good teacher to completely change your mind and concept about something, and most teachers i've had were all horrible. I believe the future of education should be the internet, where the passionate few who love to teach are making creative online visual explanations of complex topics, and millions of people all over the world can tune in at any time and learn at their own pace.

[@kventinho]

I have been that kid. Currently doing a PhD in Genetics and watching this, having the lighbulb moment to myself, made me wonder where I could have been if I have been taught maths this way. I agree with you - the future of pedagogy has to incorporate these audiovisual medium that students can digest at their own pace. Thinking back, even IF my teacher was 3Blue1Brown in a classroom setting, my personal lack of concentration would still be a hindrance. In this video, I've paused, and rewound the material a few times, just because I lost concentration and needed to get myself on track. I will not be able to do this in a classroom setting, nor should my teacher do this (it's impractical, as a typical school classroom is usually >10 students). I am so glad to be living in this era where education and knowledge is literally at the tip of my hand.

[@stephaniecurry45]

Quentin Aziz you might have adhd

[@pascalladal8125]

I relate to him and I do have ADHD. I've been self-taught for the past 10 years for this reason. School is basically the death of hyperactive minds if you ask me. I always had the feeling classes were grounding me everytime I was excited about anything I learned there. I guess it must also depends of where you come from though. Thanks to these kind of videos I got a job as a programmer without any degree. I learned finance, fiscality and accounting that way too and I'm close to be financialy free. If I was born 20 years earlier, I have no idea what I would have become without all this information easily accessible at my own pace (which strangely goes from turtle to cheetah in a matter of minutes and vice versa on realizations like: "I'm bored", "hmm that looks interesting!") Also, I"m quite certain that if those videos did exist when I was in school, I would never have dropped out of university in the first place. I would have replaced theoric classes with these videos. There is certainly questioning that needs to be done about the current method, no doubt about it. It made sense in the 1930s, not anymore.

[@stephaniecurry45]

Pascal Ladal wow props to you. I can’t imagine getting a job like that without a degree nowadays. Really impressive

[@stephaniecurry45]

Pascal Ladal and I agree that the current system definitely could use some questioning and potential revamping

[@thomasdemilio6164]

"Then, you can make high schoolers memorize this without showing them the crucial part that makes it feel intuitive" That's exactly what 99% of high school teachers do.. we're blessed to have discovered such golden content. Thank you. I really hope there'll be a huge revolution in the taching world, which will use pre-registered top tier lessons like this and integrate them in schools. Just looking at these graphs and animations would be huge... we don't even have a digital board in our classroom!!

[@beginneratstuff]

Just started watching this series as someone who has not learned linear algebra yet, and wow. These have to be the most clear and easy to understand videos I've seen from you yet. Loving the series so far; this is super fascinating stuff. Can't wait to start applying what I've learned!

[@hibak8196]

7:42 "Then, you can make high-schoolers memorize it, and hide the most crucial part that makes it intuitive!" Omg so accurate lol

[@shybound7571]

mood goddammit ms green

[@Diaboloxfan]

@@shybound7571 Camden county college?

[@shybound7571]

@@Diaboloxfan no i learned it in 11th grade

[@ChetanBhasin]

I had to pause and like the video when he said that.

[@arunjanarthanan6380]

a nail in the coffin for curriculum designers!

[@user-cr6rx1qn8v]

After decades from the video revolution, now we finally encountered the start of 'real' video education revolution.

[@wawawuu]

10/10 comment

[@joshuachoi_5609]

한국인 !! 영어 잘하시네요

[@user-ci1lm4we7q]

'찐' 영상 ㅇㅈ

[@abhijaybharathan4308]

Oh my god. This is the best video I've watched. This makes everything else fall into place like you said. Affine subspaces and all of the other concepts I just finished learning that build on linear transformations are all clear to me now. You just brought back my hunger for understanding every concept I come across that I had lost over a year ago. This is a goldmine for math intensive stem fields

[@skymanaditya]

I am coming back to these videos after completing high school, 4 years of undergraduate, several years of research-based job, and academic research, including publishing research papers, to doing my Masters's Thesis, have used concepts such as kernels, non-linear transformations, hyperplanes, and the like freely. I finally needed this for understanding robot motion in constrained 3D planes. Wonderful videos 3Blue1Brown, are very informative for any level of student/professional.

[@wensi9131]

You deserve an Oscar Award for mathematics

[@borekworek69]

And Nobel for animations Wait what

[@exismys]

Wow

[@ankushmenat]

Fields medal

[@1dan1609]

@@borekworek69 I see what you did there

[@mrdbourke]

"All of these will become easier to understand once you begin to think about matrices as transformations of space." Everything just clicked. Thank you.

[@user-xl8uo9gp9p]

wooa!! you here..man i like your work..just randomly saw your comment lol

[@robertlewis6543]

Every time I watch this channel, I think about what my life might have been like had I had these videos in high school and college. Really helps abstract concepts come to life!

[@edwardseverinsen5598]

I really appreciate these videos. I've started programming a graphics rendering engine and everything as far as the GPU pipeline; vertex shading, fragment shading, vertex buffer objects and those aspects made sense. But as far as representing a 3D image on a 2D surface like a screen with linear algebra I was lost. But this is making much more sense, thanks man.

[@JimmyDuong97]

Do not stop making these videos. they are awesome! eventho you dont have lots of views yet. visualising math is just awesome!

[@alexsere3061]

true that, I now feel robbed of the beautiful math by my teacher

[@paradoxicallyexcellent5138]

+Alex Sere interesting notion - robbed. I don't know your teacher(s) but my experience is that most, tragically, either don't know, or have forgotten, that this is how most people best learn. However, I soon found, when learning mathematics, that beautiful ideas are behind everything, and I quickly taught myself to find those visualizations myself. Practice! And if you are good at it, go into teaching!

[@renesax6103]

you can contribute to his videos via patron. Google it. Join the cause! :)

[@conphzhi]

over 100 k

[@oliverberg5738]

he said earlier he uses basic Python plotting ^^

[@midhunrajr372]

So the [ [1 0] [0 1]] matrix is called identity matrix because the unit vectors are not moved at all! WHAT WAS I DOING IN MY MATH CLASSES???

[@torwaldolafsen]

totally with you

[@odyseuszkoskiniotis6266]

Your teacher is shit

[@339059331]

same here :(

[@yeet_yeet_mcskeet9511]

Or it could be [100]. [010] [001] Or [10] [01] Etc

[@codinghub3759]

1000 0100 0010 0001

[@stolfjr]

The precise description you gave of the transformation matrix being just the endpoints of the basis vectors was PERFECT to understand what is really happening. Thank you!

[@muraadmasood2968]

1.Linear Transformation - refers to the movement of an input vector to its corresponding output vector , as whole we can imagine every possible input vector as a dot moving over to its corresponding output .This allows one to distort , stretch or compress ,elongate or shrink and rotate a plane or space. 2.A transformation is linear if it has two properties ->All lines must remain lines without getting curved ->The origin must remain in its place This keeps the grid lines parallel and evenly spaced. 3.A two dimensional linear transformation is completely described by the coordinates of where the basis vector lands.These coordinates can be packaged into a 2x2 grid of numbers called a 2x2 matrix. 4.Matrix multipication tells us where a linear transformation takes a specifc vector, by taking the coordiantes of that vector and multiplying them by the corresponding columns of the matrix and adding them together; this is actually adding the scaled versions of our basis vector i & j.

[@thomasdemilio6164]

On point👌

[@dexterlohnes]

"It turns out you only need to record where the two basis vectors...each land and everything else will follow from that". I have struggled with linear algebra and linear transformations for years, despite being a game developer and someone who (before linear algebra) felt I had an aptitude for math. This one sentence was such a light bulb moment for me. Thank you so much for your videos. You are doing amazing, beautiful, and important work.

[@joannwatu7603]

I agree. It was a light bulb moment for me too!

[@Imran52Feb]

Fantastic video and the way of you explained. This was not easy to visualise. Hats off !

[@Peter_Siri]

I believe we all have an aptitude for math; the real barrier to further understanding is that we stop learning - that we stop seeing through a lens we are not attuned with and through several other things (the Matrix, for instance) that we wouldn't really discern w/o taking a hit even if someone explains them quite clearly to us. Linear transformations really opened my eyes to how a 3D environment is rendered on a 2D screen.

[@Imran52Feb]

I am also impressed by the presentation of 3D using 2D and landings of i and j unit vectors for explaining the transformation of vectors Hats off . I learning it at 70, after retirement, for pleasure of discovering new things.

[@bubbles9443]

I'm in 11th grade right now and people have been constantly telling me that linear algebra is just something you want to pass as quickly as possible since it's really hard, but from these videos it seems too beautiful and elegant to miss out. This is the kind of understanding people need from mathermatics in schools and universities. Keep it up fam!

[@artemlyubchenko3022]

Because nobody is teaching it the right way! So many people teach linear algebra as just tables of numbers that interact with each other in some ways and nobody really knows why they do that, not paying enough attention to or even just ignoring the real meaning behind these interactions. Linear transformations and matrices as transformations of the basis vectors in my opinion are absolutely crucial for understanding what linear algebra actually is, and I have no idea why almost nobody talks about it. I'm in a linear algebra class right now and the way they teach it does nothing to make students understand what they're actually doing, so now I'm recommending 3b1b to everyone there.

[@moatef1886]

For anybody reading this who isn't sure of the connection between his geometric intuition of linear transformations and the formal/textbook definition of linear transformations it is this. 3B1B says that linear transformations are transformations in which the gridlines are kept parallel and evenly spaced (and kept as lines). This is the same notion as maintaining the scalar multiplication and vector addition properties in the definition of linear transformations i.e if x and y are vectors and c a scalar then T(cx) = cT(x) and T(x+y) = T(x)+T(y). Combining these rules we get T(some linear combo of x and y) = some linear combo of T(x) and T(y). So that's the same concept as with the basis vectors. A vector that is some particular linear combo of the basis vectors has to, after the transformation, still remain that same particular combination of the transformed basis vectors.

[@princeofrain1428]

I like this idea of matrices as transformations the way you describe it, with the transformation matrix being "where the basis vectors end up, post-transformation". It really solidifies WHY a linear transformation should be defined through matrix-vector multiplication, since whatever happens to the basis vectors must necessarily happen to the vector in question! Thanks 3b1b!

[@AlfredEssa]

Beautiful explanation. These video explanations are works of art.

[@DorFuchs]

Amazing videos! It took me so much time and afford to get to this intuition after only seeing the abstract definitions and I am also convinced, that it is better to start with intuitiv animations in R² to get a feel for it, before going abstract. Greetings from Germany! ;)

[@zairaner1489]

Wow I found Dorfuchs! I cannot imagine the quality of videos when you two would work together.

[@simonn8312]

Dorfuchs geile Mathe Songs 🔥🔥😂

[@davidhilbert8463]

Hallo DorFuchs, ich bin ein Abonent.

[@nikhilnagaria2672]

ayy look who's here !

[@marko-vasic]

Great job explaining and visualizing! Impressive, I wonder why we didn't have linear algebra explained this way in school. Everything seemed abstract and hard to understand. With geometric intuition, it feels like everything falls in place.

[@sciencedon3993]

watched multiple times, and now i have command on transforming the grid sockets in so many ways that each time i get my desired vector as correct that i feel a sense of strange happiness when i find a vector in different coordinates and grids as the same vector i start with . I can't thank you with words. It's more than a simple youtube video lesson. It's life changing video. I now can imagine so many possibilities, of doing so many things with this insight that you gave me today. I'll just wish you so many good things that are not possible to mention here.

[@tannersmith448]

Does anyone hella appreciate the alliteration in "this is a good point to pause and ponder, because it's pretty important"? Or just me?

[@brianweir112]

poetic police 👮

[@gauravluitel1344]

this pause and ponder is sticked into my head and after I see this comment. it was not only be .

[@richardaversa7128]

That might have to be my mantra for every moment of my life

[@richardaversa7128]

Ah shit, more alliteration

[@AniketKumar-ts5gt]

I skimmed past it as i have been conditioned to accept maths without thinking twice. It was a few moments later i came back to that particular instance and i was bamboozled by the gravity of the explanation. It was like finally exiting out of the tunnel to see the full view of the beautiful scenery I was taught to ignore. Loved the moment matrix multiplication as a linear transformation finally unveiled to me.

[@patrickwienhoft7987]

If anyone sees this before learning about this in school (or whenjust learning it), be grateful and keep this in mind! Interpreting coloumns of a matrix as the transformed basis vector is something I wish I knew/realized much sooner! :)

[@gnanay8555]

Yep, they are really lucky indeed.

[@pursuitsoflife.6119]

Just started matices lol, i wanted to look for some additional info on YT, glad this video popped up :D

[@rasmusbrege7433]

3:35 I absolutely love this teacher, the example given is so perfect I had to rewatch it 3 times just because of the wording. It honestly gives me shivers and brings so much happines just through a simple sentence. Amazing work

[@homunculus3646]

this 6 year old video might be the best thing i ever watched. time to binge the whole series i guess

[@bitwitch]

The idea that a matrix can be thought of as where the basis vectors end up after a transformation really demystified much of linear algebra that I had learned about but I failed to understand on an intuitive level. Thank you so much for this video.

[@ruzreuben9755]

*These videos are a MUST to really understand linear algebra.* *Blindly multiplying numbers just because that's the way matricies multiply is DUMB.* *Instead, this video made me understand that the defintion of multiplication can change.* *How is taking the transformation of two basic vectors(i and j) and finding the formula of every transformation MULTIPLICATION?* *Completley different defeniton.*

[@blobybubble268]

@Robert Raddison I'm sure Victorians thought the defecation bucket "worked" before the toilet was invented. Too many people settle with low standards. Think of where we'd be with better systems.

[@troychavez]

INDEED! What an awesome man and mathematician.

[@lilsunflower9655]

I am halfway through the semester, and I have been spending hours upon hours studying linear algebra out of my text book. Then I get my first test back and I failed (which has never happened to me in a math class ever)... After watching a few of his videos, I totally get why I failed the test, I couldn’t conceptualize what these numbers were telling me at all so I had such a limited understanding of them. I’m at university and my teacher has failed to ever present a graph in lecture. I don’t understand why, this is such an important part to understanding this shit...

[@lukedimitriades9835]

DontMaskTruth we’ve gotten this far because people broke outside of the box and thought for themselves. This just teaches you to blindly follow what others have done.

[@monny1815]

DontMaskTruth You’re right to an extent, like obviously we got so far because some great minds could imagine this stuff without needing to see the transformation with their eyes, but students, who are not all geniuses, might not get the intuition behind it because not everybody just “gets it”. This tremendously helps with the abstractness of linear algebra, since it gives you a way to visualize the concepts in 2/3 dimensions and then applying that same intuition to higher dimensions, but I definitely don’t think this substitutes a rigorous education, which is something Grant said multiple times.

[@textinface1]

Cannot stress enough how amazing your work is in helping millions of people, including myself, understand mathematics in the most entertaining experience possible. I took a linear algebra course a few years ago and it was nothing like this. A matrix was just a grid of numbers and multiplying and adding matrices was just that. Numbers that change numbers into other numbers. Memorization and no actual understanding of what the underlying concepts were actually doing. This brings a whole new perspective that correlates mathematics with the space/universe we live in (like it is supposed to)! I'm glad I came back to relearn this lin alg and found your channel!

[@kasai1575]

I just wanted to say THANK YOU for making these videos! I took an entire course on linear algebra a year or two ago and had little to no intuitive understanding of the operations we were performing. Matrix multiplication, span, eigenvalues, diagonalization, etc. were just things we did with no explanation as to *why*. Imagining matrices as "transformers" for vectors brings a lot of light as to why matrix multiplication is formatted the way it is. Hopefully I can gain more insight from the rest of your videos too!

[@hullgatt]

7:40 hahaha, what a legend! Heads up! A message of appreciation. Sometimes people ask me how I can understand certain things so well. I then tell them that It's because I was taught well by 3Blue1Brown (in this case). No one will read this but... I adore teachers so much for their hard work that I have to secrete endorphins just to manage my feelings when thinking about them. Thank you! PS. Shout out to all workers for contributing to society while blessing me with the luxurious opportunity of going to school! Thanks to all of you!

[@independentvariablez7854]

You know as a incoming math major, he does a great job to get me more excited than I already am about my major... This is cool.

[@Itasekidsteehee]

How you feelin about it now that it’s exams??

[@axeldiaz7960]

@@Itasekidsteehee he didn't make it

[@64_bit80]

@@axeldiaz7960 lmaooo

[@twgoforfun]

This is honestly the best video on linear tranformations/matrices I've seen so far and this playlist helped me understand algebra a lot better! Thank you for making these amazing videos!

[@noodlenado7827]

THANK YOU! I find the concepts of linear algebra to be very interesting, but a bit convoluted at times, and this really helped it make sense, especially since it more or less ties everything together. The visual especially are GREAT

[@Stilllife1999]

I love how you used a quote from morpheus

[@Reydriel]

I know right, it's not even the same Matrix XD

[@Peter_1986]

+Reydriel Actually, the Matrix in the movie was most likely created with the help of matrices. =P

[@Ragzzy-R]

every movement we make in this 3D space is actually a linear transformation. The Matrix movie says THE MATRIX is like a simulation(like a video game) so both the matrix are deeply related. And even in our world, Video games uses matrix multiplications to rotate move players. so we are all may very well be in a Matrix ourselves

[@k7jeb]

The quote is eerily appropriate, more so once this particular video is viewed in its entirety.

[@isnansunuyuntaro2974]

7:41 "you could make high schoolers memorize this, without showing them the crucial part that makes it feel intuitive." Damn son

[@sairajpatil940]

This is gold. Helped me so much with visualisation and conceptual understanding. Kindly also make a series on 'Essence of Abstract Algebra/Group Theory'.

[@vladimirbosinceanu5778]

This is outrageous! The clarity and intuition in these videos are outrageous. Thank you so much!

[@edsed9290]

That's what was missing from my math classes at school. I always remember myself as a hardcore visual learner that had a hard time visualizing symbols. I'm currently a CS student and these videos are a tremendous help. Keep it up :)

[@Verrisin]

This is *SO* much better than what they taught me in uni....... T-T

[@DarthZackTheFirstI]

yeah lol. in my first semester they showed us how to calculate and some math solving examples/tests. but never showed what it actually did. i mean for vectors i could figure it out myself but with matrices it wasnt so easy anymore, why that is. quite sad XD

[@kdspuh]

This is by far and away the best explanation of a linear transformation I’ve ever seen.

[@GuillermoPalchik]

I have been watching this incredible series... I cannot tell you how many "aha!" moments I have had... I took Linear Algebra almost 30 years ago and although I got a good grade, it was mostly because I memorized tricks; however, I never felt I truly understood it... it was something that bothered me my whole life. After watching (and re-watching!) these videos, I feel things finally click for me... I wish I could have watched these videos 30 years ago (or that my professors had been as eloquent as you are) This is a true gift to the world. You have my utmost respect and gratitude.

[@jaym2112]

2 minutes of this video helped me understand something that a whole semester from Dr. Ukrainian-guy and about 3 hours of various YT videos could not. God bless you, sir!

[@jordanr8745]

Lol

[@Abhinav-tk1bt]

the quote at the start is hilarious. Was NOT expecting "Morpheus"

[@hamidraza234]

AMAZING VIDEO .ALL THE ABSTRACT CONCEPTS ARE EXPLAINED SO BEAUTIFULLY .TBH IT TOOK ME A LOT OF TIME TO UNDERSTAND HOW MATRICES ARE NOTHING MORE BUT FUNCTIONS JUST IN A DIFFERENT PACKAGE.THIS SIMPLE FACT HELPED ME SO MUCH WITH LINEAR ALGEBARA .I HOPE ONE DAY YOU MAKE A VIDEO ON NON LINEAR TRANSFORMATIONS LIKE QUADRATIC FORMS

[@holo23]

I've watched this 3-4 times now and I can't believe how much knowledge is contained within this video. I can only imagine the years of accumulation of knowledge and understanding on this subject to impress even someone like me who just started learning linear algebra

[@nin10dorox]

You actually explained it! I've searched the internet for an explanation of how vectors and matrices work, and no one has done it but you!! I can't wait for the next video!

[@trollghostol]

This is legitimately an amazing video. Especially in the UK, we are just fed matrix transformations with little explanation or reasoning, they seem as abstract as can be, but this video really taught me where it comes from. Thanks a bunch.

[@3blue1brown]

I'm glad you liked it. For what it's worth, I don't think what you described is limited to the UK. It's part of the reason I wanted to make this series in the first place.

[@yahyasaad9758]

I can vouch for the US too

[@Snuni93]

+3Blue1Brown Yup, student from a german university here can confirm the same issue. I am ACTUALLY happy (smiling right now) because I now understand what matrix multiplication even means. Why didn't I feel this excitement in my lectures.... well...

[@gnanay8555]

Same thing in french universities. Uni teachers all over the world probably need some pedagogical course ^^'

[@hasaniqbal233]

same here in canada

[@yusufkor5900]

It was quite enlightening that you said and showed "Linear transform matrices show the final position of unit vectors". Using this I made my own math revolution. Now I can infer advanced kinematics equations myself. Really thank you so much.

[@liamtolkkinen5025]

This is why you are one of my favorite KZbinrs! You explain things in a way that allows me to visualize concepts that I would not have understood before hand!

[@burakcanik01]

If every "teacher" taught their subject like this, there would be Dyson spheres and whatnot everywhere right about now.

[@pranavkondapalli9306]

while I get where you're going, and most likely would be wooshed, Humans do have the technological capabilities of a lot of things rn. There are a few precursors that need to be taken for building space infrastructure, a lot of the sci-fi things can be scientifically proven possible. The reason its not done is because its hard to get govt funding for space exploration. The only reason we landed on the moon was because of the space race, and the only reason we're going out into space (mostly) is to get a one-up on your adversary

[@burakcanik01]

@@pranavkondapalli9306 Obviously you are right. I was exaggerrating but still, there would be a lot more people having success in STEM fields I think.

[@PiIsRational]

It is a shame that so much is taught in such a poor way. I was looking into what a matrix is, and almost all the results start with "a matrix is an array of number". They are confusing how we represent a matrix with what a matrix is. This video does a great job of explaining what a matrix is.

[@Arstotzka]

No quiero imaginarme cuanto tiempo me tomaria aprender algebra lineal de no ser por estos videos, gracias!

[@keithhoffmeister6722]

[@mauriziobuzzolo3386]

Ti è mai capitato di capire bene una cosa 35 anni dopo che l'hai sentita la prima volta? A me oggi.

[@richardgrosman5798]

This is by far the best visualization of the topic I've ever witnessed in my whole life. Math for everyone 🥰

[@shayboual1892]

I hate how maths is taught in schools with a focus on memorization instead of understanding. This type of teaching with showing the basic concepts that form the rules of mathematics i find much more useful as it garners a much greater understanding of the maths and makes it easier to remember since you only have to remember basic concepts (which make logical sense so its even easier to remember them) rather than seemingly random rules

[@adelfaidah3224]

5:18 i literally paused for a min before he said "this is a good time to pause and ponder"

[@manamritsingh969]

INCIDENTALLY, ME TOO! XD

[@prathamyadav3105]

Same here bro

[@ketanshetye5029]

haha same here !!

[@SreenikethanI]

ayy same

[@mohammadmohsinmohammedmohs9717]

Im not getting it x

[@minecraftmovieman1]

bro im a first year engineering student so ive been getting into how vectors relate to ideas in physics (statics so far), and these videos seem to be helping me really understand whats going on when performing operaitons. This series and essence of calculus should be watched by anyone entering into the applied sciences.

[@nerd2544]

why lenin got that kazoo

[@asatorftw]

These visualisations is mind-blowingly good. This video must have taken weeks to finish. Highly appreciate it. Thank you!!!

[@Agreedtodisagree]

I never buy any merchandise from a KZbinr, but I'm seriously considering now.

[@donaldmannikko5646]

I've taken two formal linear algebra courses. The first spent all of the time teaching mechanics without any of the underlying insight, the second spent all of the time trying to instill mathematical formalism without any of the utility. I really appreciate this approach of explanation.

[@leeris19]

I have loved mathematics for so long. I have even competed multiple times but I always looked at matrices with half of my eyes closed because I can't wrap my head about the "whys" of it although I know a bunch about its "hows". This helped me re learn my fundamentals and now I can safely say that I know matrices not because I was told but because I saw it myself! Thank you!

[@arjunramani9416]

This really was some video. I never really thought I could think of linear algebra this way. It was truly enlightening.

[@anantchopra1663]

This is the by-far THE BEST video available on the internet to understand matrices, and perhaps all of linear algebra. You're amazing, Grant! We all are so lucky to have you teach us all of this!!

[@jptuser]

3b1b please consider a series on tensors

[@Antonio-lt1sp]

Please

[@codefluence]

+1

[@zacharymiller2446]

I would also love to see this... at least a video or two!

[@ayushnema6189]

yep

[@bkzlab]

Yes!

[@hasnainshirazi166]

Goodness, that rotation matrix visualization nearly broke me. Unbelievable job explaining how the transformation of the basis is at the heart of all this, and how the whole grid shifts accordingly. And the visuals! This is what understanding something at a fundamental level must be like...

[@obakasan31]

Best series on linear algebra ever. You’re a life saver

[@b00gi3]

Just going through the play list, leaving repetitive thank yous for this AMAZING CHANNEL! I have to also say, I appreciate the dig in this video at rote learning procedures and stuff in a school setting at a factory-pace without actually understanding the underlying concept. I HATE this about school and classroom settings. I much prefer taking my time with materials like this (although there is nothing quite like this channel!) --- to actually understand. Your explanations and animations and presentations are outstanding. I really appreciate them. Thank you.

[@rrnrafael]

I tried to reproduce the 4:00 explanation on paper. On my iPad, I created the î, ˆj and ˆv. Then I copied and rotated the entire draw so that the ˆv was positioned somewhere else. By the time I tried to figure out mathematically the location of transformed ˆv through equation, the result and the position didn't match. Then I visually figured it out and realised that I made a mistake in the equation. In other words, I used the vector to see what I did wrong and resolved the equation. That was fucking awesome.

[@Charlie-rh8od]

This right here is how every child on earth should learn mathematics

[@shalomcastelino4805]

@@Charlie-rh8od Yes. I couldn't agree more with you

[@parulgupta788]

Can you elaborate how did you reproduce the transformation? I'm trying to do the same but not getting it entirely.

[@parulgupta788]

@@Charlie-rh8od so true

[@sufiyanadam]

My goodness, I felt that! That is exactly how we should learn.

[@tuwadonwang5870]

This is sensation. Even though I'm quite comprehensive about linear algebra, this series make me breakthrough the concept. I'm so grateful that there is a video like this.

[@suhaibkhan3048]

Wow! I love this! Thanks a ton for such a nice explanation. I'll recommend your channel to everyone who is having a hard time with linear algebra

[@Dademaker]

This is THE BEST mathematical instructional video I have ever had. It literally just clicked, beautiful.

[@ricciuccio]

This channel is priceless. It would be interesting to know which textbooks and materials you used in origin to come to have such a clear understanding of the topic. Thanks so much for the fantastic work.

[@zzykrkv8322]

Thank you for making this, I'm in year 10 and we just learnt about transformations using matrices, this video helped me greatly in making sense of them instead of just seeing them as numbers to memorise. I was able to see what something like [8 0 ] [0 -4] meant visually rather than just from memorising facts, which made this new concept much less confusing .

[@l0l1p0p7]

From 4 of your videos I learnt more than all my Highschool years. Thanks a lot

[@nicolaskeroack7860]

at least, knowledge that you can put in practice, and not factorize a fucken number, and then the nsa comes along and invent an encryption algorithm from that school garbage, because its so hard for a computer to compute