I don't know who you are, but these linear algebra videos are brilliant. They are pedagogically invaluable and should be incorporated into every introductory linear algebra course. I teach linear algebra and I mention these visualizations but my hand-drawn figures on the marker board, my clumsy gestures in the air, and the textbook's static graphics are all quite inadequate for most students. I will be directing my students to these videos in the future or even playing them in class. Thank you.

@Katy3865

6 жыл бұрын

I know you wrote this a year ago but as a tip, my Maths teacher often took pens to show vectors in 3D space and sheets of paper to indicate how the plains would cut. I think this visualization is important and also a good method to engage students to think about how the graphs would look without always drawing them first. The transition between 2D to 3D was hard on some students and they just couldn't visualize it when drawing on sheets of paper. This helped a lot and was fun, I still do it sometimes.

@QsHsNation1

6 жыл бұрын

Thanks for your contributions to educating the public :)

@wedeldylan

6 жыл бұрын

*pedagogically*

@seyiojewale7907

5 жыл бұрын

He is a professor. That Phd don't come cheap

@davidhofmann4857

5 жыл бұрын

Very nice!!! Thanks a lot

I'm an astrophysics phd and I use linear algebra everyday but I'm here watching these videos because they are so intuitive...

@shreyansj2703

3 жыл бұрын

true....i'm too phy phd from harvard but i like these videos too...its a good source material to give for the students.

@cephalopodtime6167

3 жыл бұрын

What do you study and/or research?? I am a high school student very interested in physics and mathematics and would like to know

@sastashroud7646

3 жыл бұрын

@@shreyansj2703 these types of videos are useful to harvard students ??

@ergpopler413

3 жыл бұрын

@@shreyansj2703 harvard is that community college right?

@isaacmandell-seaver7223

3 жыл бұрын

@@ergpopler413 nah I think it’s a middle school

1. a coordinate is a scalar, which scales the basis vector of its coordinate system 2. anytime we describe a vector numerically, it depends on the basis vectors we are using 3. a linear combination of a set of vectors is to scale them and add them together 4. the span of a set of vectors is the set of all linear combinations of the vectors 5. if a vector is a linear combination of a set of vectors, the vectors are linearly dependent 6. if each vector adds another dimension to the span, the vectors are linearly independent 7. the span of two linearly independent vectors is the 2D space, if the two vectors line up, their span is a certain line 8. when thinking about one vector, think of it as an arrow, when thinking about a collection of vectors, think of them as points 9. in three-dimensional space, the span of two linearly independent vectors is an infinite flat sheet, the span of three linearly independent vectors is the 3D space, if linearly dependent, the span is still a flat sheet 10. the basis of a vector space is a set of linearly independent vectors that span the full space

@radoslavradosavljevic7980

11 ай бұрын

thanks anime man

@XxGuiGax

11 ай бұрын

Thanks!!

@nishanthbhat6652

9 ай бұрын

❤

@porkypig7170

8 ай бұрын

You can generalize point 9: the span of a set of vectors is a subspace of a number of dimensions equal to the number of linearly independent vectors in the set

@m-sq

7 ай бұрын

7. ...if the two vectors line up, their span is a certain line, and they are linearly dependent

I am still in 12th grade and this is all I had on Google Play rewards money. But as soon as I come of age(later this year), I will save up money to send you dude. Stay at it bro

@jayvaibhawverma

3 ай бұрын

Seriously, the way he explained visually blew my mind. I was having such a hard time visualizing all the terminologies; I could understand them based on the definition in the textbook but was getting such a hard time having a clear mental image. I can't thank him enough. I would also donate, once I get a job.

@thomas_prada

17 күн бұрын

based

So many years of 'rigorous' linear algebra, but I still didn't have a good understand of the intuition behind it. Grant, you are a miracle worker. So happy to see you ended up in the math education field! fsc

@3blue1brown

7 жыл бұрын

FSC!

@iOsamaAbbas

7 жыл бұрын

Same here. I'm really impressed!

@abpnd

7 жыл бұрын

This is (understated) the most intuitive that algebra can get. Its like I am learning again and can teach my niece when shes in high school :-)

@hypnovia

6 жыл бұрын

What is the difference between rigorous and vigorous? Doesn't the latter seem more friendly?

@TheTariqibnziyad

6 жыл бұрын

rigorious 😂 😉

I don't know who you are, but I will find you and I will thank you

@xXxIMMORTALxXx

5 жыл бұрын

This is him : www.3blue1brown.com/about

@doyoulikedags3534

5 жыл бұрын

It's supposed to be a reference to Taken, Ed Ed.

@__-yz1ob

5 жыл бұрын

@@lenkapenka6976 How does that have anything to do with being Indian?

@someguy4592

4 жыл бұрын

kzread.info/dash/bejne/ipeuj9SPgNmyYrw.html That's you when you meet him

@vevo5086

4 жыл бұрын

@@Masardirasa i like people like you hhhhh

Thank you! This is probably one of the most beautifully explained videos ever, your voice and animations are incredibly helpful to understand and enjoy the video👏🏼💐

MIT undergrad here. Your video just taught me in 9 minutes what my math professor and teaching assistants couldn't in the past 2 weeks. You're amazing thank you!!

Complex Algebra: "We call the vertical unit i." Linear Algebra: "We call the horizontal unit i."

@Renisauce

4 жыл бұрын

Every person in the universe using cartesian coordinates: Horizontal is x. Surveyors: Let's say that's y instead.

@totheknee

4 жыл бұрын

@@Renisauce Are you for real serious??!

@master1900mc

4 жыл бұрын

But it has a hat in top!

@ENLY10

4 жыл бұрын

@@master1900mc big brain

@covariance5446

4 жыл бұрын

@@master1900mc Statistics: Let's use a hat to denote the idea that this is a predicted value, not an actual value!

Found this series 5 days after taking my Linear Algebra final. It's nice to finally understand what was going on the whole semester lol

@terryjones573

Жыл бұрын

Moood

@luisv8887

Жыл бұрын

duuude lol!

@Lvxurie

Жыл бұрын

I'm studying for my final now and am embarrassed that up until now i have had no idea what a basis and span actually were. They are such simple concepts but were explained to me in an unintuitive way. So glad i found this channel!

@bernardparinas4297

11 ай бұрын

totally a savior lol

@puneetmishra4726

8 ай бұрын

You bro are a legend 😂😂

I'm just gonna binge this tonight instead of netflix haha

@citrus4419

2 жыл бұрын

@Priyanshu Guha unless you’re watching Dark

@epicspartan10

2 жыл бұрын

@@citrus4419 Dark is so underrated maaaaan, its such a coincidence that people in this comment section like it too

@epicspartan10

2 жыл бұрын

@@citrus4419 es wird weider passierein

@personzorz

2 ай бұрын

My turn

"as you scale that new third vector, the sheet moves through the entire 3D space”, that’s the kind of thinking that pushes people to understand the concept for themselves. Thank you so much for your videos!

I literally left my Cornell math support tutoring crying, feeling worse, but some girl stopped me to direct me to your videos. Thank God for her and for your videos, bc I was on the verge of a breakdown

@brimussy

3 жыл бұрын

aww bless her

@syedmohammadhussain7137

3 жыл бұрын

fucking cornelll doesnt even teaches properly!?i thought it was only my college

@-danR

3 жыл бұрын

Cornell: "All your basis vectors are belong to us"

@heartbrokendra

3 жыл бұрын

@@syedmohammadhussain7137 Just bc I didn't understand doesn't mean everyone else didn't haha

@wizhaa

2 жыл бұрын

Uh oh I’m taking 2210 linear algebra next semester

The peaceful music really helps set aside the onset of anxiety that usually comes at sight of numbers and equations.

@aashudwivedi

5 жыл бұрын

Thomas Thomas I watch one of these videos before going to sleep. Love the music at start.

@1fareast14

5 жыл бұрын

you'll love this album then: vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown

@tonyvercetti2123

5 жыл бұрын

but why numbers and equations make me calm?

@EduardodaSilva00

5 жыл бұрын

That part makes the video even better

@alrayyaniQtr

4 жыл бұрын

I’m a geek so the sight of equations is what gives me peace. What’s funny is that I’m here because I had a tough day and I need to relax lol.

4:00 span of two vectors 6:50 two vectors in different plane. your span is the new plane 7:48 third not on the same plane as other two (ie their sum) means it unlocks every 3 dimensional movement in that third v direction 8:33 linear dependence

@lukes5631

2 жыл бұрын

Lets get this up voted!

@kannahsanthosh

Жыл бұрын

thank u

I'm a school dropout and I could never understand math. I thought I was stupid and had no talent for it, until I found this channel, which to my surprise helps me understand! This is better than any math book I've ever tried to conquer. The video format bypasses my mental block which interferes when I sit down with pen and paper. I feel like these videos are teaching me the general process of math and its nature of problem solving, so with this I can finally learn to self-learn. Education has made some amazing advancements, and (at least my local)school system seems to be lagging behind. I have never learned from teachers and homework, and writing with pen and paper. There must be more people like myself out there that need to be shown that there are alternate methods to learning which might suit them better.

I study at one of germanys top engineering universities and your way of explaining this is so much more superior than my univeristy professors.

@Epii_

3 жыл бұрын

Aachen? Xd

@chrissid.3763

2 жыл бұрын

KIT?

@KatzeMelli

2 жыл бұрын

I am also studying at a "Elite" University in germany and the quality of the lessons is abysmal. These videos are not only a life saver but also sparking a love for math that i never knew i had in me.

@xxmiamygirlxx

2 жыл бұрын

Haha same.

@NachoSchips

2 жыл бұрын

TU Munich or KIT?

The definition makes sense since a linearly independent vector (no matter how matter how many dependent vectors there are) unlocks exactly one new dimension. 1 vector can describe all of 1d space with a scalar of some kind, but only 1 space. Adding a linearly independent vector of that one unlocks another dimension and so on. If we were to add a linearly dependent vector we would not get a new dimension no matter how we scale it. awesome video btw (i hope my comment was readable)

@Deevil992

6 жыл бұрын

I am trying to understand the question as related to what he described about the basis at the beginning of the video. At the beginning, he described when we have two vectors, with a linear combination of two vectors, each scaled with one independent scalar, the entire coordinate system with a dimension of two can be captured by just those two pairs of vector and scalar. The span is the entire space of this coordinate system, and the linear independence is simply saying that those two vectors can do this job of capturing the whole space. Then when we think about what are the conditions to make them linearly independent, we see that they just can't be described by each other. When thinking until this point, I just realize that linear independence is probably the most efficient way to expand the space of possibilities! It's like when two people are linearly independent, they can achieve much more possibilities than if they are not linearly independent, including nonlinearly independent and linearly dependent, with decreasing possibilities! (Trying to find someone in the comments that answers his quiz and force myself to think. Thanks for the comment!)

@EmapMe

6 жыл бұрын

Couldn't a vector (x,y) reach every point in a plane? Why do you need two vectors?

@nopenopenopington

6 жыл бұрын

Jearl Price consider a 2d plane with only one vector, we can scale it however we want but it wont reach all the points in the plane, it will just create a single line across the origin

@MRxPoundcakes

6 жыл бұрын

Kristoffer: I think i see what you mean but you must be careful with the definitions. Remember that a linear combination of a set of n vectors, call the vectors X1, X2, ... Xn, is a vector Y = aX1 + bX2 + ... + cXn, where the coefficients are real numbers; that is, it is the result of scaling all of the vectors by arbitrary real numbers and then adding them together. If we have just one vector, call it X, then a linear combination of X is Y = aX, where the coefficient a is a real number. Now the span is the set of all linear combinations of the chosen vector(s). So if we just have one vector, again say X, the span is the set {Y = aX : a in R}, which is just a straight line (since scaling a vector can at most reverse its direction; it can never break out of that line). We must have another vector that is not a scalar multiple of X to break free from this line and span the plane. I hope this has helped and isn't confusing. It would be easier if we could embed Latex or somethign like that in our comments to make the notation a bit more clear.

@ainjeffery

5 жыл бұрын

It can but here we are talking about how many points in the space we can reach using the vector operations. If you were to just simple add random points and keep adding them in the ordered pairs (x,y) you can get any point in the space but you are not doing any operation here.

I would like to leave an appreciation for the fact that you start with something most students are familiar with, develop our intuition and finally provide the definition. School teachers please learn how it's done. I cannot stress enough how helpful your videos are, thank you! Greetings from Portugal

After 12 years,I realised why it's called linear algebra!! Hats off to you sir..

@lackdejuranez7084

3 жыл бұрын

@Thomas White what

@nilen

2 жыл бұрын

what

@youtubeviolatedme7123

2 жыл бұрын

what

@sinekonata

Жыл бұрын

What? He specifically said it wasn't the actual etymology though... Did you actually look up the actual etymology?

@imartinez303

Жыл бұрын

i hat or j hat?

These videos are a gem. They must be preserved for all eternity!

@Gravitraxer_AangCZ

3 жыл бұрын

68th like ...nice

@MoMo-ue9ff

2 жыл бұрын

I agree

@AA-vj3si

2 жыл бұрын

@@Gravitraxer_AangCZ 169th like …nice

@shalinidas5459

Жыл бұрын

Thats what I was saying right now! Exactly this!!!!

This channel is so damn underrated

@Nik-sv1yw

5 жыл бұрын

Agree ! But 1.2 million subscribers for a Math channel, not bad I would say.

@coffeedude

4 жыл бұрын

Math is underrated :(

@belalnoor9686

4 жыл бұрын

@@coffeedude underrated comment.

@random-0

4 жыл бұрын

Hi I'm future, don't worry he got 3M subscribers now it's not underrated anymore

@theblinkingbrownie4654

4 жыл бұрын

@@random-0 He needs more subs then T series and he still would be underrated, it's gold.

The basis of a vector space is a set of linearly independent vectors whose linear combinations can span the whole vector space.

Holy crap. I have been hammering through my textbook and lecture notes on spans, linear dependence/independence, and basis, and I feel like I’ve had my mind blown by the intuition you gave me by showing the math graphically. Everything makes so much more sense. Normally I don’t comment like this on other tutoring videos but this is soooo helpful

You managed to explain in less than 10 minutes what my professor failed to explain to me for half a semester. All linear algebra students should be required by colleges and universities to watch your videos because your videos just cut to the important stuff without any unnecessary BS.

@zokalyx

6 жыл бұрын

I love 3b1b videos too, but unnecessary BS is needed for rigorous math. Obviously, not all people want to know rigorous math, so your comment is totally valid.

@shipwreck9146

5 жыл бұрын

@@zokalyx I would like to add to this. What he's referring to as "unnecessary BS" is what visually thinking people would usually think of math that is only taught with definitions and logic, with no visual understanding to reinforce it. There are some that learn better with that method, but others that learn better visually. This is why I think that many math courses should have a visual intuition version and a logical version. Both teaching the exact same thing, but in a different way. Then students can choose which one they'd do better in. I'd imagine that at least 50% of people who say "I'm bad at math" only say that because they weren't taught it in the way that was best for them.

@hungdo6397

4 жыл бұрын

@@shipwreck9146 To even futher expand on this. I think it's wonderful to start with a good intuition before diving into mor rigorous math. However, the logical version is really necessary to understand more generalised statements, very intuition can certainly guide you but can fail you as well or might be outright impossible.

@shipwreck9146

4 жыл бұрын

@@hungdo6397 That's also a very good point. After re-reading my comment, I should have specified that you can't learn everything from math just by seeing it. But rather that by seeing it, it reinforces what you're doing with the math.

@hungdo6397

4 жыл бұрын

@@shipwreck9146 Definitely agree! For example having a firm grasp and intution in Linear Algebra certainly helps understanding Functional Analysis, but relying solely on intution will fail you there.

Best material on linear algebra. Thank you so much.

As a 41 year old software engineer I thank you for this outstanding playlist allowing me to understand things in a new, more practical and visually stunning (as well now visualizing) way!

Never could have known linear algebra without you

I'd love to see an 'essence of geometry' series

@markdzsucck1679

6 жыл бұрын

I would like some basic euclidean 2d geometry with some problems added in.

@zairaner1489

6 жыл бұрын

Euclidean geometry? Algebraic geometry? Differential geometry? Non-archimedean geometry?

@igorvinicius8087

6 жыл бұрын

I prefer abstract algebra hahaha

Let’s be honest: y’all watch this for the dialogue “I know this already” “Ah but young padawan, all knowing, you are not. A subtlety, there is.”

@troybaxter

3 жыл бұрын

I usually watch for the dialogue, but this time I actually watched for the educational content. This video made so many things much clearer for me.

The teacher I never had

im actually screaming this made so much sense like i think i get it now my mind is blown

Today's generation is blessed to have this channel.

Wow, I didn't expect the videos in this series to come out so fast.

@MCMasters4ever

7 жыл бұрын

Well, he did say for the first 5 vidoes one every day.

@Wander4P

7 жыл бұрын

Oh... I guess I missed that.

@gnanay8555

7 жыл бұрын

He said it : 5 videos in 5 days, then one every 1 or 2 weeks :D

@zokalyx

6 жыл бұрын

i think he does them all before posting them (i would do that to reduce pressure and more quality)

I have just learned about vector both in physics and computer science but no one could explain as clear and understandable as you. I love this playlist "Essence of linear algebra" and every video you make. Hope you will get more subscribers, I will share this channel to my friends.

I want all my mechanical engineering brethren to watch this whole series. 6 years and still the most intuitive linear algebra material I've ever come across.

all linear algebra lecturer should teach us like this so that student can get the picture about it ,not just read the definition.

@instaminox

7 жыл бұрын

I agree, I wish they did ... but since the educational system has not changed since the big bang than the chances are very small that it will be applied for the future generations. #sadtruth

@roielia2

7 жыл бұрын

AOJ keygen I very much disagree, I think it's wrong to think about vectors in the geometric sense in the beginning, you need to start from the abstract to the geometric examples, otherwise you won't be able to think about the abstract cases correctly.

@PENDANTturnips

7 жыл бұрын

Better yet simply replace all of these leeches who, regurgitate the same poorly thought out material every year, with khan academy and this dude.

@PENDANTturnips

7 жыл бұрын

+ועי איליה I couldnt disagree more. Most people arent smart enough to deal with very abstract topics from the getgo and it all just boils down to a series of uninteresting operations that they forget as soon as they pass the course. I also dont understand your point about learning geometrically first being a crutch when moving to the abstract, that makes very little sense. People like seeing the real world aplication before they move onto abstract stuff. Not to be rude but its people like you who are responsible for poor education in the world, too many professors teaching difficult abstract topics first before showing the real world use of them, which is why videos like this exist.

@TheIsrraaa

7 жыл бұрын

You have to think of it in an "abstract way" too (or abstract algebra way) because in a space greater than R3 it's actually imposible to visualize it.

One 10 minute video explained the basics of Linear Algebra better than an entire semester at college. Thanks Grant!

Many beautiful words were said about how helpful your videos are. I'll just say, this is one of the first times when I'm watching an "educational" video without having my eyes wandering to the "recommened" videos. You sir are doing a better job fighting ADHD than any pills I've tried!

I studied linear algebra in one of my college courses. Towards the end I did manage to figure out what all these things mean and how to apply them, but holy shit why can't we just receive an intuitive explanation first and then go into the math.

The lecturers on the linear algebra course at my university recommend this series as supplement. And I see why. It especially helps me to visualize what's going on.

At the time I was at grad school, there was not a KZread channel like this. You have no idea of the valuable public service you are doing with these videos. Thank you a lot and congradulations on the great job!!

I am a computer science student and never once in class the professor told us how and why we use vectors in CS. So it used to be something I dread cause of the lengthy and complex definitions they give. You are a life saver sir. I can't describe how much this helps me.

This man singlle handly carrys every 1. Year Physics Student during pandemic through their exams. Thank you a lot!

You opened my 3rd eye, enormous respect and love for you teacher.

I am 40 yrs old learning this for the first time....this brings a sense of completeness and tranquil to my life.....even if so in tiny bits. You should be proud of your service to mankind (and kids kind !!)

6:35 that flat sheet visualization really hammered home the span concept in 3D space

I love how the intro video to this series and this series you pointed out the difference of how physics, math, and CS students view things. I love how you break these things down bc it helps me digest this so much easier!

I have never let my schooling interfere with my education. ~Mark Twain (1835 - 1910)

@tishaanants

3 жыл бұрын

Omg words of wisdom 😂👏

I'm currently taking a linear algebra course in college and was trying to visualize how vectors act in 3 space, especially when you say it's dependent or independent. The book doesn't help and the instructor can only do so much, but I'm glad I found your videos. You do an amazing job and I'll keep watching as I learn the new topics to better visualize, and I will definitely recommend this to my classmates

As a person in architecture + design, I am so grateful for these videos. I missed math after so many years of not needing it, and wanted to teach myself linear algebra. These videos show that thoughtful design can enhance everything that we do, especially learning. Imagine if every class we ever took taught concepts like this... The movement, simplicity, and clarity made the topic not only easy to understand for a visual learner, but also soooo fun to learn. THANK YOU

You can't imagine how much this was helpful fo me! I'm a chemistry student and I'm studying linear algebra and through your beautifully made videos with these really convincing animations you have helped me a lot. I was just studying the maths behind vectors, matrices, span, basis etc.. without something visual I could refer to. Thank you 3blue1brown, I think we are all really happy to have you for free on KZread going through all this work to help us understand better. We love you man!

This is an absolute pleasure to watch. While taking linear algebra, I had to form most of these mental models on my own-glad to see them illustrated so well!

I've watched this series out of interest about a year ago. I did not see how helpful this would turn out to be! (Taking Linear Algebra 1) THANK YOU, thank you, a thousand times thank you.

@MAGINOKU

6 жыл бұрын

Dor Peled me to im talking linear algebra right now and I have to show a proof on the relation of the range being a subspace of the span and I had to go back to these vids to get an understanding

every video, I try to write each valuable Information to craft it in my mind, I end with writing each possible word you spoke.

This has definitely opened my eyes, I love how you explain how linear algebra is being used and how you demonstrate each aspect of it. This definitely going to help me throughout my class.

These videos are absolutely incredible-you’re spoiling us! Also, someday, for the coders among us, you must show us your workflow in detail! Please?

@3blue1brown

7 жыл бұрын

This is definitely the question I get asked most. Perhaps one day I'll make a video on it, but the workflow itself changes. Once I'm finally at steady state maybe...

@larrykaufhold6108

7 жыл бұрын

You know a steady state will never happen! You can't get nothing from something! Schrodinger. That's why there are three blue and one brown!

@TJGalloway1

7 жыл бұрын

What language do you use?

@GriffinCalme

7 жыл бұрын

It is python, his GitHub is github.com/3b1b

@TJGalloway1

7 жыл бұрын

Great. Thanks for sharing.

the number of "eureka" moments in your videos is incredible ^^ these videos are brillant

The way you know somebody is good at teaching their subject is when it can make a newfound appreciation and love in their students for the subject. I really love math when I see it through your lens, and I'm so glad I got to be able to have you share this perspective.

finally, after graduation, i got intrest in mathematics👀😃😃

@naveenkumardasari

3 жыл бұрын

same with me broh

9:02 "If each vector adds another dimension to the span, they are linearly independent"---LIGHT BULB moment for me. Uni professors/GTAs couldn't explain this simple thing. Hats off to you sir!

This is the best way to learn linear algebra. Thanks for uploading this series. Wish there were some videos for multivariate calculus on your channel.

@harshrajkamal3943

5 жыл бұрын

Sorry for the late comment, 3b1b has the multivariate calculus covered in Khan Academy.

I wish I had you as my Linear Algebra teacher. You explain so clearly in 10 minutes a concept which I did not get the entire semester, despite being able to mechanically solve more difficult math problems.

The imagery and verbal descriptions are so beautiful and smooth. I love the way you paint these topics

Pure gold! Thanks for these!

I never fully visualized everything I've learned in math through high school and college until now, your videos are amazing and saving me for my linear algebra class!

Your end slide captures the essence of what makes your videos (3Blue1Brown) and another KZread channel, QuantumSense, so brilliant. You start off presenting the intuitive concepts, and then carry the viewer into the fuller picture. This is THE right way to teach. Too many others begin with abstract, sterile definitions plucked from nowhere, proceed to prove abstract theorems, and only then give the student any sense of what all of this means. No wonder students find these subjects daunting.

This is so much better than what most books and lecturers do: throwing a rigorous definition at you, and then working on examples with the definition that we barely understand, hoping that we can learn as they do the examples

@hhtd4554

2 жыл бұрын

I really hope those lecturers can put down their pride and learn from this video. Or even just show this 10 minute video to their students. Saves students’ brain cells and saves their office hours

You just took us to another dimension. thanks for your efforts.

These videos are absolutely amazing. They're making the geometry of linear algebra not only clear but beautiful.

You're opening me a world. Definitive explanation of this subject, nothing more needed. Humanity should gratefully thank you!

This is a beautiful series. I have been studying linear algebra for the last couple of months, and while I did understand the theorems and the algebra itself, I really did not understand what it all means. Those geometric representations really gave me a better understanding of what I am actually doing. Thank you so much!

7:54 really helped me visualize that concept

Next one's gonna be sweet, can't wait!!

This series of videos has increased my understanding of linear algebra more than a whole week of my college class. I cannot stress enough how fantastic these visuals are. This video has cleared up days of confusion.

This video made things 10 times easier. Many thanks.

The animations leave things waaaaaaaay more understandable

Because basis vectors are linearly independent, does that mean basis vectors always have to be at right angles to each other?

@3blue1brown

7 жыл бұрын

Good question, if two vectors are "linearly independent", it doesn't mean they are at a right angles to each other, it just means that they *don't* lie on a single line. So two vectors at a 5 degree angle from each other, for example, would still be a basis for 2d space.

@JM-lh8rl

7 жыл бұрын

frozenbagel16 No, it just means they are not collinear. When you talk about 3D space, it means they are non-coplanar.

@hehythrjkljwqoi

7 жыл бұрын

I had to go so down to find this question, pin this comment, will be helpful to others.

@MAGINOKU

6 жыл бұрын

frozenbagel16 I totally agree with this question I had the same thoughts in the video but awesome question

@NicolayGiraldo

5 жыл бұрын

The answer to that question is not, but you can manipulate a basis and change the vectors so they are perpendicular one to another. This is called orthogonalization.

I've been watching this series in awe for at least 4 years, and now I'm finally far enough in my education that I am taking linear algebra and I feel like I'm only reviewing stuff because of you. thank you

This man is carrying me in my college Linear Algebra course.

pure joy who needs netflix!? perfect! 20 years later after my studying but it's what I should have watched back then!

Took two semesters of linear algebra and honestly passed those classes by just memorizing the patterns in how to solve the problems. Never actually gleamed any knowledge on any of it which is a shame. Thanks for these vids.

Amazing to see someone make math interesting

I'm a student from Vietnam, we study math rigorously throught tests and exams but I still can't learn such concepts as intuitively as watching your videos. Thank you, for your effort.

Thank you for incorporating images and animations in your videos. It is impossible for me to understand a linear algebra concept without me seeing it in a geometric form.

Amazing! This single video has taught me things I was unable to understand in 3 hr lectures of linear algebra class. Thanks a bunch! P.s The quiz was very helpful in deepening my understanding.

Theses videos should be mandatory in any Linear Algebra class! It's amazing to see that linear algebra is way more than a bunch of abstract concepts. Finally understanding linear algebra, thank you so much!

I have never commented something like this on an educational video, but this is insanely helpful. I've always been good at math, and after taking through calc 3, linear algebra is the first course where I REALLY need something like this to help me visualize it to understand it. Nothing like this is ever shown in the textbook or in class. I'd be stuck with mindless memorization without this! Thank you

I love that classical music. It adds even more joy to the joy of actually understanding the simplest Linear Algebra concepts :)))

the quote from Angus K. Rodgers in the beginning of the video make me cry. it was just beautiful

My teacher suggested your channel. Been watching your videos in regards to machine learning like a few years ago if I recall correctly. It's awesome how teachers are learning to suggest videos like these. Since I bet you have used quite a bit of time to figure out how to come up with an easy to show solution and short and precise description. It's a lot better than a teacher who stumbles while speaking since it's a lot different to present live than on a video.

Man.. more than 20 years after leaving school, watching a video in a foreign language, I understand more than I ever did in school in my native language. You are a genius.

This is amazing. Thank you so much, sincerely, from a student eager to learn but didn't know where and how to start at linear Algebra . Thank you

For the first time I understand linear algebra intuitively. What a great tutorial.

You are the definition of what a teacher is supposed to BE. God bless you sir. I finally am understanding this. Thanks

Simply "WOW"! I've been working with matrices (from a pure mathematical perspective) for a long time, without actually fully understand the intuition behind the terminology and operations. Now I'm super impressed. That's really an innovative way of teaching! Best wishes :) You're a genius!

Got a test in linear algebra this friday and u literally just went over basically everything on it 10x better than my professor.

Thank you for your high-quality video plus the explanation was so easy to follow. I wish this was how I had been taught in college, then I'd have not forgotten those valuable concepts!.

Holy moly for a whole year I had to memorise this without understanding it until now wow! I had to sit there with my jaw dropped for a few minutes after you mentioned linearly dependent vectors 😭 It all makes so much sense now!!! Thank you for the clear and very intuitive explanation!

This video has singlehandedly revived my choking intuition while wallowing in the swamp of a disorganized linear algebra textbook. I almost cried when it finally clicked for me. Forever my favorite channel on KZread!