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Matrices can be thought of as transforming space, and understanding how this work is crucial for understanding many other ideas that follow in linear algebra.
Full series: http://3b1b.co/eola
Future series like this are funded by the community, through Patreon, where supporters get early access as the series is being produced.
http://3b1b.co/support
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3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted about new videos, subscribe, and click the bell to receive notifications (if you're into that).
If you are new to this channel and want to see more, a good place to start is this playlist: https://goo.gl/WmnCQZ
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Website: https://www.3blue1brown.com
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Text Comments (1426)
## Would you like to comment?

Jacob Hegnsvad (1 day ago)

Are you kidding me. I wanna cry about all the wasted time in class outperformed by a YouTube video.

MAST INSAAN (2 days ago)

Can u please explain the difference between linearly dependent and linearly independent vectors with a totally new video.....I beg of you for this.🙏🤲

George Henderson-Walshe (2 days ago)

To any person who went to school in America. I was curious as to what age this subject is usually taught in your education system. I'm from New-Zealand and I think the standard here is to study this when one is 16-17 but I'm not sure. I ask because my brother and I have been in advanced mathematics classes for pretty much all of our time at school so far but I get the feeling our standards are behind other wealthy nations so I'm not sure how much that fact is really worth.

Just for you (2 days ago)

I THOUGHT I KNEW ABOUT VECTORS BUT IN REALITY I NEED TO GO BACK AND STUDY BASICS FIRST

Daniel Bourke (3 days ago)

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

Alejandro Uscátegui (3 days ago)

You're showing us the art of the universe in such a way, I feel way too much more in love with maths. Thanks. I'll definitely will pantreon you.

Vishnu Prasad (3 days ago)

hey brown and folks please clear my doubt in this video. I haven't understood this transformation concept clearly 1. how really the i & j hat grid lines lands along with the transformed vector 2. how to compute function required to get the transformed vector

Vishnu Prasad (3 days ago)

hey folks and brown thanks if any one worked or read my request. i hit the bull eye. i broke down my understanding and found my fault where i went wrong. thanks brown.

MedTheRockstar (4 days ago)

Your videos are always beautiful!

Bhaumik Solanki (4 days ago)

What will be the new vector, if my new ´i´ will be (0,2) and ´j´ will be (2,0) and my first vector is (2,2).

Hivlik (5 days ago)

That pi student without eyes was some scary shit

Anthony Tonev (5 days ago)

so ihat and jhat are like these arrows in animation programms that indicate x,y (and z) axis ?

himanshu atri (5 days ago)

You are best thing ,I have found on YouTube.

ILoveEggs (7 days ago)

Your videos are gems. They unlock the viewers's ability to see the poesy of our world. Thank you for that.

Sonu Das (7 days ago)

7:43 and that's how you make children fear maths👍👌

Ishan Sinha (8 days ago)

You are the Michelangelo of educational film. Beautiful!

Chris Dejong (8 days ago)

7:40 "Then you can make high schoolers memorize this without the crucial part that makes it intuitive!"
Damn, that was one deep jab against common core.

Tanner Yilmaz (11 days ago)

I just want to say I am so thankful for you. You should be just as famous as Elon Musk. You have helped so many people with your work. Run for president

Gova Murali (11 days ago)

Your videos are amazing..do you have plan to do probablity?

susanius (11 days ago)

The dislikers will never understand liner algebra

Dhineshkumar R (12 days ago)

Oh my God!!!!!! This made my day.

Daniel F (13 days ago)

HOLY SHIT I UNDERSTAND

Den Den (16 days ago)

Бля)) Нахуй я обучался 15 лет?????

윤종문 (16 days ago)

Most helpful movie understanding matrices and transformations

Asfand Nasar (17 days ago)

all this beautiful math content for free. Much love <3. It all finally makes sense

Ehit Karim (18 days ago)

Lets get married

Faizul Haq (19 days ago)

https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/ This course from prof. gilbert strang is very intutional, and there is more mathematics. https://www.youtube.com/playlist?list=PLE7DDD91010BC51F8

Billy Kotsos (19 days ago)

But what is the point in studying linear algebra if you don't know this stuff???
Can you s\really say that you understand the subject just because you know how to use the formulas??

Billy Kotsos (19 days ago)

Took me 10 watches to understand this

이인섭 (20 days ago)

youguys are great. literally.

Syed Inamul Haq (21 days ago)

I found a beautiful corner on youtube

Vu Thanh Phan (21 days ago)

i suddenly love my mathematics bachelor especially linear algebra so much. Haha thanks a lot

Yang Li (22 days ago)

think of matrices as transformation of space

S Walkers (24 days ago)

Beautiful

syed saad (24 days ago)

Doesn't i^ land at (1,-2) and j^ at (3,0), how come at 6:16 it becomes (3,-2) for i^ and (2,1) for j^ ?

Martin Contreras (25 days ago)

i watched this to remember linear algebra from college, but instead i learned more in this shorts videos than in a entire semester.

Martin Contreras (25 days ago)

i watched this to remember linear algebra from college, but instead i learned more in this shorts videos than in a entire semester.

Karan Karnik (25 days ago)

I was confused between thinking of x,y as coefficients of linear combinations of the matrix vs as the general vector in a transformation.. i guess they're different things

Zeineb Haouari (25 days ago)

brilliant! extremely brilliant!
thanks a lot for this beautiful video :)

Vlad Baje (26 days ago)

I just downloaded Jiu-Jitsu into my brain.
Apoc, 3d space simulation please!
Medetation really helps keep focused and imagin all what he said.
And it feels like paradigm shift.

Aviram Yitzhak (26 days ago)

Thank you so much for these videos! they are really interesting and clear. Can you recommend any online source which can be used for practicing the subjects of this series?

Hardik Sachan (27 days ago)

Do you use any library to do these animations or you have coded it on your own?
\

Chris Morgan (28 days ago)

I love you.

Rishabh Chandra (29 days ago)

goddamn give this man a nobel prize. truly amazing.

Ariel Hernandez (30 days ago)

Dear 3Blue1Brown, at 7:40 you said, "Then you can make highschoolers memorize this without showing them the crucial part that makes it feel intuitive." I feel your pain. So WHAT HAPPENED to math education that its so awful at times? I'm a senior Physics undergrad, and sometimes feel like my education and intellect has been intentionally sabotaged my whole life since I was in elementary school. What do you think are the causes? What are the factors?

Quantum Plex (30 days ago)

0:05 lol

AlchemistOfNirnroot (30 days ago)

Think in FP1 (A Level) you do learn matrices in this way (although I got kicked out before I got to that part).

Gao Zhan (1 month ago)

In my country linear algebra is also taught in a disgusting way. But many students know to find and read a lot of good linear algebra books that are out there and also a lot of enlightening videos like this. Youtube and channels like this are really contributing to education.

Shahzodakhon Rustamova (1 month ago)

after watching this video,finally can imagine what i am solving.Thanks, you r great

Isaac Restrick (1 month ago)

Excellent video. I am glad to discover the intuition behind matrix algebra.

Porqueeeporqueporqueporque (1 month ago)

6:23 I came

Porqueeeporqueporqueporque (1 month ago)

"Unfortunately, no on can be told what the matrix is. You have to see it for yourself." really thanks for this :c

Porqueeeporqueporqueporque (1 month ago)

I'm subscribing.

SOURAV MAZUMDAR (1 month ago)

You are God of All Things!!! I am your fan!!

Terry N (1 month ago)

Thanks for the efforts u put into the graphics of these videos for better visualization

APARUP BEHERA (1 month ago)

Thank You Grant :D

V (1 month ago)

I'm amazed that I've been transforming vectors as far back as [Edit: systems of linear equations in] elementary algebra.

my account (1 month ago)

I recently finished a basic course into linear algebra. I love how this video causes the material to actually become intuitive and make sense!
Looking forward to watching the other videos on this playlist :)

Shojib Miah (1 month ago)

I wasted a whole semester not understanding the basics of linear algebra until now. Mind-blowing.

Teddy Juma (1 month ago)

I'm so glad i found this channel. I understand mathematics better than what i'm taught. First year mathematics students. Thank you teacher. You're the best

Ms J (1 month ago)

This is a good video thanks, but there is definitely a gap in the intuition at 7:45 which is how you get from the middle to the RHS geometrically. I don't understand why when transforming planes, a vector's x co-ordinate will move only according to the x co-ords of the transformed defining basis vectors i and j, multiplied by its's own x and y co ords... and likewise for the y co-ord. A visualisation of this would be really helpful!

Aiman Al Masoud (1 month ago)

7:45 Exactly!

Brennan Noble (1 month ago)

Thank you! Thank you! Thank you!

Karan Singhasane (1 month ago)

I am thankful to god that I found this channel at the right time (Engineering degree first year)

Arnav Das (1 month ago)

Thanos should see this video, he will definitely return the infinity stones then

Ujjwal Gupta (1 month ago)

Can u please give a note on Transpose of matrix

Emily Polanco (1 month ago)

When you finally understand Linear Algebra three days before the final XD

jitendra singh (1 month ago)

I love it
Please sir make series on complex number

Jorge Mercent (1 month ago)

@6:10 when you start to use another example of transformed unit vectors, you should state it clearly.

Ax H (1 month ago)

Great video!

Adam Bergman (1 month ago)

wow.....

Nisarg Jain (1 month ago)

9:30 mind-fucking-blowed

Sarah C (1 month ago)

@7:21 what does the ol' mean in "Any ol' vector"

ali_p_q games (1 month ago)

I passed Linear Algebra at uni just by learning the algorithm to solve the questions, because I could not understand what a linear transformation actually was or what use I would ever have for it. Yeah, uni was bad. Great video, though!

Thundergear007 (1 month ago)

9:38
what the actual fuck man, this makes so much sense i am legit mad not being taught this

George Soden (1 month ago)

You are amazing dude

Fransiskus Winaris (1 month ago)

This is the most beautiful video I've ever seen.

Daniel Rio (1 month ago)

😮 🤯 🤩 😅 😭

Jim Williams (1 month ago)

Since I basically live in OpenSim (virtual reality) I find his explanations beautiful, but I kinda wish he would show some of this stuff in "real" terms of what the eye sees in a projection. I don't know how many times I have had to deal with people orienting their houses off the axis and then gone mad trying to put a picture on the wall. (I guess that would bring up the subject of object occlusion...)
We even deal with quaternions...

Shaswat lenka (1 month ago)

WOW! You taking us back to high school again and at the same time proving that we didn't learn a thing! Amazing!

gopsdv (1 month ago)

❤️

Robotic_Llama (1 month ago)

In high school I've taken all of the highest level math classes including calculus (disguised as IB Math 12 tho) and we never learned what a matrix actually is. sad

Hiếu Võ (1 month ago)

I am really really thankful to the people for making this video. I studied these thing at school to do excercise without knowing what it is and where it comes from. That was so meanningless. Now i understand its origin. It's wonderful.

Binary Quantum (1 month ago)

Wow! Gave me goose bumps! :D

Jean diaz-chantraine (1 month ago)

It was a wonderful intuition of what does a linear transformation really represent !! Thank you so much for that!
I actually try to understand what a bilinear transformation represent. Can you help me for that ?

Gagan Narang (1 month ago)

Attention video may gives orgasms...

Jeffrey Wong (1 month ago)

Welp, my exam on this was yesterday.

S D (1 month ago)

You are the best teacher I know

rabago85 (1 month ago)

Wow, I've taken two Linear algebra courses, one computational, and I skirted by linear transformations without really conceptualizing them in a deep way. It finally clicked!

Juan Mora (1 month ago)

7:48 yes,it is.

Juan Mora (1 month ago)

thank god you exist.

Aa Hemmitt (2 months ago)

initially i was like "what mathematician is Morpheus"

cenk cafer (2 months ago)

This shits on khan academy

Selcuk Caglar (2 months ago)

In my university no one could teach that to us.

Damian (2 months ago)

You sir are brilliant

manos sideris (2 months ago)

Great videos!! One remark... In the videos there is movement shown but in "reality" the transformation is only a start - end thing. I was wondering, would it be meaningful to express the orbit of such a "movement" by using functions of coordinates instead of numbers in the matrix? (given that the orbit is a function). For instance the orbit of the "shear" (8:31) can be written as [1 a; 0 1]*[x; y], with 0<a<1 and the resulting family of vectors is [x+ay; y]... just saying...

Aubree Swart (2 months ago)

This channel is so amazing! Thank you so much for working so hard on these superb videos!!

Dan Qin (2 months ago)

Could you explain why the transformation keeps lines straight?

Shikhar Arora (2 months ago)

i don't get it, why does our brain analyze linearly and sense non-linearly (human senses are logarithmic). Is evolution be-fooling us or our education system.

Parthiban Rajendran (2 months ago)

can you please provide a python API (saw your github), where we simply enter the matrices as list, and it outputs the related animated transformation of space? Please please.

yannick tafrey (2 months ago)

the best Explanation ever. could you please make a Video about Affine Transformation, about the 4by4 Matrix that encode Rotation, scale and translation together and what homogeneous coordinatesystem really ist. thank you in advance

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