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

16658 ratings | 861042 views
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 ------------------ 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 Various social media stuffs: Website: https://www.3blue1brown.com Twitter: https://twitter.com/3Blue1Brown Patreon: https://patreon.com/3blue1brown Facebook: https://www.facebook.com/3blue1brown Reddit: https://www.reddit.com/r/3Blue1Brown
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Text Comments (1278)
Kate Yeh (15 hours ago)
These videos are AMAZING thank you
shreya rajput (21 hours ago)
Beautiful!!!
Aniket Sharma (1 day ago)
The perspective you show to any mathematical phenomenon is just mind-blowing and so interesting. I never thought matrices could mean so much more than a bunch of numbers representing an equation. Please write a book on Mathematics
Yusuf Limdiwala (1 day ago)
Absolute Delightful i feel after seeing your amazing videos. Thanks a million Sir.
Rahul Deora (1 day ago)
[[3,0],[0,-2]] This matrix is adding a to y component to the unit vector [1,0] but how is it giving it a direction that our original never had? Also the matrix [[3,0],[0,-2]] transforms [0,0] to [0,0] and [0,1] or any [0,a] to [0,0] therefore points in this new space are not unique. How is this consistent with the stretching of the axis metaphor in the video?
lahiru udana (1 day ago)
Thank you
I cannot comprehend at 8:34
Shi Rou (2 days ago)
if this guy had been my math teacher, i might have (very possibly) went to a much better college.
Tiexiao Wang (4 days ago)
PLEASE BE MY PROFESSOR!!! MY CURRENT ONE IS NO WHERE NEAR AS GOOD AS YOU!!!
Damn Slash (4 days ago)
this blows my mind!
Mohammad Jasim (5 days ago)
woooow i learned matrices at the school and at the university they were very komplex und important and i had no idea what the fuck are they. All I knew that they are some numbers in brackets! Thank you very much.
Sebastián Alegre (5 days ago)
i love u <3
Krishna Kumar (5 days ago)
who the hell are you man! Beautiful mind!!
Ariade White (6 days ago)
awesome and hats off..thats the best 11 minutes i ever had with these arrows
At 9:14 I believe I just understood what makes the determinant negative.
Matan Kribus (7 days ago)
so the transfomation is a change inn the space where the vector "lives" yea? how do i distinguise from a vector and a tranfomation?
Aditi Mishra (7 days ago)
yess...thanku so much sir ..u give me right idea of linear algebra...thanks alot..its really beneficial for me ....
Crazy World (8 days ago)
One word:WOW
AmbieWolfie (8 days ago)
I love how your videos have the extra bits at the end... like when true fans wait for the "after credits scene"!
thebullybuffalo (9 days ago)
This guy gives great intuition but I wish he connected the math to the intuition more. After I get the intuition from him I always have to look at the math and see how his intuition falls from the math. It's an extra step for me that he could explain in his videos. For instance, why does being closed under addition and multiplication mean to that the grid lines must be straight and parallel?
Swarnaditya Maitra (10 days ago)
I really don't understand why anyone would dislike this video....
kirienitram (10 days ago)
holy shit my teacher sucks
Ho Ting Tang (10 days ago)
i freaking love these videos!!!! it makes so much sense than what uni teaches me
Pranjal Daga (12 days ago)
I LOOOOVE THIS!!!
Orocimaro Say (13 days ago)
such a good video i really like it
Jr Animator (13 days ago)
my mind is totally blown.
Osir isrex (14 days ago)
It’s sad how many math students will never be able to appreciate the beauty of linear algebra beyond just lists of numbers in square brackets. Videos like this need to be taught more often. Anyway, great job a usual!
Stefano Bregante (14 days ago)
I passed this course with out knowing the beautiful intuition behind it. To put it bluntly, I passed it with out knowing shit, just memorizing what and when I have to compute in order to get the right answer. Now I have built a basic intuition, I find this subject fascinating. Great teacher. Thanks!!!
Ajit Nepal (14 days ago)
I think this is basic behind Minkowski Spacetime diagram🤔... It took me almost one hour to go through this video
Isnan SunuYuntaro (14 days ago)
Those who dislike this video deserve a place in hell
Isnan SunuYuntaro (14 days ago)
This video is my holy grail
Sushanth Kapisthalam (14 days ago)
Any prerequisites for this playlist?
Nithin Prakash (15 days ago)
You are awsome!! all your videos which i watched are so cool!! things explained in a way human mind can process.. great job and many thank you.
KingMJAH (15 days ago)
if i understand this right than you could you also divide the matrix to find out where the untransformed point would have been ? and if so what happens if you pick a matrix that transforms 2d space into a line and than pick a point not on that line to divide by the matrix? dose it involve the 3rd dimension in that case or is it impossible ? does it have any practical implications?
birawaich (15 days ago)
Got you recommended by my Linear Algebra assistant. Linear Algebra has never been that relaxing!
MRGCProductions20996 (15 days ago)
Educations systems have so much to learn from you. I dream of the day all teachers evolve into teaching like a father teaching his son how to make a tent.
EPISTEMOLOGIST (16 days ago)
just curious, how long does it take you to create the visuals for one video. This is some serious quality work!
siddhartha chatterjee (16 days ago)
would love to see something on complex no
Max Jönsson (17 days ago)
This video blew my mind, simply amazing!
Anonymous (19 days ago)
I WILL SUCK YOUR DICK YOU SAVED MY LIFE!!!
ffakke (19 days ago)
That video is just wonderful.
Anupam Raj (19 days ago)
At 9:10 when you said to imagine the transformation yourself, I was far away from the right movement...damn these animations are a gift.
Alexander Engelhardt (19 days ago)
How would the function for the graph at 3:05 look?
bigolcookie99 (20 days ago)
Anyone else just watch these videos for fun?
ProfX (21 days ago)
Every time he mentions transformation reminds me of TRansFormer.
benjahnz (22 days ago)
A masterful reduction of linear algebra. Truly great work.
yugal sharma (22 days ago)
Sir thank you very much You are doing a very very very great job I was taught matrices earlier but No one told me how beautiful they are.
Neato Electro (23 days ago)
Why can it not make this much sense in class?
Dias Dauletov (24 days ago)
Why should we transform plane? While we can change the values of scalars and indicate any direction in plane??????????????
Piyush Majgawali (25 days ago)
why is middle pi at 1:14 not smiling?
Ankush Girotra (25 days ago)
7:40 lmao for real though I had no idea why matrix multiplication worked the way it did until I watched this video.
laghoule (26 days ago)
infinite grid? WTH All the people here are full of shit. Your explanation sucks, not intuitive, waste of time.
Sarvesh Thakur (29 days ago)
How to interpret matrices which are 1) Not Square Matrices or 2) Of order higher than 3X3?
Samin Chowdhury (29 days ago)
Is this how activation functions in Neural Networks bend the feature space?
Betty Barrett (29 days ago)
It seems that n by n matrices describe linear transformations in n dimensional vector space. However, if matrices are fundamentally based on describing linear transformations, than what linear transforms do non square matrices represent and how? I think perhaps non square n by m matrices describe transforming a m dimensional vector into a n dimensional vector. If this is the case, are all the basis vectors of the input space linearly independent to any basis vectors of the output space? Is it impossible to visualize transforms described by non-square matrices as points sliding between points?
Sumit Limbu (1 month ago)
It would be better if you give problems related to the topic to have better understanding... great explantion💚💚
PrAndonuts (1 month ago)
I could watch your videos all day
Siebe Albers (1 month ago)
Linear transformation==linear combination?
Redchicken (1 month ago)
The animation helps a lot.
Debapratim Paul (1 month ago)
U are the best YouTube er who has ever happened to me...
Yanna (1 month ago)
It just clicked in my head. 5:58 saved me. Thank you.
Sami Mentes (1 month ago)
Now I understand why multiplication with identity matrix give us the same result, this is because identity matrix has the coordinations of unit vectors i and j, which by definition is the same vector as it is. I rearly comment on youtube videos but sir, you have my full love and respect for sharing your wisdom with us. Thanks for your efforts.
Marco Villalobos (1 month ago)
I'm really thankful for this beautiful way of learning algebra
S.M Hement (1 month ago)
You really changed my life completely, thanks 3blue1brown
kristoffer kulju (1 month ago)
07:40 the sass😍
Bo Do (1 month ago)
Wow what an intuitive way to think of matrices. I can seriously see some of the transformations in my head now.
Marcos Pimentel (1 month ago)
Sorry if im making a stupid question, but if matrices are like functions, which take vector inputs and outputs, can you build the graph of that function, and can you use calculus to find the derivative?
sadik orange (1 month ago)
Complex Analysis please......
Shane Bennett (1 month ago)
What monster was teaching this to high schoolers
Cecilia Osorio (1 month ago)
Wow, great teacher! Very clear explanations! Thank you!
Suketu Bhuta (1 month ago)
just amazing !
Lakshmi H (1 month ago)
None of my teachers ever told me that vectors and matrices are this beautifully linked. Why? Seriously, why? Thank you so much for helping me visualize this concept. Things make so much sense now.
Sam Wainwright (1 month ago)
This was great, but I tripped up on the idea that we're describing where the "new basis vectors land" in terms of the original basis vector's coordinates.
Gaurav (1 month ago)
Simply Wow!
Sreekar Parimi (1 month ago)
Ecstatic.love the video and wish I had this in college
ramit verma (1 month ago)
Great Video..But why the name 3Blue1Brown??
Praduman kumar (1 month ago)
Bro its a awesome video. What we have memorized as a formula is now seeming to come from intuition .
Ray Alibux (1 month ago)
I think I'm slow. I still don't get any of this
Majd AL Kawaas (1 month ago)
oh man, most of the comments are about how poorly we learned about math thank god these videos are here to the rescue
Shamim Yeasir Saion (1 month ago)
U are aweeeeeeesooomeeeeeeee❤❤❤❤
keshav L10 (1 month ago)
🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥
young eun Oh (1 month ago)
This is so crazy!!! Thanks
Abhishek Wagh (1 month ago)
Very resourcefull !
Ari Papagianis (1 month ago)
If a matrix is defined as the transformation of the unit vectors, what can you say about the bigger matrices such as 5x5 or 1000x1000?
Murlidhar Gore (1 month ago)
How do you visualize all this
Keyser Söze (1 month ago)
Can we interpret a 2x3 matrix as a transform of a space?OR for a matrix to represent a space transform does it have to be a square matrix?
MuffinsAPlenty (1 month ago)
It is a transformation of space, but a kinda weird one. It transforms 3-dimensional space into 2-dimensional space. It takes the first basis vector for 3-space and sends it to the first column vector of the matrix, etc.
Adwait Shinde (1 month ago)
Loved the Matrix reference
Swathi Upadhyaya (1 month ago)
Amazing set of videos! The visualizations are really great! Thank you for uploading and making it accessible to everyone for free :)
Manik Aggarwal (1 month ago)
I love this channel. Thanks for making maths so fun and easy.
happywin (1 month ago)
Great video but why did you zoom the graph when you showed the transformations. Got me all confused. Thanks
Batuhan Evren (1 month ago)
i am transforming right now!
Laura Santiago (2 months ago)
this is beautiful. thank you so much for the help!!!!!!!!!! :D
Abhishek Joshi (2 months ago)
man you are awesome, you are living the dream.
abhishek G.Y (2 months ago)
I felt like I had been trapped in a dark cave of eternal ignorance prior to discovering this gem of a lecture. Thank you so much for shedding new light on this abysmally dreaded topic. I'm looking forward to quality videos on tensors and tensor calculus :)
Sana Shahzadi (2 months ago)
I love your channel.. I can’t express how good your channel is..
Mycotina Fulica (2 months ago)
In my highschool I always wondered why matrix multiplication is defined in such a weird way.... and it all makes sense just NOW, WTF
Chi Nguyen (2 months ago)
I appreciate what you did and the lesson is easy to learn. So wonderful
Andrei (2 months ago)
It's not very clear for me why the transformed vector can be expressed as that sum of the transformed basis vectors times the original vector's coordinates. In the video it started with "it turns out that..." but I didn't get the explanation for why. Could someone explain?
Diaming787 (2 months ago)
Omg I wish math was taught like this 😍😍
草本柚子 (2 months ago)
中文繁體的翻譯真差啊

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