Nonsquare matrices as transformations between dimensions | Chapter 8, Essence of linear algebra

A brief footnote on the geometric interpretation of non-square matrices.
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[@quentinwach]

100/10! I am currently studying physics at university. How can it be, that this is the only place I can find that shows this explanation? No book, no video, no script, no teacher, tutor or professor have ever explained linear algebra via this eye opening geometric interpretation. I struggled with remembering all the different proofs and definitions. Now all my problems vanished. Simply by working through this series. Most of the time I can now instantly see and explain the solution to 99% of the problems in my Linear Algebra class. - All praise to you!

[@FajarSuryawan]

Actually, Prof. Gilbert Strang of MIT has explained this. Have a look at his lectures. You come to 3b1b for intuition and animation, you come to Gilbert Strang for intuition and computational know-how. Here: kzbin.info/www/bejne/kHyWgGdmZ9yZZ8U

[@oxman5646]

100/(10!) ? How come you didn't rate it highly? =P

[@non-inertialobserver946]

@@oxman5646 I knew someone was gonna make a joke about that

[@rickole7404]

@@oxman5646 beat me to it

[@valentinazito9385]

SAME MATE, SAME

[@MATHBOSS287]

imagine if everything you study at school had a video made by 3Blue1Brown available on KZbin. At that moment Education would be the most enjoyable journey in our lives.

[@KK-rg3nj]

i couldnt agree more

[@vikasmanav3715]

At that time, universities will cease to exist...

[@paprikar]

@@vikasmanav3715 so?

[@rohithkumarbandari]

@@paprikar education would become universal. It wouldn't have any financial barriers.

[@dark6.63E-34]

imagine our curriculums actually focusing on us understanding math instead of teaching us crunching numbers, something that we invented a tool for that does it flawlessly.

[@jackkennedy98]

You inspire me to do mathematics.

[@3blue1brown]

This is what I love to hear! And from a president, no less.

[@srinivasams9323]

@@3blue1brown what president

[@pralaykayal2476]

@@srinivasams9323 I don't know 😂

[@debluflame1899]

@@srinivasams9323 he proly changed his name now. Originally, jackMcJackJack had named himself as a US President [dont remeber which] which was the basis of Grant's joke

[@alihasanamu]

Really! His math is so tough. Haha

[@AgglomeratiProduzioni]

2:35 It's always funny to spot Pi

[@SomeRandomFellow]

oh wow i never noticed that and i do stuff like that all the time

[@anselmschueler]

where?

[@SomeRandomFellow]

schuelermine look at the thumbnail. look at the numbers

[@heldergeovane]

What about 'e' at 3:15?

[@tonyennis3008]

Good catch

[@NovaWarrior77]

A special thank you for the "gridlines parallel and evenly spaced" idea. I don't know if you made it up but it was extra thoughtful to come up with a illustration of linearity in this context that was somehow easy and mathematically verified.

[@mayankbhatt1308]

That was the only idea that enabled me to understand everything from there on

[@Interista697]

I've been attending a Linear Algebra course at university for six months now but I couldn't quite wrap my mind about what vector spaces actually are or why matrices work the way they do...but since a friend of mine suggested watching your videos my view on these topics changed completely! Not only I went from learning the rules by heart, without having the foggiest about what I was actually doing, to understand the theory behind them and be proficient at solving exercises but I even started taking a liking to Linear Algebra,and for that I have to thank you and your explanations!

[@zhengqunkoo]

With regards to the quote at the start, if the determinant is the measure of how much vector space is deformed, then the determinant must be meaningless when deforming from one dimension to another. Intuitively, one can imagine going from 3D space to 2D space: the 3D space, with nonzero thickness, is squished into a 2D space with infinitely small thickness. How would one represent that extent of transformation, with a real-numbered determinant? Learning about linear algebra this way makes it so much easier to understand ideas... never really thought about why determinants only apply to square matrices. Thank you so much for this illuminating series of videos.

[@zhengqunkoo]

Then again, square matrices with 0 determinant, geometrically, is squishing vectors into a lower dimension. I guess the inverse implication (squishing vectors into a lower dimension, implies 0 determinant), is not true?

[@newsoupvialt]

Zhengqun Koo although couldn't you argue that there should be a way to calculate the determinant for tranforming from a lower into a higher dimension like 2D to 3D, since the output will still be a plane that you can view as a 2-dimensional object? Edit: after writing that I immediately remembered that that's what the (length of the) cross product is, and how similar the cross product calculations are to calculating a determinant. That's kinda cool.

[@anshumanagrawal346]

The key thing you're missing here is to *NOT* think about all the 1-D, 2-D, 3-D etc space as existing in the same Plane (For the lack of a better word) of existence. Because then the determinant of a Non-Square Matrix makes sense, it's just 0 as you can just fill the remaining spaces with 0 so that it becomes a Square Matrix. Instead, think about all the different N-dimensional spaces as existing in different universes

[@NeoAemaeth]

@@zhengqunkoo Intuitively your second statement of squishing vectors into a lower dimension implying 0 determinant, does make sense to me. You can just add zeros to any nonsquare matrix to make it square, which add no extra information to it and just create |n-m| extra empty, "virtual" dimensions or |m-n| null vectors representing the squished vectors. Calculating the determinant would always result in 0.

[@binnypatel7061]

Nobody: 3blue1brown: So if you see a 3 by 2 matrix out in the wild

[@dailymemigzugxoyditsi3273]

sorry to say but @3Blue1Brown be like : I have archived Comedy .

[@1conscience0dimension]

pac man world

[@JK-jt3lr]

I literally went to the comments to write this :-D

[@RyanTosh]

One of the lesser known pokemon

[@shradhachhajed]

Why haven't I discovered you earlier 😭😭😭, the way you teach mathematics make me cry...I always loved mathematics but now I can clearly see it's beauty. Thank you so much ❤️

[@NovaWarrior77]

I'm certain that others have pointed this out but explicitly: the 2x3 matrix he uses has the first 6 digits of pi.

[@turuus5215]

Cool, you're so observant

[@garyindianna1]

"squishification" ha! love it

[@nischay4719]

Garrett Rathke you stole my comment

[@abdul-kareem4429]

Mine too, only after reading it I realized this =D

[@martind2520]

From this I deduce that although multiplying a 3x2 matrix with a 2x3 matrix will give a 3x3 matrix, that 3x3 result must always have rank 2, yes? Or have a made a mistake?

[@3blue1brown]

Indeed! I think that's a great example of something much easier to reason about from the transformation point of view.

[@ElchiKing]

Almost. The rank could be smaller, e.g. if you have lineary dependend columns in the 2x3 matrix.

[@martind2520]

Yes, I noted that after. Here's another interesting example: Let A be the matrix 1 0 0 0 -1 1 and B be the matrix 1 0 0 1 0 1 Then A and B both have rank 2, but AB is a 2x2 rank 1 matrix.

[@JoeJoeTater]

Multiplying with a 3x2 represents moving a plane out into 3D space, while multiplying with a 2x3 represents squishing 3D space down into a plane. If you put them together like [3x2]*[2x3], the result is a 3x3. Normally, a 3x3 represents moving stuff around within 3D space, but we know that the 2x3 portion looses some information, so to speak, and the 3x2 has no way of recovering that information. So, we know that our resulting 3x3 will "loose information" by squishing stuff into a plane.

[@JoeJoeTater]

***** Yup. You've got the definition for linear independence flipped around though. If the 3x3 only gives you a plane, then it can be described by two independent vectors in that plane. So, it can be described by just two of the columns of the 3x3, which means the remaining column can be described by those two, since it is also in that plane... It can be described as a "linear combination" of the other two columns, so it "linearly depends" on them. The columns are "linearly dependent".

[@anneblythe10]

I never 'got' linear algebra until I saw this playlist. Thank you so much! Supported you on Patreon!

[@northdankota]

the real curiosity is how you learned this knowledge ? , must learn this for applying own learning path , @3Blue1Brown

[@user-jj5se2is1w]

He studied at Stanford and then did a lot of research on his own

[@jk_ordeanneil3783]

Plus, there’s a beautiful series of video lectures by prof Strang at MIT open courses available on KZbin. Professor does a wonderful job in making these concepts intuitive. The greatest contribution by 3Blue1Brown is in creating the actual visuals, he used python to create the graphics.

[@becastockman981]

I am jumping from a math-lite public policy bachelor's program to a math-heavy economics master's program. When I say this series is currently saving my life... I am not joking even a little bit. Thank you so much for these excellent explanations!

[@AamirAliDrigh]

Due to him, i have jumped into masters in data science program. previously done bachelors in supply chain with 8 yrs experience.

[@Mrsangiole]

This is truly an amazing learning source. I thought I knew my linear algebra quite well and yet these videos have been eye-opening and made connections I have never truly thought of! Thank you, thank you, thank you!

[@quentinjanicot8350]

We saw you, yes you ! Pi hiden in the 3x2 matrix : 31 41 59

[@adamraiyan]

irrational comment

[@NamanNahata-zx1xz]

​@@adamraiyanreal ideas

[@mohitajay69]

You have the best kind of vocabulary to easily convey the intuitive nature of mathematics but at the same time maintain the simplicity of this. Thankyou

[@santialterman3066]

this video helped me understand matrices, finally. I now understand that they literally mean linear transformation. square matrices are linear transformations that keep the dimensions unchanged, while non square matrices are linear transformations that change dimensions

[@esper4605]

As an EECS major, this not only makes linear algebra more enjoyable, but more understandable. Still studying from the book of course, but at least this explains it in a different way that puts less emphasis on raw calculation.

[@andrasfogarasi5014]

A vector's matrix representation is just the transformation matrix required to transform the 1x1 matrix of [ 1 ] into that vector. Furthermore, were we to make the assumption that a scalar "x" and a 1x1 matrix "[ x ]" are equal for all possible values of x, absolutely nothing would change in mathematics. Thank you for coming to my TED talk.

[@hugorobalino6803]

I'm doing my masters in Computational linguistics / machine learning. I really needed this series to understand what was really happening with my vectors when I coded. I'm immensely grateful!

[@nerd2544]

u done with that masters?

[@hugorobalino6803]

@@nerd2544 not yet but almost

[@nerd2544]

@@hugorobalino6803 I thought masters only takes a year or 1.5 years 🤔

[@nekoy2010]

u done w/ your masters now?

[@alessandro_natali]

It takes genius to discover these mathematical concepts, and it takes a similar amount of genius, of a different kind, to be able to present such mathematical concept in a manner understandable to neophytes. I study architecture and I failed many times at math, and this is the first time someone really makes me grasp linear algebra. Thank you.

[@anantchopra1663]

Such a simple and wonderful explanation to something I've been struggling to understand since a very long time! I wish your videos were compulsory in freshman year at all universities in the world. Everyone would be so much better off with their understanding and intuition of concepts then!

[@superbart3000]

I was about to drop my linear algebra class because I was so lost. It seemed like the professor was just randomly making up names for ways to compute numbers. I honestly had no idea, that this is what we were working on. All we were doing was Gaussian elimination and echelon. But I had no idea why, or to what end. Seeing it visually like this allowed me to see it's purpose, to understand the reason behind it. I'm a computer science major, and all I could think was, I could write a super basic program to run these numbers for me, why do I need to learn this. But now, after seeing these videos, I truly understand the benefit of linear algebra. I wish I had found these videos earlier. I may possibly need to retake the course at this point, but these videos have given me a fighting chance at understanding and passing the class. Truly thank you for making these. These should be mandatory viewing material for anyone in a linear algebra class!

[@corexdraconem8632]

At 1:11 : It's not animation laziness, I just want to prove my point... (continues at 3:33 to animate 2D to 1D though, hmmmm :D)

[@rekarlopunzalan]

Animations in 3:33 operate in a 2d scene in manim, the animation program 3b1b made himself. Meanwhile animations in 1:11 work in a different 3d scene which requires Grant to redo the 2d vectors in this particular scene, so I guess there was indeed "animation laziness" haha

[@briandwi2504]

Love the way you teach. Seeing the solution of a simple linear equation in the light of a 2d to 1d mapping that takes a line to a point via two 1d basis vectors is such a beautiful and powerful idea. My thanks to you.

[@anonymous0x0]

so i just completed basics of matrices and these videos are now giving me a new perspective, all the things that i memorised makes so much sense now. Mathematics is really beautiful

[@stefluyten712]

these videos gonna solely carry my lineair algebra exam in a few days! keep up the great work and thanks a lot!

[@MrLyonliang]

Got it! Nonsquare matrix is used to lift lower dimension to higher dimension (row > column) or projection from higher dimension to lower dimension (row < column). Amazing Intuitive!

[@BharCode09]

You know in Hindu tradition we have the personification of learning/knowledge, Goddess Saraswati, who removes obstacles in learning and helps you learn things better. You, sir, are the GOD!

[@camkiranratna]

Bhargavi Suhas I totally agree with you.

[@shanmukeshr1696]

Yup we know he is

[@frankielee1836]

Why so many people just ask questions and not thank to the author? Anyway ,this is an awesome masterpiece about linear algebra!!! Love u!!!

[@차정민-b1z]

Because everyone already do this in their mind

[@stultus938]

because the "like " button is here for that purpose and the thanks comment are really annoying for someone who is looking for interesting questions/thoughts in this section;)

[@That_One_Guy...]

李会平 why are you so cocky whatever your name is ? Didnt you also ask question when you dont know / understand something

[@paulfoss5385]

Asking follow up questions is the highest praise you can give a mathematician. It means you think what they are talking about is worth thinking about.

[@robertlis8841]

I am kind of blown away, that 10 years after leaving the university, having memorized all this, I finally understand it.

[@OptimusPhillip]

I think this has helped me understand matrix multiplication a lot better. I now know why the height of matrix A must be the same as the width of matrix B for the product AB to exist.

[@LAnonHubbard]

I am familiar with linear algebra. I first learnt from the Khan Academy videos. That took quite a while because a) I was new to the topic and b) Khan Academy goes into the number crunching quite a bit so it can be difficult to see the wood for the trees. I persevered and grasped a lot of what the videos were saying. Next stop was MIT's Linear Algebra videos here on KZbin with Gilbert Strang. I *strongly* recommend people watch these. Search "MIT 18.06". Simply excellent! Just like 3Blue1Brown, the conversation is conducted at the higher level where the important concepts are grasped. Linear Algebra is best understood when looking at matrix-matrix or matrix-vector products without drilling down into the component-wise arithmetic which gets verbose real quick.

[@AndrewT]

Wow. You did it again. I've always thought of the dot product as a projection of a vector onto another vector. But the idea is finally being cemented here. Where were you when I was studying physics and math at university?? lol this stuff is game changing..

[@kas8131]

I had a mental block trying to think about inputs when looking at a matrix, this helped to clear it up

[@Yunas_Jet]

Thanks for doing this little side note! I probably would have eventually figured most of it out myself, but given that little prod at 0:37 it made so many things taught on my linear algebra course click for me - Things like why you can only multiply certain nonsquare matrices together and such. All stuff I knew how to *do* beforehand, but now *understanding* it finally is one of the best feelings in the world!

[@bidish2224]

I can never pay back for the help of these channel

[@majjikishore8689]

this kind of teachers make fall in love with maths,with the one who hates maths

[@joulesjams20]

creative numbering at 2:33. Also if you use a 3x2 matrix would the column space be linearly dependent (the output all be on the same plane is what I mean, trying to use the jargon)

[@3blue1brown]

The way you'd say it is that the columns are linear dependent, or that the columns space has a lower dimensions than the input space.

[@joulesjams20]

+3Blue1Brown thank you. Why columns not column space. I thought column space was the output of the linear transformation

[@paradoxicallyexcellent5138]

Actually, you mean 2 x 3 matrix. This may come as somewhat of a surprise, but when talking about matrices, the number of rows "height" (dimension of output space) is said first, and the number of columns "length" (dimension of input space) is said second. Also, you're right, the column space is the range ("output") of the linear transformation. When there are more columns than the dimension of the column space, that means the columns are linearly dependent, and, as you noticed, that must be the case for the transformation represented by a 2 x 3 matrix because you cannot have three linearly independent vectors in two-dimensional space. Good questions!

[@GrEEnEyE089]

Your Essence of linear algebra Videos are Great. Please make more of them

[@kjekelle96]

0:00 intro 0:15 what about non-square matrices 0:44 transformations between dimensions 1:21 encoding these with matrices 2:17 2D to 3D 2:37 3D to 2D 3:14 2D to 1D spits out numbers 3:58 outtro

[@ProfessorElvisZap]

At about 3:03, the illustration on the right would have been a little nicer if you demonstrated where the unit cube landed. The 2x3 matrix can then be thought of as a drawing of a cube into the plane. Otherwise, this is a great video!

[@3blue1brown]

Hmm, great suggestion. I'll put that into consideration if I find myself animating any more 3d-to-2d transformations.

[@Robocat754]

@@3blue1brown There's an error here right? Your 3D basis vectors after transformation are not in the right place in 2D space. Only the first basis vector i is in the right place after transformation.

[@sofiamonroy7435]

These videos are absolutely eye opening. I actually understand what I do what I do when I solve problems!!

[@bhocatbho]

I was pondering on the fact that a nonsquare matrix has not a determinant. Here my tentative answer. The determinant of a matrix is a number. This number tell us how to scale an area (in 2D space) or a volume (in 3d square) or an hypervolume (in n space). If you go from a 3D space to a 2 D space, there is non scalar that can scale a volume into an area and vice-versa.

[@ismailsevimli6424]

Thanks really, I am dedicated to watch and grasp all of your videos. Then understand the math by intuition theen some practicee.

[@amit-mishra]

Richard Feynman had said whenever people tell me, on what they are working I ask them a example instead of going through calculations I visualise it through that example can simply interpret them or tell where the problem lies. You are developing the same visualisation within us. We are living matrices.

[@alanwolf313]

I never got the opportunity to study linear algebra (I changed schools a lot), but I've always wanted to because of how versatile it seems. Couldn't ask for a better teacher, thnx Grant

[@Warwipf]

I love you so much, really man. You helped me quite a lot when I was trying to understand backpropagation for my seminar paper on Connectionism and now this. You're just an incredible teacher. You don't care about flaunting your knowledge, you just want to help people understand.

[@naveenverma2951]

interstellar music in another tab and yours in running in front. This is ultimate level of madness the way way you crack my fav matrices. Keep going

[@ismailbachchar1577]

0:48 it's a little bit confusing if u think about the input for example as a matrix 2*1 (1d colomn space) so in this case we have a transformation of a 2*1 matrix to 3*1 matrix

[@lakshandissanayake4348]

Unmatched ❤️ way of teaching. Clear intutions. Lead me to understand a few theorems(clearly) that our lecturer had omitted the proofs of. Thank you.

[@mahmedaa]

Great set of videos. I started to just like the videos even before I start watching them. Please keep it coming.

[@ChumX100]

No, if you see a 3×2 matrix out in the wild, you razz-berry it throw an ultra ball and hope that it stays, obviously

[@pranshusinha7266]

Or use the master ball

[@kartiksama5467]

I finished a math course in Linear Algebra in my graduate school. But it is only now that I realise that I understood nothing. The essence, the intuition that I now am getting by watching these videos is invaluable. Thanks to you sir.

[@fooltard]

squishification..... my vocabulary has been expanded.

[@NavyaVedachala]

Your teaching is magical

[@devanandanmanoj7882]

I don't understand everything you say but I am getting 70 percent of what you are trying to say. Usually I only understand 40 percent of what teachers say. Thank you

[@alleksandrs]

Never say "matrices" to Mr. Lambert, otherwise he puts a bag over his head. Your videos are really inspiring, thank you! If you write a book some day, I will buy it with certainty.

[@jdlopez131]

Just wanted to say I went to sleep and woke up watching these videos... Hope you keep making these

[@MaesiHunter]

Non-square matrices can represent linear transformations between spaces of different dimensions, like from 2D to 3D or 3D to 2D. The number of rows represents the dimension of the output space and the number of columns represents the dimension of the input space.

[@Lashistoriasdelilith]

yesterday I finally passed my algebra test and I didn't even know it was so so soooooooo interesting and thirilling. I'm so happy to have found these videos!! All this information seems magic!! to analyse and understand space like this. I wish these videos continued longer!! Thanks!!

[@orti1283]

man, you're really changing my life with these videos

[@tymothylim6550]

Thank you very much for this video! It is a new perspective on helping me understand matrix transformations in the case of non-square matrices!

[@DutchDread]

Ok, so you said I should have the basic understanding now to think about this, I thought about the quote in the beginning, and I had a chuckle, since I realized a 2x3 matrix would squish everything to a 2d plane, so the determinant presumably is 0. Then you proceeded to explain exactly that. You, good sir, are doing a good job teaching.

[@sketchbook8578]

Thank you so much for making these wonderful videos. I always thought linear algebra was too abstract, but not anymore.

[@AxisAngles]

Just to comment on your starting quote: Well, there is a useful way to define a determinant analogue for non square matrices. For example, say you have a 4 by 2 matrix, you can get the area spanned by the two column vectors by doing: det(transpose(A) * A)^(1/2) This is always positive because 2d rotation doesn't make sense in higher dimensions, because you can just change your perspective.

[@saustin98]

Ab. So. Lutely. Fabulous. Thank you for understanding how we learn.

[@julesthomas3335]

2:17 Perfect ! Indeed I usually see 2×3 matrices in the forest around my house

[@TheTahoeGinger]

Damn it man, this series is so good, I have 132 subscriptions, most of which pump out videos, but I have literally gone on KZbin just to see if the next video in the series is out

[@asadkhansohail]

wow amazing, really appreciate your work. Thanks for sharing

[@tohtorizorro]

To all these comments about "how can it be that this guy explains this stuff so eloquently and intuitively while my maths courses do not", I'd like to share some thoughts. There's no denying that the 3b1b videos are very good, in fact I'm here chasing the intuition as well. But I'd like to cut the teachers and professors some slack. While studying engineering / applied sciences in Finland it occurred to me that for some reason (I guess it all boils down to money) the time spent in class with one's teacher has been cut down severely. My past professor actually explained me that since the 90's the lectures (maths, physics, and the like) have been reduced to 25% of what used to be. Of course 3b1b videos are very concise and to the point, but it would be unreasonable to expect all teachers and professors to come up with such material. Or is it? Is that their job? I don't really know. I do think that curriculums should be updated rigorously, trying to incorporate new promising methods like this that are emerging throughout the webs. My point is, while teachers and professors could be required to come up with solutions like these videos to make the teaching more effective, the real culprits are the decision makers. And of course the times we are living in where everything is valued by it's efficiency in producing material wealth and this is where it all goes wrong. By cutting down the lectures and challenging students to graduate faster and faster leaves little if any space for innovations to rise, when most of the people lack a basic intuition of what is going on in their field. The one eyed pursuit of making everything more efficient (a fundamentally good goal) turns out working against itself. Vote against Trump.

[@jyotigoel4387]

you make me enjoy maths your glorious human being

[@080396001006]

this is amazing! even though my professor is great and explains everything amazingly, I still wish he would incorporate your videos in lectures to help us visualise. your videos are the gems of the internet!!

[@gamaray6136]

I give extra credit for the initial quote, that was a good one

[@maximuskolbycantos6065]

You make such a positive impact on our lives and help so much, youre a legend tbh

[@SimmySimmy]

So why we can't calculate the determinant of a nonsquare matrix, e,g the 3x2 matrix A mentioned above. like TA(x), before the transformation, X represents a plane in 2D space, then, after transformation, it's still a plane in 3D space. in this sense, we should be able to calculate the determinant of it.

[@rolandheinze7182]

How can you calculate the scaling factor of an object in 2d when it is in 3d, even if it is only a plane? What if the plane has shifted from the x-y plane, then it wasn't just "scaled", it was "twisted"... So scaling doesn't make any sense

[@tigeruppercut7]

Hey thank you this video has changed my life

[@joseluismatabuenasedano6881]

is there any way to extend the definition of determinant to non square matrix? I mean, as it was exposed befote, the determinant is related to the channge of "volume" (or its analogous concept in the proper dimenssion), what prevents you to say: case A, mxn matrix, being m>n (you are going to a higher dimenssion). The imagen of this transformation is a subspace of dimension n in a m-dimensional space (let m=3 and n=2, in the transformation R2->R3, all the R3 imagenes lays onto a plane defined by the colums of the matrix), so you could still aply the concept of "change in area" (as the output is a subspace of the same dimension as the imput). case B, mxn matrix, being m

[@samvelminasyan3393]

I thought a little, and I think in the case of 3x2 matrices we can still calculate the determinant(how much the initial area scales). But I also think that I'm missing something, correct me if I'm wrong please. Despite the fact that it moves vectors from 2D to 3D, the two vectors are still in one plane, because they intersept at 0.0. For matrix [x1 x2] [y1, y2] [z1, z2] The determinant is equal to = sqrt(x1^2 + z1^2) * sqrt(y2^2 + z2^2) I came up with this formula using just basic geometry and intuition.

[@samvelminasyan3393]

MAYBE it can work for higher dimensions too, for any non square M x N matrix, where M > N or something like this. These are just thoughts :)

[@manojbhumanand5592]

No words to explain my gratefulness!!!

[@MrPabgon]

This is fascinating. I'll on my first year of engineering degree (in fact, tomorrow I have the first semester math final xD) and linear algebra was normal. I mean, I didn't dislike it, but it wasn't something special for me (I do like calculus more). But now I know why I didn't like like it. It's because I didn't know *why* xD. Now that I understand what is everything instead of just the formulae and those things to remember, it's way, way more fascinating.

[@vrnkasi]

Amazing series! Please keep it coming..

[@yehuditflax204]

thank you so much!! I wish all teachers explained this way!

[@dmitriykhvan2035]

this stuff is mindblowing

[@jonsnow8543]

It would be awesome if you came back to Linear Algebra again and covered stuff you missed and applications. Examples of using this intuition with problems helps test and cement that intuition. Also would like to see rotations using quaternions

[@r1a933]

finally it all make sense now. Thank you 3Blue1Brown

[@sumitftr]

1:58 if it is so then column matrix is a transformation not a vector where i^ lands at the given vector

[@ВладимирКильчевский]

to be honest, i was very impressed, that was the best explanation that i have met about this topic, i wish i would have had this kind of explanation at school

[@rommeltito123]

This video series should be the pre-requisite for any linear algebra course

[@Ashitaka255]

I'm already dreading the end of this series and going back to the crap my prof teaches. Please make more videos. Especially ones about computer science.

[@himanshubhusanrath8036]

You are no lesser than a God to me. Never imagined this kind of explanation is possible!

[@masibhai339]

I use to wonder what matrixes actually used for after this series I came to understand all the magic behind matrix. Thanks sir really appreciate your hard work. God bless you.

[@TheKillaCake]

Years ago I was watching these videos, I liked them but I couldn't really follow the subjects. I'm now taking my first year of a mathematics degree and knowing to watch these is a godsend.

[@Noah_Krakatoa]

Moving down dimensions makes more intuitive sense than up, since you could represent a projection from 3D to 2D as a 2x3 matrix.

[@happyhayot]

This series of videos is incredible. It would be perfect that you made one about analysis of differential elements to build integrals and form objects

[@RobertMilesAI]

So if you have a camera looking at a 3D scene, you could describe that with a matrix of three columns and two rows? That would map where each point in the 3D scene ends up in the 2D image plane?

[@honeyjars]

Love these videos although they may be a fair amount above my level! I'm wondering what programs you use to edit these though! They are very sleek and beautiful! Thanks! :)

[@mastershooter64]

manim

[@yemaneabrha6637]

You are a lifesaver you make me spend at least 15 min of ma day, in doing mathematics I am a Computer science student, thanks to Andrew Ng who helps me to know you