The Big Picture of Linear Algebra

[@ksbalaji1287]

Prof. Strang gave his final lecture before retirement yesterday. He has touched the lives of millions of students. Please watch his final lecture just to see the emotion filled comments of his students from countries across the world. Thank you, professor!

[@maxlee6986]

China along has nearly 3 million views, he's a legend.

[@solaokusanya955]

May the universe keep and always bless prof strong, You see that part 5:20 when he rhetorically asked that question was as of he saw my soul... Only today , have I fully intuitive understood the whole idea of matric maths... I am an autodidact, polymath in the making that I am trying to switch to all tech and maths science as I see that this is the present and the future, and slowly I am getting all the "first principle" of things so I can be relevant in life.... Hopelly I get a scholarship to help my journey too... I keep learning...

[@yael3326]

when he emphasized "M-I-T blackboard" i felt that flex. my professors only use regular blackboards.

[@eyelss5659]

In my school, they use a whiteboard

[@bngr_bngr]

Yael Manlangit my professor used an overhead projector.

[@uzairakram899]

I hate black boards

[@gold4963]

Uzair Akram Don’t be racist.

[@studentcommenter5858]

He wasn't trying to flex. He was just trying to explain concretely. This happens when you spend a lot of considerable time in "maths world"

[@vibhugupta1082]

Prof. Strang is one of the best if not the best professor when it comes to linear algebra. He makes you visualise all the stuff. Thanks for MIT open courseware for providing this damn good content for free.

[@muttleycrew]

Yes it is wonderful to have an ivy league professor as a personal lecturer for free, just remarkable.

[@Peter_1986]

One thing that I really like about Gilbert Strang is that he has a very easy-going and friendly style. He doesn't feel like one of those intimidating super-serious professors, he feels like someone who genuinely enjoys having a chat with the students and showing cool things about his courses, in a way that also happens to work as actual course lectures.

[@astrophysicalevidence]

Wow! I was using Strang's linear algebra book back in 1979 in college! Glad to see he's still going at it!

[@JohnSmith-qp4bt]

He’s on life support, though. You should have gone with Howard Anton

[@absolutezero6190]

@@JohnSmith-qp4bt how do you know that?

[@noonansean1979]

Wish I had seen this when I was taking Linear Algebra. Wonderful short lecture.

[@nicbouchard153]

When you didn't study for your final exam and you have 16 minutes left

[@kaoutharasma6567]

x"D !! Damn

[@harishussain764]

I feel your pain

[@jennyispoop4]

Me right now haha

[@gogatesudeep]

Sames XD

[@clementchidozie4009]

lol

[@daylyskater]

best part at 04:40 "not very thick, is it? because it's just a line!" x'D

[@roman51220]

Wink, Wink

[@zachcheu4403]

wink wink

[@lambda494]

Gil being Gil

[@pkgamma]

you're just as thicc as the null space

[@xiaozhengshang855]

this is whom I call a good teacher... brief, thorough and "BIG PICTURE" indeed

[@RobertMJohnson]

who

[@lookupverazhou8599]

@@RobertMJohnson The guy in the video.

[@pkgamma]

this lecture was in the null space of my brain but now it's in the column space

[@chotenque6877]

This man will always hold a special place in my heart

[@小明早上好]

He is a wonderful gentleman and a great prof.

[@mamadetaslimtorabally7363]

He is indeed. An amazing professor. He is intelligent, mathematically conceptual and didactic.

[@ricciuccio]

Here is one of the best lecturers in the world.

[@IanRichardArko]

Maybe the people at MIT should invent a blackboard that can fit an infinite plane?

[@atimholt]

You could fit more branching ideas if you use the hyperbolic plane. If you can map all the concepts to a tree graph, every branch could have more than enough angular space to expand upon the idea with. If you want to illustrate Euclidean principles (or just keep text legible), you can stick your figures into a circle that internally scales linearly in a Poincare disk embedding. Or you could also stick more Poincare disks inside your linear circular regions for asides. Or use apeirogons for enumerating countable sets, etc.

[@donfox1036]

Sure, they have. Not only that but the infinite plane can accommodate both blackboards with infinite space left over.

[@jacob9673]

atimholt nah. Sounds like it would be inefficient to write on and even more difficult to create, to a smart person efficiency trumps conjecture and uniqueness.

[@atimholt]

@@jacob9673 Computers make any math thing easy. UI’s just been stuck in a rut since Xerox.

[@chriskoperniak784]

Wow you’re funnie

[@q44444q]

Thank you Dr. Strang!!! You've helped me tremendously. I have never seen such a clear and concise explanation of linear algebra. Just a couple notes for anyone still confused: - column space is also known as range - null space is also known as kernel - the row space is sometimes called "null space perp(endicular)" - the SVD computes a basis for each of these four spaces - the row space is the set of all input vectors v for which Av =/= 0 - the left nullspace is the set of all vectors that A cannot produce in its output range - vectors in the row space in the domain "get sent" to the column space in the codomain - vectors in the null space in the domain get sent to the zero vector in the codomain - In this example, the domain=R^2 and the codomain=R^3. If you haven't learned these terms yet, go look them up. Matrices are linear functions, so they can be described using the language of functions, such as domain, codomain, range/image, preimage, injective/one-to-one, surjective/onto, etc. Note that here I use range to mean the same thing as image, so in my terminology the range/image is a subset of the codomain.

[@3x3-x3x-oXo]

"- the row space is the set of all input vectors v for which Av =/= 0 - the left nullspace is the set of all vectors that A cannot produce in its output range" Plain wrong. The complement of a subspace is never a subspace because it never contains the zero vector.

[@digitalconsciousness]

He seems like he is very straightforward and to the point, which is great. Jumps right into it and doesn't waste time.

[@ToniSkit]

I love this Professor. He is amazing and I'm really grateful that he did this , and grateful that MIT hosts it. Thank you

[@donfox1036]

This guy is too amazing. Just when you think he didn't know how to raise the first board, he does it.

[@mathewschau9361]

Is there a reason why the null space is perpendicular to the row space? 5:23 The explanation only proves that it is perpendicular but not why it would be in the first place.

[@LisaCoffee-i4s]

Vector product

[@MarkusReinert]

Not quite what I expected from the title, but he is great! Watching the video(s) is very entertaining and informative. Thank you!

[@Uncertaintycat]

Best book I have on Linear Algebra is by Mr. Strang. Well worth the read!

[@Rayquesto]

I love Gilbert Strang's commitment!

[@GeorgeZoto]

Amazing professor and well explained content! "Oh just by beautifulness, general principles of elegance..." -Professor Strang talking about the left null space

[@TonyTheTerrible]

havent done linear algebra in almost 7 years, this brought back a lot of the basics very fast

[@JustinGarfield1]

I don't have any degree in math but I have been studying it for 2 years. I just bought a linear algebra book and I find this stuff so fun to do.

[@Regular.Biceps]

Which book is that? I'm also trying to study myself to have a shot at becoming data scientist

[@lambda494]

I have Gil's latest book ("Linear Algebra and Learning From Data"). It's just like listening to him talk.

[@loct824]

from OCW 18.06, glad to see Professor Strang again.

[@wadehe740]

When you go with a 1.5 OR 2 speed, dear Prof. Gil is quick and vigorous like a young man! It's the fascinating part of online learning.

[@onetwo1817]

!!!

[@David-km2ie]

Haha, you know how it works. Everyone can learn at its own pace. (Up to 2x speed ;))

[@han5vk]

@@David-km2ie Not necessarily just up to 2x speed :) I often watch videos at 3x or therabouts. Just a simple case of running [document.getElementsByTagName("video")[0].playbackRate = 3] in the browser console :)

[@bngr_bngr]

Wade He old guys last longer.

[@devvv4616]

@@han5vk you can just download an extension for that in chrome webstore

[@strnbrg59]

Must-see video for the rest of us, who were taught that linear algebra is about manipulating matrices.

[@zacharyadler4071]

The reason it is called the left null space is because it is typically obtained by analyzing vA = 0 which is essentially the same thing he did to calculate the left null space but he presented it differently with transpose of A

[@MrSimmies]

Thank you. Could you tell me why the null space MUST be perpendicular to the row and column spaces? Thanks!

[@luisluiscunha]

15:51. No: thank YOU, Sir!

[@georgesadler7830]

This lecture is a beautiful introduction to linear algebra.

[@lordlem]

Dearest Gil, It is beautiful and wonderful to watch you work. You are truly a gifted expositor in Mathematics. You bring richness to students of all ages. John M.

[@zachgrant7809]

It's kind of amazing that there are so many professors out there that don't succeed in explaining this to people, but someone that watched this video could take the information learned and explain it to a fellow student successfully in a short amount of time. I will admit it is possible that youtube videos such as these are doing no more than filling in gaps of professors, instead of informing the student 100 percent more than they were before they watched it.

[@SpaghettiToaster]

Well, Strang's entire MIT linear algebra course is also online in the form of "videos such as these", so it's in fact informing the student 100% and not leaving any gaps!

[@SebWilkes]

His book on Linear Algebra is great fwiw

[@hermilomoreno6847]

Thankiu very mocht Teacher Gilbert Strang for your excelent explanecion about of the Sistem’s Linear ecuations for any dimensional’s Spaces.

[@kenpachizaraki4184]

My class had a teacher with a thick accent, was very softspoken, and always blocked the board when writing down concepts while explaining them. I literally learned like less than 10% of the stuff he was teaching. If it wasn't for these lectures, I would have never made it through that class. I didn't even need to show up to class since the grade was 50% midterm and 50% final. I'm paying for an education that I'm not getting. I'm getting it from these videos.

[@yinghaohu8784]

Nice Tutorial. I read this in a Linear Algebra book, about 3 years ago. Fortunately I didnt forget them. Until I watch this Video, I understand the concepts and relationships between these spaces.

[@adurgh]

At 6:35 he says "zero combination of the rows" but that's incorrect. It's zero combination of the columns; isn't it? Additionally, what are m and n. I'm assuming the matrix A dimensions, but they were used without being defined!

[@MrSimmies]

Phenomenal lecture as usual from Professor Strang. Thank you sir for your contributions and enjoy your retirement!

[@nandakumarcheiro]

Kindly raise the first board to give more descriptions for our better understanding of electrodynamic principle projections along perpendicular null space .

[@MissionMan]

I took Abstract Algebra, and at least 2 other math courses that required Linear Algebra as prerequisites, without actually taking Linear Algebra. Watching this 15 minute video helped me realize what I missed. Thank you!

[@zenchiassassin283]

Watching it nearly two years after the exam. Well at least it's nice to refresh these stuffs

[@alexandersavadelis8121]

This was amazing. Thank you for the clear and concise explanation!

[@mancinellismathlab7451]

Excellent summary of those key topics in linear algebra. Also, thank you for the well written textbooks!

[@fastacelzapacescu5445]

Dear Sir, you could easily teach other teachers how to teach. There are a few like you, but not as many as we need. Dear MIT, Out of the ones on the internet, MIT has the most gifted teachers as teachers, before being experts in their areas. As far as I am concerned. So please continue to do what you are already doing.

[@johntrolle8935]

I attended his linear algebra class in the early 00s before the OCW. Such a legendary professor.

[@lozotau]

Thank you, prof Gilbert Strang.

[@yourface4248]

the null space, now available wherever vectors are sold.

[@JalenF]

The math he did at 4:00 blew my mind

[@PranavPandey]

To the point explanation and I m also surprised with such recalling capacities of the things 🙏Thanks Sir!

[@Phi1618033]

The way linear algebra is taught is absolutely baffling. If every course began with just this simple 15 minute overview, I'm sure it would spare a lot of students a year of hell.

[@chensun2427]

i began to find Linear Algebra an interesting Subject after watching this open course!

[@teuzzxv4676]

simplesmente Dr. Strang melhor prof de algebra linear do planeta

[@IMadeOfClay]

Professor Strang is amazing. The best. 5:20: The way he says "You wanna know why?" Like he was about to say something really naughty 😂

[@gunhasirac]

It seems legit to watch this video while learning Fredholm Alternative for compact operator. It always amazes me how mathematicians can go such far and I think there’s more generalized theory for normal operator.

[@ColeTroi]

It’s not just a blackboard. It’s an MIT Blackboard.

[@DelikatesyLafuente]

Wonderful ❤

[@nandakumarcheiro]

This directional changes from clockwise to anticlockwise with null space projections in a way represent an electrodynmics combinations as inspired by Professors explanation.

[@jposadalcs]

Brilliant overview of at least half of intro to linear algebra (18.06).

[@derekrieger4699]

I feel special when he winks at me. ;)

[@vishal9030]

😉😉😉

[@InspektorDreyfus]

Don't feel special, he's just ironic. 😉

[@francislin9575]

Internet is amazing. I watched this man teaching linear from 2009 to 2016 to 2020.

[@solfeinberg437]

Thank you. Studying this now in conjunction with the singular value decomposition of an m x n matrix and least squares - trying to gear up for generalized linear models. This was very understandable, but complex enough, and I'm getting an inkling that it's fundamental to linear algebra - in fact I heard it described as the fundamental theorem of linear algebra, i.e. important enough, that's it's very interesting.

[@q44444q]

Make sure to learn how the SVD decomposes a matrix into bases for all four subspaces (partitions of the left and right singular vectors). This will really make this all click for you, from a modern, computational perspective. Most presentations of SVD don't even mention this

[@solfeinberg437]

@@q44444q Thank you. I'm right at that point.

[@changxiao2687]

Thank you, prof Strang. Really hope to have you in our school to give a lecture on Linear Algebra.

[@poohjh0515]

This is an excellent explanation. I wish I had gone to MIT for the lecture and then entered into my alma...

[@Fishofrank]

dang right at 15:15 i could feel everything all coming together

[@CapAnson12345]

Now I need someone to give me the big picture of the big picture so I can understand what he's talking about.

[@kwokhoilam2451]

Good to get such short lectures to introduce such difficult topics to me

[@PhysicalMath]

I love how Professor Strang makes LA fit together into a comprehensive and comprehensible picture.

[@omegapointsingularity6504]

3:50 whats happening, "one of the third one of the second" and so on, how can you make a 3d vector like that.. how can you multiply a 3d vector to a 2d plane like that?

[@SxC97]

Watching this video after taking Linear Algebra is like looking up lore videos after beating a game. Ohhhh, now I understand what I was doing...

[@BackToBackSWE]

Every youtube video on linear algebra I make a connection to something new.

[@FasAntick]

How does the row space make a plane if the vectors have 3 elements? Shouldn't it make a volume?

[@Qazdauysty]

Hi! Not sure if you are still in need. But this video helps kzbin.info/www/bejne/jZ6VaIxsnd2ViNU. You can skip to minute 5 i think where we have 3 variables. Cheers

[@muttleycrew]

I love how messy his board gets as he talks, each lecture is a demonstration of high entropy.

[@twk844]

At the end of the video, shouldn't left null space go together with row space and null space with column space?

[@fooger]

Prof Strang just rock..

[@royxss]

confusions untangled with simplicity. Thank you so much. You are a gift to humanity

[@marilenapantziri3688]

Wish i was given the opportunity to be taught by him!

[@movement2contact]

Okay, where do look for videos that start from something more like "2+2=4"..?

[@zhuyixue4979]

Insightful: row space is perpendicular to null space, so is column space to left null space.

[@OrionConstellationHome]

Very important topic! Only Gilbert Strang’s book and course has it early in the course, that is the only way to teach it. Thank You! It is unfortunately missed in most of the other textbooks early in the course, where it should be, and showed randomly in different chapters, so students never get a complete picture. But in MIT OCW by Gilbert Strang it is done right at the right time in the right place! Right On Gilbert Strang! Respect! Thank You!

[@keithdunn6306]

'The Big Picture' but I can't see the whole picture as the board is partially obscured by the subtitles. I wonder if anyone needs these (not perfectly accurate) transcriptions; Prof. Stang enunciates very clearly. Does anyone know whether one can get these MIT OpenCourseWare KZbin clips without the overlaid captions?

[@mitocw]

The captions can be turned off. If you're on mobile tap on the 3 little dots in the top right side of the screen. There you should see a captions option, so tap on it then turn captions off.

[@keithdunn6306]

@@mitocw Excellent! I am so pleased. Thank you very much for the information and the promptness of your response.

[@chuckstarwar7890]

Statistics and linear algebra are the most interesting topics. These two math classes can truly help you have more fun, logic, and money.

[@DoctrinaMathVideos]

Having an in-depth knowledge of these two topics can prepare you for the study of Artificial Intelligence.

[@bluecollar8525]

I mean, thia is pretty intuitive, and my textbook covered visualizations and vector spaces in their readings. I think the issue is if you had a textbook that didn't, which hinestly doesnt surprise me, since its almost a point of prode in math for teachers to intentionally make things mroe difficult on purpose so that the smart kids can figure it out amd they can keep a bell curve. Its always "we dont want to spoon feed you, you have to learn how to lrarn by yourself" which basically is just code for "im going to neglect to show you this stuff because i dont care to put in the effort"

[@Jab_hutt]

Just watched it, mind wondered off at the end, but was just pleasant to look and hear. Thanks!:) Legend.

[@solaokusanya955]

Wjat does the "perpendicularity" mean? What does it matter?

[@simondemarque2826]

for a foreigner, if I understood well, the 'left' means, remaining ? right or false ? but 'remaining' takes much more place to write on the blackboard, that might be the reason of using the vocable 'left'

[@420_gunna]

Late comment, but you are correct. "Left" as in "Left Over" or "Remaining"

[@TrevorKafka]

It refers to left as in the opposite of right. The left nullspace is set of row vectors x that solve xA=0, where the multiplication is done on the left side of the matrix instead of the right.

[@kaiwenyu8354]

isnt [0 0] a 2d zero vector, why is it written in that 3d space?

[@SuperMaDBrothers]

Maybe you should talk about why the rank/row space/column space actually matter, other than they are random quantities you defined which have a surprising relation.

[@pauldorman]

Excellent summary of a subject I've long been curious about. Not sure if Professor Gilbert is kidding or not though.

[@martyglacerda]

What did he mean by a perpendicular line is an object in ONE dimensional space? Wouldn't it be two dimensional? Height and width?

[@alfonshomac]

a line is a 1d object that can be described as related to 3d space. Imagine a taut cable of 0 thickness going from a pole to another, the cable itself is a 1d object but it's position is 'embedded' in 3d space and in order to place a line in 3d, you need extra information to get to it and then to describe its direction, but once you're in it, you just need a single parameter multiplied with said "unit vector" to reach any point in that line. The plane is 2d in the same sense a line is 1d, in order to reach any point on a plane that lies in 3d space, you need 2 non-parallel vectors to describe the plane and then 2 parameters to traverse it. Now imagine there's a line perpendicular to that plane, like a post sticking out of the ground, in order to place that line somewhere in 3d space you need some extra information but the line itself, like any line, is a 1d object. once you're confined to it, you'd only move back and forth along it. is it better or worse? hahah

[@zvxcvxcz]

+Marty Lacerda A line has neither height, nor width. Maybe you're thinking of a line segment. That will have a length, but to have a height you would need to define which way is 'up,' e.g. like for a non-equilateral triangle as you rotate it, the height changes. A line itself is one dimensional and extends infinitely. Think of it like the 'number line.' The only thing you need to find your place on it is a single number, it goes infinitely in both directions, now just take that line and point it in a direction in N-dimensional (think of say 3-D for the moment) space. The vector is only used to define that direction the line runs in. So the vector is like a mapping from the number line to a line in a particular orientation in the N-dimensional space.

[@johnsmythe9058]

Suppose you have a vector v=[1,2,3], then the line is av, where a is a scalar and v is a vector in R^3. It is one dimensional since you have 1 free variable, a. The line would be all the points p = [a, 2a, 3a], for a in R. There is only a single variable a, even though the line is in R^3. You may be thinking of a line y = mx + b that you see in algebra classes and think this is 2 dimensional ie. x and y. However y = f(x) = mx+b. This is a function of a single variable, i.e. x. It is a different way of thinking than what is presented in beginning algebra classes. If you wrote it in parametric form it would be [x, mx+b]. The only variable is x. You cannot ask y, because y has vanished. There is only one independent variable.

[@clapdrix72]

"Only has 1 puny vector..." Love it

[@bmw123ck]

So the cross product of the vectors of the row space is the vector of the nullspace?

[@salbrinkman5126]

I love it! Wonderful people like you make me want to study at MIT!

[@MrBINGEBOY]

Very clear and direct. Good teacher.

[@quarterday]

This lecture would have made Axler much easier to understand! Never thought I’d say this, but I’m excited to pick up the book again.

[@hari8568]

Officially mind blown with these patterns!!! How did I not observe any of those!! Need to improve observation skills

[@tonymontana9221]

One thing that always bothers me about linear algebra is how can the data we collected turns into a matrix in linear algebra. Take machine learning as an example, supposed we have a data set with N features and hence we have 1 N*N matrix to work with, what's the mathematical proof that justifies the idea that by doing the calculation of eigenvector, eigenvalue, and another critical concept, we can get the feature of the data.

[@Phillip.K.J]

I love Gilbert Strang

[@epicdoik]

this legend got me my A+ for linear algebra bless him

[@antonellomascarello4698]

This lesson is a masterpiece 💜