1. The Geometry of Linear Equations

MIT 18.06 Linear Algebra, Spring 2005
Instructor: Gilbert Strang
View the complete course: ocw.mit.edu/18-06S05
KZbin Playlist: • MIT 18.06 Linear Algeb...
1. The Geometry of Linear Equations
License: Creative Commons BY-NC-SA
More information at ocw.mit.edu/terms
More courses at ocw.mit.edu

[@mitocw]

Audio channels fixed!

[@cpcheats2003]

Awesome! :D

[@BentHestad]

God Bless You, MiT and Gilbert Strang! You deserve a lot of praise for this series, now in even better quality.

[@santanumisra4190]

Thank you MIT OpencourseWare

[@rj-nj3uk]

Mit please provide a series on propositional logic and predicate calculus.

[@giorgostseligas37]

Many thanks to MIT and Professor Strang! We love your work

[@scottbrandon6244]

This was filmed in 2005 when Gil was 70. He just retired from MIT this month (May 2023) at age 88 after a 62 year career as a professor. What a great career.

[@mgh7818]

Wow, he looks so much younger than 70 in this lecture

[@Peter_1986]

​@@mgh7818 I would have guessed maybe 55-60.

[@henryfordsonofindia8399]

Was he 70? And retired at 88 this year 😲😲😲

[@Peter_1986]

@@henryfordsonofindia8399 Yeah, he seems to be in very good shape for his age; he can walk around in front of the board, and he can juggle around matrix algebra with ease.

[@lowlightevangelist9431]

Fastest moving, smartest 70 year old, prolly ever.

[@sanchayanmaity5731]

In two years of junior college and 4 years of engineering, not once did anyone gave the column picture explanation. Mind blown. Thank you, Dr. Strang.

[@BC-hg6dz]

Thank you Dr. Strang. I graduated from high school functionally illiterate in math. In my journey to teach myself math you have become a great mentor.

[@artherladett442]

I second this! I was exactly the same!

[@georgecarlin2656]

Me too! Here on Mars good math professors are a scarce resource too!

[@popeagapitusi8]

Good luck!

[@David-km2ie]

Good luck in your amazing journey

[@maheshnahak9510]

@@georgecarlin2656 uhm...MARS????

[@gerardomoscatelli9035]

I wish I had teachers like Prof Gilbert Strang when I studied linear algebra. What a difference makes to have someone dedicated to explaining the geometry of what is going on.

[@nickyork8901]

33 years ago I was told to read his book on linear algebra and its applications, my first experience of learning this stuff...did not always work for me, but I persevered with his approach and got there in the end. It's amazing now to see what he actually looks like.

[@NazriB]

Lies again? Gay Of LA

[@PhucLe-qs7nx]

00:00 Course overview 01:00 Lecture overview 04:25 Row picture 2D 08:35 Column picture 2D 15:38 Row Picture 3D 22:00 Column Picture 3D 27:55 Can I solve Ax=b for all b? 32:25 "Visualise" 9-dimension 35:33 How to multiply a matrix and a vector?

[@arisoda]

​ @MIT OpenCourseWare please copy this info to the video description so it becomes visible in the video timeline

[@MrJ691]

Someone do this for every lecture video

[@aadarshmishra2504]

The cameraman here is now a Senior ML Engineer at Google, California.

[@aadarshmishra2504]

@Bob Smith It's a sarcasm to the fact that how good these lectures actually are. 😅

[@prashantdhameja5778]

@Bob Smith That kinda dumb is acceptable in the nerd academy so no r/wooosh for you.

[@user-yk6cz3dh9i]

@@aadarshmishra2504 very useful

[@jainpranayr]

r/woosh

[@sigmaswan2969]

😂

[@shutupimlearning]

its insane that this video is 15+ years old and still so valuable to someone beginning to learn linear algebra like myself

[@diegoferrari3341]

WTF? How old are you? 10? What 15 miserables years are to the Greatness of Mathematics!??? Math rules the Universe!

[@AbdullahAlToyori]

Same here!!

[@Dontpushyour_luck]

Because mathematics doesn't change

[@davyroger3773]

@@Dontpushyour_luck Until another Leibniz comes along

[@tchaivorakfauresohnsieg9532]

Have you finished it, was calculus prerequisite?

[@adarshkumar7238]

Professor strang is master of teaching He is an artist. His way of teaching is a masterpiece. Thank you Sir. Respect, from India

[@maan92cu]

I'm learning from Saudi Arabia. And I couldn't agree more with what you wrote! God bless him!

[@SolidSiren]

Hmm...I wouldnt go that far. Imho, he is a good teacher from what I've seen here. He clearly cares about teaching, and he engages his students the way a good teacher should. He starts from first principles and assumes next to nothing about his students. But I wouldn't call it artistry or a masterpiece of teaching haha.

[@MLFranklin]

Despite his humble comments, he actually is a Rembrandt.

[@adarshkumar7238]

@@MLFranklin so true

[@HOWYOUDOIN884]

Yes... it's interesting how you can't really just rely on a book, and why it helps to watch a person actually doing it. And the style totally is important.

[@Christopher-mi6qe]

Took 18.06 in 1992 and he taught it. Great stuff, and an awesome professor.

[@wlycdgrTheEverydayWorld]

Kept 1st lecture intro to one minute. Bless this man.

[@matematik-ks3860]

Please spread this school on your wall and your friends because students need to see it. So will you. There are more than 4 000 tasks and theorems for primary and secondary school students. Thank you for your contribution. Respect! ........................................... .Të lutem shpërndaje këtë shkollë në murin tuaj dhe miqve tuaj, sepse nxënësit kanë nevojë ta shohin. Kështu do të kontribosh edhe ti. Janë më tepër se 4 000 detyra e teorema për nxënës të shkollave të nivelit fillor dhe të mesëm. Faleminderit për kontributin tuaj. Respekt! kzbin.info/door/ELXh5_dCYX6umKP3qlOL9g

[@suckmyartauds]

This lecture plus 3blue1brown's videos are getting these concepts to stick for me. Thank you Prof. Strang!!!

[@douglasstrother6584]

It's so important to develop a geometric view of linear algebra.

[@MelodyMcQwerty]

We're using Professor Strang's book in my linear algebra class and I really cannot understand my professor's teaching style. My friend pointed me to these lectures instead and I already feel like I actually understand a lot more from this one lecture than I have the past two weeks in my class. Thanks Professor Strang!

[@SatireDaily]

humans will start learning machine learning here.

[@bouthaina7846]

Yes, I'm here because of machine learning

[@rezaadventures]

@@bouthaina7846 hi can you tell me what is the application of Linear Algebra in ML ? I really confiused whit vector space and Scalar field and what we exactly model and do in linear algebra

[@bouthaina7846]

@@rezaadventures actually I'm so new and just started studying machine learning but for what I've seen in neural networks, PCA and a lot more things everything is about matrices and linear algebra so I've decided to firstly understand linear algebra then everything else will be quite easy ...

[@moonriver7632]

@@bouthaina7846 Do u study computer science before or directly do machine learning ?

[@bouthaina7846]

@@moonriver7632 yes I studied cs first

[@mathyscrete9469]

For everyone begining this lectures that have no notions of linear algebra, I reccomend you to watch 3Blue1Brown series before because he put a lot of image on the word. Sorry for the mistakes, I am not an english speaker.

[@osxs333__7]

That series was amazing

[@HenryTechBoy]

Currently watching that series myself and would recommend it as a phenomenal starting point for Linear Algebra @3BLUE1BROWN

[@osxs333__7]

@Chris Metcalf i got my eyes on landing a job in data science for myself, just brushing up the mathematics behind machine learning but definitely not planning on doing deep theoretical tinkering and what not

[@nerdpotato9368]

what should i watch? the essence of linear algebra?

[@mathyscrete9469]

@@nerdpotato9368 yes

[@FHB71]

That makes me remember the first semester linear algebra lecture. He certainly takes his time to explain, what the interpretation is and that is wonderful. We were not so lucky unfortunately.

[@fernandozolubaspreto6976]

Hi professor and online collegueas. I'm a control engineering student, so Linear Algebra is the essential of my field of study. Despite I've taken a linear algebra course before and even sophisticated discussions in control systems, I never truly understood until now why it is better to think using columns instead of using rows. But now it's totally clear in my mind, so thank you very much, professor Gilbert. I'm anxious for the next classes.

[@marco.nascimento]

An amazing first lecture, glad to see the audio is fixed, thanks for taking care of this course. It's beautiful to perceive people from all over the world gathering here to gain knowledge, Professor Strang is a great teacher

[@randomdude354]

There are so many moments when I wanted to give this video an extra LIKE! He's such a good teacher!

[@omeryacine8281]

about to cry of how beautiful this is

[@graphchick]

as a 1st year grad student a very long time ago, i took a course in linear algebra using Gil Strang's book. learning about the relationship between the four fundamental spaces of a matrix felt revelatory at the time, and i still have that book. he's such a wonderful expositor.

[@achievemindsetmastery]

Preparing for IIT/IISC M.S interviews. IIT Madras, Electrical Engineering department had recommended this lecture

[@dddiyamirza]

I am preparing for Gate(next year). How much have you done?

[@chellingiganesh6111]

Me too preparing for Gate exam

[@inalambricoteseo3082]

thank you Mr Strang for your lecture!!! i don't understand yet the fact that with 2 millions subscribers, only 4000 folow your lecture!! Your chain is one of the best i found regarding basical science, congratulations!!!

[@muzou520]

This is an amazing lecture for Linear Algebra. Thank you Prof. Strang!

[@homoperse979]

i watched these now for the second time, it's still awesome. oh strang, you are great lecturer!

[@acho8387]

So many questions answered in 40 minutes. Thank you so much, Professor Strang.

[@wxie7689]

Even though I've learned Linalg from my undergraduate course, it was ages ago. Thanks to Prof Strang, the explanation of the geometry meaning of the matrix is much clearer than what I've ever been given.

[@kirkyrussell]

I love the quirky teaching style of Prof. Strang, especially how he feigns to encounter something surprising, whereas in fact it is well-rehearsed.

[@solfeinberg437]

This makes me want to applaud. I watched his summary lecture that reiewed this course and I damn near clapped in my home. I should buy his book - I bought another one that's working fine, but didn't emphasize the row space column space null space left null space fundamental theorem of linear algebra picture that this course develops.

[@ahmedatiatolba]

This man is the best LA professor ever and these lectures are perfectly made. I really appreciate Prof. Strang and all the team involved.

[@douzijin6541]

For a person who is not really good at math, I hope this will really help me to develop a big picture of what linear algebra is. Thanks Prof. Strang.

[@feruzaerkinovna6052]

I always dreamed (and still continue) about studying at MIT, but I am not a genius and not a daughter of some deputy, so instead of university lectures I watch this course and imagine myself studying there))

[@starzandearth]

hopefully you reach there :)

[@lucianoinso]

There's not that much difference on content on the subjects between good universities and the ones at the ivy league, just study for the love of it.

[@lizzy1138]

Luciano Rodriguez this is true. I attend an Ivy League university and paradoxically, the intro level math classes are awful, I study from MIT materials

[@TigerAlert]

I mean you also have to think about the school fees and see if it's worth it. I don't think it's as black and white as you may think, it's probably more grey.

[@akbarrahmatullah6701]

That's passion. Great thought.

[@douglasstrother6584]

I like the column perspective.

[@dhairyashah1746]

Dr Strang is a Legend, Thank u Sir and MIT for this awesome course

[@BACPEP]

One of the best teachers! As a viewer of this course you feel the passion and love for math. He feels the math and can transfer his knowledge to other people. Well explained! Top!

[@MLFranklin]

I had linear algebra over 40 years ago, but this was the first time I'd seen it presented this way. I've done fine in my engineering career, but this kind of thinking would have definitely made a few things easier.

[@paulmerritt418]

I can see why Dr. Strang is a professor at MIT. The best of the best.

[@aaronbell5994]

Wait, this was published 4 days before I decided I needed this? Thank you!

[@ath216]

i guess they have re-uploaded the video again after fixing the audio channels. check their pinned comment before 1 month ! enjoy :)

[@mrnobody1321]

Can't remember last time i was so pleased learning maths, apart from a few 3b1b vids. Thanks Sr.

[@LordMgls]

10 years ago I skipped Lin Algebra class and learned by watching this fantastic series of lectures. Moved to have learned that Prof Strang just retired. A big thank you to him for delivering such well thought through and well paced lectures.

[@tomshabalin9775]

I'm a student in Ariel University at Israel. I'm studying computer science and math, and I just want to say that you are a terrific teacher! I will recommend your lectures to my friends for sure! Thank you a lot.

[@bazmira3283]

Free Palestine

[@animalsilvestre]

Soy arquitecto, antes de ingresar a la facultad ingresé también a matemática pura, lamentablemente dejé las matematica por la arquitecturas, ahora devoro libros de matemáticas, espero algun dia retomar mi camino, mientras tanto me divierto con sus clases....y la pasión aumenta. Saludos!

[@frankgao1265]

This is best Linear Algebra Course one can find on the internet

[@deepakkumarkhatri]

Really got the points! First time in my life I have completed a lecture on math and also learned multiple new things! All concepts got cleared. Thank you Gilbert Strang for introducing me to LA. Respect++;

[@testus86]

I had 3 semesters math in informatics study. This guy is way better than my prof years ago. He explains it way better

[@lovemoremadzongonye2624]

Linear Algebra runs in his blood (Prof. Gilbert Strang). I now understand Linea Algebra thanks to his amazing lectures and his too much insight into this subject!!! He opens mind and unlocks concepts which are not naturally easy to grasp.

[@coemgeincraobhach236]

This might help others trying to learn this material. I found it easiest to watch a few lectures, but without being overly concern about learning exactly whats going on. This gives you a basis to work from. Then get the text book and work though it slowly. Write down the examples in the book as you encounter them and get used to forming a mental picture for whats going on. The column picture is key. The row picture is probably what your school math teacher taught you. This just confused me a lot. I don't like dot products, but the "combinations of columns" is very intuitive once you can visualize why it works. The reason I found this easier is that the concepts introduced in the videos are a bit loose, its just an overview. The real learning comes from the text book.

[@elangochellappa9113]

Fantastic explanation for column based geometry.

[@nicholasmohr1619]

Today was Gil’s very last lecture and retirement day. Happy Retirement Dr. Strang!!!

[@jacobsabanin8128]

Collect Mr Strang's lectures like a puzzle now (Because he's really good)

[@faizanmohsin3685]

Linear algebra had been mystery for me until I saw these types of lecture online. Thanks

[@piotrek0207]

I'm really happy that great lectures are available for free. Young people should appreciate it that they have internet and they can learn from materials from other universities especially from other countries from another part of the globe.

[@MarceloSantos-hz1bg]

Gracias Dr .Strang ,yo estudie algebra lineal con su libro ,pero verlo a Ud ,explicando en ingles , me llena de emoción , desearóa estar en su clase ,es un genio ,en persona es mas de lo que pueda decir su libro ,Mi admiración y respeto , Que calidad magistral para explicar algo tan ABstracto ,lo hace ver facil y sencillo,tanto que hasta mi hija de 12 años que esta en primaria lo entiende ,es un superhiperExplicador

[@pabloganoza2720]

Es un maestro a carta y cabal.

[@dbsxogh2]

MIT 이래서... 최고의 인재들이 태어나는 군요.

[@samiulislam6527]

The Business Analytics admission director for McCombs school of business told me to watch these videos for a understanding of linear algebra.

[@hidetsuguhiraki3307]

When we get linear equations, Ax=b, where A is an n x n matrix, and x and b are column vectors with n components, we can see b as the linear combination of column vectors of A and components of x is equals to b. That is, column 1 of A * 1st component of x + column 2 of A * 2nd component of x ... + column n of A * nth component of x. This view is new for me. Thank you, Prof. Strang.

[@melontusk7358]

This is so helpful. I adore professor Gilbert so much. It was a bit surprising to go straight into linear equations at first, but it worked out in the end.

[@matematik-ks3860]

Please spread this school on your wall and your friends because students need to see it. So will you. There are more than 4 000 tasks and theorems for primary and secondary school students. Thank you for your contribution. Respect! ........................................... .Të lutem shpërndaje këtë shkollë në murin tuaj dhe miqve tuaj, sepse nxënësit kanë nevojë ta shohin. Kështu do të kontribosh edhe ti. Janë më tepër se 4 000 detyra e teorema për nxënës të shkollave të nivelit fillor dhe të mesëm. Faleminderit për kontributin tuaj. Respekt! kzbin.info/door/ELXh5_dCYX6umKP3qlOL9g

[@secheledelaruse1937]

attend to the strang paradox; a vector field of unfinished sentences that form the basis vectors for a stunning range of interesting algebraic problems

[@vensoltface4506]

I'm studying in McGill and this man is saving lives. Ofc, cramming before the finals :D God bless this man. Watched his videos on 1.25x speed

[@emilefortier1688]

Nice to see a fellow McGillian! I graduated in 2016.

[@kriztoperurmeneta7089]

It does help to have a geometric view of it. Great lecture!

[@fireblizzard2287]

I came here just for fun, but end up knowing something new. A new and easy method to multiply matrix. Thank you so much 🥰

[@marinabaskakova2333]

Thank you Professor for an amazing lecture and MIT for publishing!

[@metacortex4140]

Well boys, future starts here.

[@hobao4965]

I love this lecture so much. I turned from hating math to see a useful part of its application. Thank you OCW

[@biswabismitabag9017]

Thank you MIT OpenCourseWare..Prof. Strang is God, Not only MIT students the whole World deserves a great teacher like him. A student from India

[@aarayan111]

this course is good for under graduate student

[@okarakoo]

This is a seriously gifted teacher

[@HWPcville]

I enjoyed this lecture. Professor Strang seems like a congenial person truly interested in imparting his knowledge. Some professors seem to look down on their students and have the attitude that if you don't get it, tough.

[@sannyii1098]

SO fantastic！！！THE Right way to open Linear Algebra!!! 我是来自中国的学生，当我认真学习线性代数的第一堂课后，我被深深的shocked到了！让我重新理解了维度！ N-D space is a combination of N-1-D sapce and non-N-1-D vector!

[@NinjaRunningWild]

What an excellent teacher! I have that edition of his book (& the next). They're recommended texts in game programming.

[@indrashispowali]

Thank you very much, Professor, for such a beautiful explanation of the subject

[@shailendersahu5814]

Respected sir,you are the only one who made me fall in love with matrices and liner algebra,Thank you so much for your service ❤️

[@elamvaluthis7268]

Superb clarity.world renowned professor of Mathematics.see how he acts on the stage one should inspire.I grace your heels.

[@immortalhades1609]

Man this is the definition of LECTURE.....❤️❤️

[@reubenlemba388]

I like the the way you are analyzing the questions

[@TheWonderworld10]

You know a man's the GOAT if lectures from almost 2 decades ago with chalk and a blackboard are better than 90% of the content online. Thank You, Professor Strang and MIT for bringing such amazing content to the world...

[@hassanlaghbi7047]

Watching Dr. Strang is never ever boring

[@piotrek0207]

Thank you for great linear algebra course.

[@sumanmighty]

I wish I could have found these gem of videos much earlier in my life

[@someoneyouIoved]

thats the greatest explanation of geometry of linear equations i have ever seen, thank you for everything sir, greetings from istanbul technical university

[@jaydean5243]

Prof Strang really LOVES teaching, his soul is part of it-they say a genius make poor teachers- he is the one exception- Thank you for your efforts to bring people into your world

[@FlanzShirt]

Wow! Thanks so much for uploading this! What an engaging professor.

[@matematik-ks3860]

Please spread this school on your wall and your friends because students need to see it. So will you. There are more than 4 000 tasks and theorems for primary and secondary school students. Thank you for your contribution. Respect! ........................................... .Të lutem shpërndaje këtë shkollë në murin tuaj dhe miqve tuaj, sepse nxënësit kanë nevojë ta shohin. Kështu do të kontribosh edhe ti. Janë më tepër se 4 000 detyra e teorema për nxënës të shkollave të nivelit fillor dhe të mesëm. Faleminderit për kontributin tuaj. Respekt! kzbin.info/door/ELXh5_dCYX6umKP3qlOL9g

[@joaocabral3541]

I olny had math until 6th form (around 12th grade) and i was invited one day to see a matrix class. And let me tell after the first 14 minutes of this video i really felt in love with maths

[@BuddyNovinski]

So, that's the reason for the row versus the column combinations! I could've used this concept in late 1976.

[@horaciormartinez1551]

Professor Strang : Thank you once again !!!

[@slaozeren8742]

I love him with my whole heart.

[@thomasfisherson]

I love this dude. I love his unique way of talking.

[@GiangTranAI]

Dear Prof. Strang, you’re a great teacher

[@dipayandasgupta7506]

sir you are a living legend! Cannot thank you enough.

[@quirkyquester]

amazing lecture, I thought its gonna be crazy hard to understand, yet, I can follow up :) thank you MIT thank you Professor Gilbert!

[@luisbernier326]

God bless him. Thanks x 100 times for sharing this video. I watch all the videos about Professor Gilbert Strang.

[@Alfonsoltd]

I'm taking this course at IU using the book Dr. Strang wrote and this is extremely helpful

[@naosouumrobo8086]

Amazing! I like linear algebra a lot, I already had this class in college, I keep reading about it and ... I didn't even notice the passing of 40 minutes of the first class you! No wonder MIT is a world reference!

[@hasrulmuhammad8271]

1 o n 2[. 77|530

[@ericlindal8008]

Seems like a great instructor.

[@OBGynKenobi]

Clear and straight to the point!!

[@ShivamAnandSpeaker]

My term 2 (semester 2) is about to start. There we have maths 2 and to prepare myself for that I am here.My Senior suggested me this. I am in foundation level (1st year) of IIT MADRAS (India) BS in Data Science and Applications. Seems good one! Yet to explore more! By the way Thanks to Respected Prof Gilbert and MIT.

[@akshatgupta8199]

Hii! I'm the same case as you, preparing for maths 2

[@marcosgoldin5623]

Great lecturer . Thank you Dr. Strang

[@Ernesto1317]

I can see in the class of this excellent professor and mathematician what the real problem in mathematical education is: our obsession to cover the topics always skips the necessity of the students to form the proper intuition of a topic. Intuition takes time, but above all takes proper activities to develop it. His talk is basically about the generation of spaces through linear combinations, but honestly could anyone sitting down there and listening reproduce those ideas without the help of proper representations, say, software, or at least good pictures? In mathematics, physics etc, intuition is the most important, but scholastic education has always tortured them with logic. That is totally unfair and a waste of talented students.

[@davidjohnson497]

Why u say it is devoid of intuition when he is showing exact steps of logic? Sure, u may miss understanding the steps of logic, but nothing he is saying depends on intuition! Follow the formula correctly, get the right answer! Make a mistake in calculating, get the wrong answer!

[@NinjaRunningWild]

@@davidjohnson497 🤦‍♂️

[@brahimoukhouya5888]

U can pause and think for yourself.

[@ibadurrahman5954]

page 35 of Deep Learning by YannLecunn brought me here the best explanation linear combinations and this gave a completely new perspective on how to look at matrices

[@thomasbates9189]

What a beautiful explanation!

[@YGHF]

This is awesome. That column picture analysis is new to me. Great lecture my dude