Thanks to Prof Strang and MIT, for putting so much work into this, for the benefit of the entire world, while asking nothing in return. Thanks from mankind.
@@yoyolol22 I too hated matrices and geometry (here in Brazil we study GAAL, which means, Analytical Geometry and Linear Algebra). But think of matrices as a collection of vectors, say column vectors. What I mean is, instead of studying or using a vector at a time we use a collection of them at once. Think of a vector as a single, very well defined and important, purpose. With this collection, say a toolbox, you could do amazing things. It may not help, but it did help me start loving LinAlg. Again, reinforcing the idea. Instead of using only one tool (a number), we can use a lot of them at once. That's why matrices appear everywhere, even in probability and statistics. An example is the multivariate Gaussian distribution.
I can't imagine trying to learn this stuff in a school environment! I'm learning or at least familiarizing myself with all of the math I will encounter when I start pursuing a bachelor to master next year.
Lecture timeline Links Lecture 0:00 Permutations 1:17 Possibilities of permutations 7:23 Transposes 10:15 General formula for transpose 11:38 Symmetric matrices 12:43 RᵀR is always symmetric 15:06 Chapter 3: Vector spaces 20:12 What "space" means 22:03 Why is Origin necessary in Vector spaces 25:33 Most important thing about Vector space 28:29 A case that's not a Vector space 29:41 All possible subspaces in R² 35:54 All possible subspaces in R³ 39:04 Subspaces come out from Matrices: Column Space 39:45
I didn't notice that because I was too immeresed in his lecture. Thanks for pointing it out. It does feel like he's teaching in front of the whole class.
That explains why there is no noise of people leaving classroom before the class even ends like before. I didn't notice that without your comment. This made me feel sad a little bit.
@@lightgaming7618 there's no one (except the camera person, probably) there. He probably had to record this one again or wanted to I don't know. But still, he is teaching as if there are students there. Professor Strang is just this awesome.
There is noone in front chairs. But it is possible that there may be some backbencher present in class. Because professor strang is looking not on camera but on chairs gives hint that someone may be present there.
These lectures are so good that I am falling in love with studies, specifically Linear algebra for the first time. I always dreaded the topic in my college days, just wish there was someone who could have referred me to these lectures back then.
“Take a breath and listen up!” - lecturer’s perspective starting vector spaces. 👍 I just love Gilbert Strang! I completely agree. I will tell that to my students tomorrow. 🙏
If I told younger me that I was going to be watching MIT videos to actually learn the content they are teaching, I would never believe it. yet here I am studying for an exam.
he is my hero ! that man made me love linear algebra :) i can watch his lectures every year ! thx professor Strang! you are the best teacher we're all dreaming about
My teacher took entire semester and was not able to explain me vector spaces and he did that in 40 mins.. Damn.. Hats off sir .. Kudos.. Love from India
@@nakamura7346 subspace is like a bit of space that lives within a more extensive area, and it obeys all the same rules as the ample space, making it a sort of "mini" version of the larger size.
@@nakamura7346😅I can explain to understand space and subspace u should understand what is dimensions (components)which is basically a way to mathematically measure information on that domain for example in an area of a square there are two components that's define that square that component is there in cube , etc in every thing think about it dimensions is like a axis which just means measures information n-dimension means n different information 2dimension means two different measure of information so I think u got this now u can better understand what is vector space focus it's important a vector space is a collection of vector that's it and it should follow some rules 1 it should contain 0 vector the components wise sum or multiplication should be in that space that's it now let's understand what is subspace which is a space inside a space . A space which follows rules like components addition and components multiplication is simple know as a sub space think about it.
Professor Strang lays out the concept of vector spaces in almost tedious detail. The result: I finally confidently understand vector spaces at a very basic conceptual level and I'm already beginning to extrapolate for myself. I imagine that this firm foundation will make the next few lectures that much easier and that much more rewarding. Thank you Professor Strang and thank you MIT.
I paid a ton of money in college went there every day for several hours and learned nothing. And here i am learning free of cost from the comfort of home from the best professors in the world. Respect and love to professor strang.
This lecture is so different from the math lectures I had (Germany). Our lectures consisted of the prof speed scribbling down Lemmas & Theorems, barely any examples, and they would ask random questions from time to time while everyone was trying to catch up with the blackboard. This seems to be the total opposite, much better for learning.
I just love Prof. Gilbert Strang. One of the best teachers humanity has to offer. To all those who haven't watched this playlist - This is the best way this subject can be taught.
Gilbert Strang, you have my love. I love you. The devotion you show for the subjects you teach is rare. One can easily see why an institution like MIT would like to keep you for the rest of your life. Thank you.
Before the internet, we spent our days eating each others hair. Thanks, MIT and Professor Strang, for making these videos openly accessible. Learning online has really changed my life.
I would assume that someone noticed the (bad) quality of the last lecture video but didn't find a fix for the problem in time for this lecture. And if that's true, I would assume he was asked to do the lecture again for the camera to continue/complete the video series.
***** Dustin Charles Lecture 5 had recording problems during class and was re-recorded in an empty classroom a short time later. The problem for lecture 4 is different. These videos were originally published in 2005 as quarter screen (320x240) RealMedia files. Years later we remastered them in Standard Definition. The tape for lecture 4 was lost, so remastering was not possible. Sadly, the quarter screen version is all we have left of it.
Know what... Actually it's not just the guys sitting in the class who learnt this.... Students from different corners of the globe, from over a decade's gap learnt it... Truly amazing 👍 thanks sir
This is awesome. We use the fourth edition of Dr. Strang's Intro to Linear Algebra in class and I am so glad these videos are available online. I found vector spaces to be a tricky topic in class, via my notes, even in the book. This lecture was executed beautifully. Makes so much more sense now. I am definitely the type of learner who really needs to have a deep understanding for the concepts to stick. Thanks so much!
Thank you so much Sir! Making a lecture without students at that time. Seeing your lectures is way easier than those college lectures that directly start with vector spaces second day.
Anyone going through these beautiful lectures and finding difficulty in visualizing some of the stuff - I would strongly recommend you watch 3blue1brown's series on linear algebra. After watching that, try to visualize what Prof. Strang is saying when he is talking about vectors and vector spaces. That will help you to grasp the key concepts.
😭😭.. Linear algebra was never understandable to me before, I always it only there to make my life difficult... This series saved me. So much love and respect Prof.Gilbert. Bless you. 💜
OCW has actually posted a copy of that video that is more or less the same quality as the rest of the videos. It's linked from a few places, I think, but the link is pinned near the top of the poor-quality video.
I do see your point, but not many great researchers even make it as a half-ass instructors. I studied & worked under/with the creme de creme of research, who I would not bother to ask a question as most live inside his own brain. This professor is worth asking questions & even complaining about. So, I have not complained; however, those who complain obviously care to dare. He is a great researcher & teacher at the same time. So, think of the complaints as love expressed in frustration. I wish I discovered this 10 years ago. And, I did make a small contribution to MIT Cloud Learning Foundation. If anyone reading this has not then pls care do donate. MIT is going for 1000 contributors (any amount) until June 28th. Participation Drive 721 Donors 72% Towards goal of 1,000 Donors $39,710.00 Raised 4 DAYS LEFT
My observation is not a comment about the quality of instruction. I really enjoyed these lectures, and I really appreciate Strang's instruction. I thought it was interesting that there was no one there, I wasn't implying anything.
thanx alot for this great lectures, i´m from Portugal and I´m amazed with this oportunity that M.I.T. gave to every one around the world M.I.T.believes in a better world where everyone can get free high quality education. With this actitude of M.I.T., i´m starting to baleave myself that a better world is possible.
I love that many universities and colleges do this sort of thing. Looking at different points of view on the same subject is very helpfull. This is extremely well explained, thank you.
Ok two things. First - That is one giant piece of chalk. Second - To be able to lecture this well with nobody sitting in front speaks volumes about the qualities of this great teacher.
Que manera de explicar tan simple los espacios y subespacios vectoriales , cuando los vi en la universidad no entendía de que se trataba. Muchas gracias.
I've been using the terms "Closed under..." in two of my midterms now without caring of what it meant exactly graphically...its only now that I understand it. Thanks
THANK YOU VERY MUCH FOR YOUR GOOD LECTURES.IM WIS wishing you a very Merry Christmas and a HAPPY new year.YOU DESERVE ALOT OF DONATIONS.THANKFUL FOR YOUR LECTURES.
Thanks very much for those videos. For the first time in my life I finally understand those concepts. Hopefully, statistics textbooks will become less scary from now on.
"You sure have to see it in R^3, because we're going to do it in R^10, and we may take a combination of 5 vectors in R^10, and what will we have...? God knows..." LOL!
I love how he's so polite in his lecturing: "Please can I show you this?". Even in his books he's like "I hope you will let me show you this".
These videos follows his books in chronological order as index page ?
I wonder what would be say if the class said "no don't show it to us" 😂
@@rahulsharma-ul9li then he'll say, "Congrats for an extended year with me(Failed the person)".
Thanks to Prof Strang and MIT, for putting so much work into this, for the benefit of the entire world, while asking nothing in return. Thanks from mankind.
stole my thoughts. But thanks for writing it before i could. Well, i'm learning machine learning so just refreshing my knowledge in mathematics.
@@infonothing7807 do you need linear algebra for machine learning?
@@yoyolol22
Yes, you do.
You need linear algebra for computer vision. Computer vision falls under machine learning.
@@judepope6196 damn I hate matrices
@@yoyolol22 I too hated matrices and geometry (here in Brazil we study GAAL, which means, Analytical Geometry and Linear Algebra). But think of matrices as a collection of vectors, say column vectors.
What I mean is, instead of studying or using a vector at a time we use a collection of them at once. Think of a vector as a single, very well defined and important, purpose. With this collection, say a toolbox, you could do amazing things.
It may not help, but it did help me start loving LinAlg. Again, reinforcing the idea. Instead of using only one tool (a number), we can use a lot of them at once. That's why matrices appear everywhere, even in probability and statistics.
An example is the multivariate Gaussian distribution.
god bless gilbert strang
and also thanks to MIT for putting them on the web
what on earth did we do before the internet ?
I can't imagine trying to learn this stuff in a school environment! I'm learning or at least familiarizing myself with all of the math I will encounter when I start pursuing a bachelor to master next year.
I did have Linear Algebra in college and, for me, was on of the hardest subjects i attended to.
before the internet we would go outside
We went to the library
You can ask what would thousands of lecturers round the globe do AFTER the internet. Possibly lost jobs forever...
Lecture timeline Links
Lecture 0:00
Permutations 1:17
Possibilities of permutations 7:23
Transposes 10:15
General formula for transpose 11:38
Symmetric matrices 12:43
RᵀR is always symmetric 15:06
Chapter 3: Vector spaces 20:12
What "space" means 22:03
Why is Origin necessary in Vector spaces 25:33
Most important thing about Vector space 28:29
A case that's not a Vector space 29:41
All possible subspaces in R² 35:54
All possible subspaces in R³ 39:04
Subspaces come out from Matrices: Column Space 39:45
Solomon Xie Thank you!
thanks!!!
谢谢课代表！THX for the time pins !
thank you
i love you
So much energy for a recorded video with an empty class. Thank you so much Professor Gilbert!
I didn't notice that because I was too immeresed in his lecture. Thanks for pointing it out. It does feel like he's teaching in front of the whole class.
He is teaching infront of a class
Everyone is silent
That's how good he is
That explains why there is no noise of people leaving classroom before the class even ends like before. I didn't notice that without your comment. This made me feel sad a little bit.
@@lightgaming7618 there's no one (except the camera person, probably) there. He probably had to record this one again or wanted to I don't know. But still, he is teaching as if there are students there. Professor Strang is just this awesome.
There is noone in front chairs. But it is possible that there may be some backbencher present in class. Because professor strang is looking not on camera but on chairs gives hint that someone may be present there.
These lectures are so good that I am falling in love with studies, specifically Linear algebra for the first time. I always dreaded the topic in my college days, just wish there was someone who could have referred me to these lectures back then.
“Take a breath and listen up!” - lecturer’s perspective starting vector spaces. 👍 I just love Gilbert Strang! I completely agree. I will tell that to my students tomorrow. 🙏
Self teaching linear algebra is 100x easier with these lectures!
how's linear algebra going now?
@@avidreader100 MIT ocw + text book + discord group, is way better than my actual "engineering" linear algebra lecture.
@@briann10 wait, what's the discord group? I also wanna join and ask my doubts as studying all by myself is hard.
@@avidreader100 I’d say it is self-learning. You can’t ask questions or get any feedback.
And the audience was speechless.
Literally
I LOVE how he realizes the concept of vector spaces is difficult for most people to understand!
Legendary. No other words. And I’m only 5 lectures in! Can’t wait for more. Thank you to everyone involved in making MIT courses so available!
If I told younger me that I was going to be watching MIT videos to actually learn the content they are teaching, I would never believe it. yet here I am studying for an exam.
"I never drew a pi-e before..." GOD I love this man!
very good service given by MIT for other students who cant study at MIT
Thanks to MIT
he is my hero !
that man made me love linear algebra :)
i can watch his lectures every year !
thx professor Strang! you are the best teacher we're all dreaming about
Prof. Strang is such a good professor! Good pace and he explains it in a very clear and easy to understand way. Thank you so much!!
You save my life. You save my whole grade. I do think my professor should have watched this. Thanx.
My teacher took entire semester and was not able to explain me vector spaces and he did that in 40 mins.. Damn.. Hats off sir .. Kudos.. Love from India
I still can't understand what is subspace
@@nakamura7346 subspace is like a bit of space that lives within a more extensive area, and it obeys all the same rules as the ample space, making it a sort of "mini" version of the larger size.
thank you! @@rashid_iiitd
@@nakamura7346😅I can explain to understand space and subspace u should understand what is dimensions (components)which is basically a way to mathematically measure information on that domain for example in an area of a square there are two components that's define that square that component is there in cube , etc in every thing think about it dimensions is like a axis which just means measures information n-dimension means n different information 2dimension means two different measure of information so I think u got this now u can better understand what is vector space focus it's important a vector space is a collection of vector that's it and it should follow some rules 1 it should contain 0 vector the components wise sum or multiplication should be in that space that's it now let's understand what is subspace which is a space inside a space . A space which follows rules like components addition and components multiplication is simple know as a sub space think about it.
thank you very much, I didn't expect someone would explain this to me so patiently @@anuj7876
My heartfelt thanks to Prof. Strang. You have been forever immortalized in the minds of students by your stupendous teaching!
Bless his soul. I do not understand his textbooks at all but listening to him explain actually makes sense. This is singlehandedly saving me from an F
I find the way he explains the material absolutely amazing!
I don' even know why I go to my linear algebra lectures anymore. This guy is amazing.
Professor Strang lays out the concept of vector spaces in almost tedious detail. The result: I finally confidently understand vector spaces at a very basic conceptual level and I'm already beginning to extrapolate for myself. I imagine that this firm foundation will make the next few lectures that much easier and that much more rewarding. Thank you Professor Strang and thank you MIT.
I paid a ton of money in college went there every day for several hours and learned nothing. And here i am learning free of cost from the comfort of home from the best professors in the world. Respect and love to professor strang.
It was Prof Strang's lecture that benefits me to review and understand more about linear algebra. Thanks to these high quality public courses.
Thank you, thank you a lot. I live in Spain and now I can studied Algebra thank to this videos.
Gratitude, respect and love to Prof. Strang ❤️
This lecture is so different from the math lectures I had (Germany). Our lectures consisted of the prof speed scribbling down Lemmas & Theorems, barely any examples, and they would ask random questions from time to time while everyone was trying to catch up with the blackboard.
This seems to be the total opposite, much better for learning.
Graduate linear algebra is theorems only.
I just love Prof. Gilbert Strang. One of the best teachers humanity has to offer. To all those who haven't watched this playlist - This is the best way this subject can be taught.
"Pi is about 3 and a little more, e is about 2 and a little more"
24:30
that's more badass than anything else 😂
He used the fundamental theorem of engineering
Gilbert Strang, you have my love. I love you. The devotion you show for the subjects you teach is rare. One can easily see why an institution like MIT would like to keep you for the rest of your life. Thank you.
That moment when I realized teacher is giving so much energy and teaching with so much passion under a 3 by 3 space known as "Empty Room Space". 😢
I have an A in my linear algebra course and it sure isn't because of the Textbook, thank you MIT and Gilbert Strang for making these videos free
What a great man and a teacher, wish i had the privilege to be on one of his classes just once
Before the internet, we spent our days eating each others hair. Thanks, MIT and Professor Strang, for making these videos openly accessible. Learning online has really changed my life.
THANKS, Professor Gilbert and MIT for such a great course and clear illustration of Linear Algebra thanks is not enough though
Amazing Lectures. If you think its taking too much time - Watch it at 2X speed. 2X Speed still sounds very clear.
Anyone noticed that there were no students (at least in the front 6 rows) in this lecture?
I would assume that someone noticed the (bad) quality of the last lecture video but didn't find a fix for the problem in time for this lecture. And if that's true, I would assume he was asked to do the lecture again for the camera to continue/complete the video series.
Dustin Charles
If so, he is a phenomenal teacher. Lecturing to an empty room can really throw you off.
***** Dustin Charles Lecture 5 had recording problems during class and was re-recorded in an empty classroom a short time later. The problem for lecture 4 is different. These videos were originally published in 2005 as quarter screen (320x240) RealMedia files. Years later we remastered them in Standard Definition. The tape for lecture 4 was lost, so remastering was not possible. Sadly, the quarter screen version is all we have left of it.
He said "don't all speak at once", but nobody's there. That gave me a chill :(
phung trung I guess it was a joke :D
The way the ideas were tied together in the end - simply beautiful.
My algebra teacher uses his book to teach, while I learn from the author himself. 😂 Upgrades.
Well, he explains so beautifully that you can teach your teacher
@@palashrajput428 yeah I guess lol
Teaching in empty class and still remaining cool. Great professor!
Know what... Actually it's not just the guys sitting in the class who learnt this.... Students from different corners of the globe, from over a decade's gap learnt it... Truly amazing 👍 thanks sir
This is awesome. We use the fourth edition of Dr. Strang's Intro to Linear Algebra in class and I am so glad these videos are available online. I found vector spaces to be a tricky topic in class, via my notes, even in the book. This lecture was executed beautifully. Makes so much more sense now. I am definitely the type of learner who really needs to have a deep understanding for the concepts to stick. Thanks so much!
Oh how I wish my Linear Algebra professor had been this clear !! Thank you! Thank you! Thank you!
Thank you so much Sir! Making a lecture without students at that time. Seeing your lectures is way easier than those college lectures that directly start with vector spaces second day.
19:10 "What is R transpose transpose?"
>empty classroom
"Well, don't all speak at once"
🤣🤣🤣🤣
professor you have my deepest thanks for these wonderful lectures. you will always be remembered by your lovely lectures
Thanks so much for video taping it and sharing it. This is helping so many people!
I want to thank MIT for putting this up and Sir Gilbert Strang for such an amazing lecture
I just want to say thank you to Sir Strang for making such an amazing series.I am really helped a lot by his series.
Coming from lecture 4 I feel like my ears are being sensually caressed.
Thank god you saved my life. Its mysterious how people could understand this kind of things without internet and youtube!
Anyone going through these beautiful lectures and finding difficulty in visualizing some of the stuff - I would strongly recommend you watch 3blue1brown's series on linear algebra. After watching that, try to visualize what Prof. Strang is saying when he is talking about vectors and vector spaces. That will help you to grasp the key concepts.
Prof. Strang telling an empty room not to "all speak at once" at 19:07 gives me life.
I just can't stop watching this series. They are brilliant!
i am in love with gilbert
thx
this is such an awesome lecture, especially after I notice that Processor Strang was doing a re-recording without any audience
Absolutely legendary. Professor Strang explains linear algebra 10 times better than my own professor at school. : )
This is truly golden knownledge. Gilbert Strang is incredible. I love this linear algebra course so much
😭😭.. Linear algebra was never understandable to me before, I always it only there to make my life difficult... This series saved me. So much love and respect Prof.Gilbert. Bless you. 💜
That was a beautiful ending. It made me cry
So good to be back do decent quality after Lecture 4!
OCW has actually posted a copy of that video that is more or less the same quality as the rest of the videos. It's linked from a few places, I think, but the link is pinned near the top of the poor-quality video.
This is another excellent video on transposes, permutations and vector spaces. This is the real part of linear algebra.
When he said this lecture is the Beginning of Real Linear Algebra,I got goose bumps
Thank you MIT, thank you Mr. Strang, this is the way EVERYTHING should be taught!!!
These lectures will live on forever.
What an amazing lecture and the explanation of vector spaces, and subspaces is just so clear and understandable it's poetic.
Knowing that the class is empty, 19:09 becomes incredibly hilarious.
That was awesome :D Gotta love this guy
Surprised that Prof. Gilbert could finish this lecture without any mindblowing which shows how professional he is
That moment at 19:08 when Prof. Strang says, in response to his question posed to the empty lecture hall, "Well, don't all speak at once" 🤣🤣
Oh my god I just realized that haha, he’s such a sweetheart
Paragon of pedagogical excellence. Infinite gratitude.
you have changed lives of millions.
The akward moment when free knowledge is given and you complain about quality of video
not only free knowledge but also that quality of free knowledge is given. It's a must to be respectfull
I do see your point, but not many great researchers even make it as a half-ass instructors. I studied & worked under/with the creme de creme of research, who I would not bother to ask a question as most live inside his own brain. This professor is worth asking questions & even complaining about. So, I have not complained; however, those who complain obviously care to dare.
He is a great researcher & teacher at the same time. So, think of the complaints as love expressed in frustration.
I wish I discovered this 10 years ago. And, I did make a small contribution to MIT Cloud Learning Foundation. If anyone reading this has not then pls care do donate. MIT is going for 1000 contributors (any amount) until June 28th.
Participation Drive 721 Donors 72% Towards goal of 1,000 Donors $39,710.00 Raised 4 DAYS LEFT
The quality of this video is an insult for this respectful man,
uza abi
@@bilaltekin1531 The quality of video is not so bad. It just reflects the 4:3 aspect ratio which was common in the time of shooting, year 2008 or so?
WOW HIGH QUALITY THANK GOD
+Kelvin Lau I know! can't believe I survived the horror of L4
+Raed Khader L4 is the devil itself
He reveals the beautiful face of Math. I am not only learning but also I am really enjoying it. Thank you
45:24 column space with 2 linearly independent vectors in R3 form a column space which is a plane through the origin
My observation is not a comment about the quality of instruction. I really enjoyed these lectures, and I really appreciate Strang's instruction. I thought it was interesting that there was no one there, I wasn't implying anything.
thanx alot for this great lectures, i´m from Portugal and I´m amazed with this oportunity that M.I.T. gave to every one around the world M.I.T.believes in a better world where everyone can get free high quality education. With this actitude of M.I.T., i´m starting to baleave myself that a better world is possible.
After watching Prof. Strang classes, I have decided to become a lecturer or a teacher . He is a true definition of passion.
I love that many universities and colleges do this sort of thing. Looking at different points of view on the same subject is very helpfull. This is extremely well explained, thank you.
Ok two things. First - That is one giant piece of chalk. Second - To be able to lecture this well with nobody sitting in front speaks volumes about the qualities of this great teacher.
This man works miracles.
THANK YOU PROFESSOR GILBERT.
Whole Class Is empty ...
Even though He Lectured in such a way that Class is full of students
Awesome
Que manera de explicar tan simple los espacios y subespacios vectoriales , cuando los vi en la universidad no entendía de que se trataba. Muchas gracias.
I've been using the terms "Closed under..." in two of my midterms now without caring of what it meant exactly graphically...its only now that I understand it. Thanks
Thank you for being the best teacher on the planet.
My greek professor should watch this. Thanks to this Professor Gilbert Strang!
I’m in the top engineering school in all the United States and have already learned this yet MIT still makes it intimidating
Hailing from the horrible video experience that is Lecture 4, I feel at peace finally
I am skipping that lecture, hopping to do the concepts from the book itself.
The only problem with this video is that it feels a bit lonely and desolate, since there are no students there.
But that's not really a big deal. =P
it was reuploaded btw
actually enjoyed it
This is brilliant! I can see why MIT is one of the best universities in the world!
17:26 "That 7 is no accident...that 11 is no accident..." And I live over a 7-11...spooky, Dr. Strange.
Thank you MIT and Gilbert Strang sir,you made my studies simpler
THANK YOU VERY MUCH FOR YOUR GOOD LECTURES.IM WIS wishing you a very Merry Christmas and a HAPPY new year.YOU DESERVE ALOT OF DONATIONS.THANKFUL FOR YOUR LECTURES.
Thanks very much for those videos. For the first time in my life I finally understand those concepts. Hopefully, statistics textbooks will become less scary from now on.
“Well don’t speak all at once” 👍 gotta love his sense of humor !
"You sure have to see it in R^3, because we're going to do it in R^10, and we may take a combination of 5 vectors in R^10, and what will we have...?
God knows..."
LOL!
Truly amazing teaching skills. I want to see how well he would have explained it given more time.
self-learning algebra here... thanks Prof Strang and MIT.
And we are so thankful that you did re-record it! :D
Thank you MIT. It is helping me to understand the fundamentals of linear algebra.