Prof. Strang certainly sensed the central role of Linear Algebra in the ML/AI development and produced this series to help those who want to tap into the area. When everyone is complaining about the fast moving world and criticizing how short-sighted people have become, Prof. Strang is answering people's demand by distilling his 18.06 course into a even shorter introduction, with fully updated digitalized slides, AND it's FREE. What can you say? RESPECT.
houkensjtu “FREE” vs. ‘open source’ are different concepts. Printed course materials still must be purchased and I’m sure there’s an option for donations. Perhaps you’re familiar with ‘shareware’?
“ Can I tell you all right from the start?” Yes, Only you can, Dear Dr. Gilbert Strang, I miss your chalk board and your passionate handwriting, big matrices, lines and planes drawings. I think, you miss it too when you are trying to show lines and planes with your hands. Please let him write on white board or even better a wall and keep it there forever like an art! He is the best improvising, jazzing through the lecture, taking it in unpredictable direction and finishing on the most interesting moments, that you just can’t wait till the next episode, I mean lecture. You inspire me to teach Linear Algebra your way. I am so grateful!
Clearly, Professor Strang is one of the best teachers- maybe the best teacher- on the planet. As a PhD student (not at MIT), I bought and read all his books. His book- Introduction to Applied Mathematics- is a marvelous reference for understanding the fundamentals of many areas of applied math. I spoke to him on phone in late 1990s regarding his book on Wavelets. He is also a very nice, caring person.
Omg 4 years ago i just start the same course in youtube, but i given up in half way .Cant imagine now i get the 2020 vision, and you just passion as the old day . May be i should start all over again and finish it this time, you are the fortunes of humanity.
▲ Oh, how nice to see our dear professor again in a new and modern environment, but most importantly, with the same zeal for teaching math. I wish you all the best, Professor Strang. Sincere greetings from Bosnia and Herzegovina.
I still cannot believe I was using his book back in 1978 in my Linear Algebra class that is right next to me now on the shelf....just reached over and touched it!! Great to see you still out there Prof. Strang!
Dr. Strang: On slide 4, perhaps observe that the third column is the sum of the first two, so when "filling in" between the first two, we don't go in any genuinely new direction, and therefore we have a plane, not all R^3. (Clearly the first two columns are not in the same direction because you can't rescale one to get the other, so we have at least a plane.) You discuss this on slide 5, _after_ you've already made use of it to claim a 2D column space. (And a big THANK YOU for these videos! They're excellent!)
Very smart way to find Reduced row echelon form. I watched the original videos two times and it helped me enormously in my engineering career. I really can’t thank you enough.
I'm currently taking L.A. and physics at University and those playlists are a true blessing, paradoxically I'm learning much better now during lockdown then when I did at normal lectures,Prof Strang and Prof Lewin have become mentors of mine...you guys are revolutionizing education , i m sure that those lectures will reach one billions of views
what i like in your videos not the way you explain them but the motivation to explain this course another time with a different way you give me a motivation so not to get bored while watching Cheer ! :D
4:40 Can someone help me understand why Ax (All linear combinations of columns of matrix A) fill out a plane in 3D-space? What I understand: - Like Mr. Strang explains, if you have two crossing lines that meet at (0, 0, 0) and fill out the space between them, you get a plane. What I don't understand: - How do you get from "(1, 3, 2)^T*x1 + (4, 2, 1)^T*x2 + (5, 5, 3)^T*x3" where each vector obviously has 3 coordinates to the understanding that every linear combination of them resides on the same plane?
Just those 3 particular vectors are coplanar (e.g., via triple product, y × z is orthogonal to both y and z, but if x ∙ (y × z) = det(x, y, z) = 0, then x is orthogonal to y × z, hence it is coplanar with y and z). In general that isn't true, for example, for I₃ = diag(1, 1, 1) we have I₃ x = x, which gives us the whole ℝ³ back.
Wow looking at the response, it feels like people go out of their way to make things waaay more difficult, or to sound smart. The prof didnt really explain it very well and I had trouble with it at first. Visualising it helps tremendously. Imagine two vectors in space starting from zero, with a small angle between them, Like say the angle between your index and your middle finger. Now imagine a third vector that adds those two original vector, (it of course also starts from zero, its a simple addition). What youll find is that with these three vectors, you've lost a dimension (the third) and are operating on only two dimensions, i.e a plane. This is entirely because that third vector adds the first two. I have another silly but probably effective example, that will help illustrate why the domain becomes a plane instead of remaining 3d. Let me know
The third column is a linear combination of the first and the second (C3 = C1 + C2), which means that the columns of the matrix are not linearly independent. The first two columns represent two vectors which define a plane in R3. Since the third column can be expressed in terms of these two column vectors, it is on that plane.
For those of you who miss understanding and rationale behind the very basics of matrices and vectors, I highly recommend "3 blue 1 brown" youtube channel. You won't regret it, the guy is a genius and explores the math the way it was explored by pioneers. You will like it.
Its a really intense and distilled video and I don't think I understood everything but I still love it! I am going to use it as a start, especially to get some basic ideas, for which it is amazing, go to the longer videos and then come back! Thank you very much
I really like Prof. Strang's lecture. It is enlightening for beginners like me. Maybe drawing on a blackboard will help to explain the concepts better than PowerPoint slides 😄
why is the linear combination from minute 5 only taking up a plane? I would think because it has 3 vectors of 3 components, it would fill r3... I am new to this study though so some clarification would be really helpful
@@microndiamondjenkins566 Are you still single because you are very pretty and I am single if is there any scope of .......you know what I am talking about
Excellent first lecture. That CR factorization was new to me. There is just one detail I don't understand. Probably I am just a bit tired. We have r independent rows in the R matrix. The row space of A lies in the span of those rows. But how do we know the rows of R don't span a larger space than the row space of A?
6:04, why are there only 2 lines that meet not 3? Wouldn't each of the column vector represent a line, in which case the 3 vectors define the whole R^3?
@@harshitjuneja7768 I believe I understand it now. A single column vector represent a point not a line. A line has to be defined by 2 vectors, one indicating any point along the line, and the other the direction.
@@harshitjuneja7768 Imagine two vectors in space starting from zero, with a small angle between them, Like say the angle between your index and your middle finger. Now imagine a third vector that adds those two original vector, (it of course also starts from zero, its a simple addition). What youll find is that with these three vectors, you've lost a dimension (the third) and are operating on only two dimensions, i.e a plane. This is entirely because that third vector adds the first two. I have another silly but probably effective example, that will help illustrate why the domain becomes a plane instead of remaining 3d. Let me know
I like that you're explaining column spaces first, but i tought it was a little confusing to jump into an example straight away on slide #4 since it's unclear which parts of what you're explaining apply to this matrix, matrices like it, or all matrices. i think it would have been better to have a quick summary of how different kinds of matrices result in the different kinds of column spaces, and then give examples of line, plane & R^3 to illustrate the differences.
@mitopencourseware, Thanks so much for providing us with a brand-new Lienar Algebra course! Really appreciate your effort to provide free, quality educational content during these difficult times. I have one question, though; is this course meant for first time LA students, or is this primarily for computer scientists who need refreshers on relevant topics? Thanks in advance.
my undergrad linear algebra course used his text. he was a cult figure among math majors. in a just world, and a civilized society, Gilbert Strang would be a household name, and cardi b would be unheard of..
I don't like how the video keeps switching between a version of the slides with Dr. Strang and a version without him. Very distracting :( What's the point of switching the views around if we can already clearly see the material from the screen Dr. Strang is using? MIT please minimize unnecessarily context switches in future videos. It's mentally taxing for some of us. Thanks!
Thought this course was gonna be on factorising quadratics, expanding triple brackets, completing the square etc... then I read the channel name "MIT"... probably some university level stuff 0of
Gave up after five minutes. It was wandering and disjointed with no attempt to introduce concepts in order. I've done some linear algebra previously so I'm not a total newb, but this just didn't help.
Yeah I think this video is really for those who have taken a full study of linear algebra already. He clearly presupposes knowledge of linear combinations, spaces, matrices, etc. I wouldn't recommend this if you're just starting out. My study of linear algebra bounces between abstract and concrete. My first introduction was from Lay/Lay's textbook and I just didn't gain much from it. I've been reading Sheldon Axler's textbook (far more abstract) and it's the first time I actually understood what a vector space was, what a linear map was, etc. He goes into what a matrix represents, too. Really, a vector is just any member of a vector space. We can represent them geometrically (having magnitude/direction) or computationally (just a collection of values describing some feature), but they go far beyond that abstractly. What's helped me is not getting so much on what a vector "is" and the hardcore definitions of certain objects, but the relationships between objects. Who cares what a vector space is? It's anything built on a field that's closed under addition and multiplication and possess certain axioms. Then, whatever that object is, its elements are vectors. This isn't a really rigorous explanation of any of this, but this has helped me out. I definitely share your frustration as it seems a LOT of resources out there presuppose a great deal of mathematical thinking and rigor before diving it and it leaves people thinking that can't understand the subject when instead there are often tons of unspoken prerequisites going in, even if they say otherwise.
For links to Professors Strang’s related courses on OCW, visit the Related Resources page on the full resource site: ocw.mit.edu/2020-vision.
Prof. Strang certainly sensed the central role of Linear Algebra in the ML/AI development and produced this series to help those who want to tap into the area.
When everyone is complaining about the fast moving world and criticizing how short-sighted people have become, Prof. Strang is answering people's demand by distilling his 18.06 course into a even shorter introduction, with fully updated digitalized slides, AND it's FREE.
What can you say? RESPECT.
18.06, you mean. I thought for a moment there you meant Strang had a lectuere in the 19th century. I certainly agree with you though.
@@chaebeomsheen5875 should have put a period in there 18.06
余宗翰 that’s not a period, it’s a decimal point: 1, 8, 0, and 6 are all numbers.
houkensjtu “FREE” vs. ‘open source’ are different concepts. Printed course materials still must be purchased and I’m sure there’s an option for donations. Perhaps you’re familiar with ‘shareware’?
@@HighestRank You're right. Decimal point is the proper term.
Listening to this makes me so unbelievably happy, I could cry. Gilbert Strang is the math teacher we always wanted.
“ Can I tell you all right from the start?” Yes, Only you can, Dear Dr. Gilbert Strang, I miss your chalk board and your passionate handwriting, big matrices, lines and planes drawings. I think, you miss it too when you are trying to show lines and planes with your hands. Please let him write on white board or even better a wall and keep it there forever like an art! He is the best improvising, jazzing through the lecture, taking it in unpredictable direction and finishing on the most interesting moments, that you just can’t wait till the next episode, I mean lecture.
You inspire me to teach Linear Algebra your way. I am so grateful!
I miss him working on the chalk board as well.
The Legend is back!!!
The Legend never left.
Legends never die!
Clearly, Professor Strang is one of the best teachers- maybe the best teacher- on the planet. As a PhD student (not at MIT), I bought and read all his books. His book- Introduction to Applied Mathematics- is a marvelous reference for understanding the fundamentals of many areas of applied math. I spoke to him on phone in late 1990s regarding his book on Wavelets. He is also a very nice, caring person.
This guy is excellent. He's making me interested in this, and without even any fun visuals or anecdotes, but purely from his high energy.
Omg 4 years ago i just start the same course in youtube, but i given up in half way .Cant imagine now i get the 2020 vision, and you just passion as the old day . May be i should start all over again and finish it this time, you are the fortunes of humanity.
▲
Oh, how nice to see our dear professor again in a new and modern environment, but most importantly, with the same zeal for teaching math. I wish you all the best, Professor Strang. Sincere greetings from Bosnia and Herzegovina.
Protect Dr. Strang at all costs during this epidemic!!!
I still cannot believe I was using his book back in 1978 in my Linear Algebra class that is right next to me now on the shelf....just reached over and touched it!! Great to see you still out there Prof. Strang!
Dr. Strang: On slide 4, perhaps observe that the third column is the sum of the first two, so when "filling in" between the first two, we don't go in any genuinely new direction, and therefore we have a plane, not all R^3. (Clearly the first two columns are not in the same direction because you can't rescale one to get the other, so we have at least a plane.) You discuss this on slide 5, _after_ you've already made use of it to claim a 2D column space. (And a big THANK YOU for these videos! They're excellent!)
Very smart way to find Reduced row echelon form.
I watched the original videos two times and it helped me enormously in my engineering career. I really can’t thank you enough.
First did Linear Algebra in the early 70's, never really understood it until I found his video's. An awesome teacher/lecturer.
I'm currently taking L.A. and physics at University and those playlists are a true blessing, paradoxically I'm learning much better now during lockdown then when I did at normal lectures,Prof Strang and Prof Lewin have become mentors of mine...you guys are revolutionizing education , i m sure that those lectures will reach one billions of views
The immortal legend. His enthusiasm for the topic is so damn intoxicating
Gilbert Strang's brand new video. This is a treat. I've been following him on MIT Open Course for about 10 years.
Nice! New linear algebra course from Professor Strang!
Good to refresh my knowledge in linear algebra even after my ML exam😄 Remember, linear algebra is the king in ML and calculus the queen
Agree😆
With this man's help I'm filling in the small gaps in my computer science degree.
what i like in your videos not the way you explain them but the motivation to explain this course another time with a different way
you give me a motivation so not to get bored while watching
Cheer ! :D
He made my day whenever he says "Thanks"
Dr Strang is the best thing ever happened to Linear Algebra for me ... because he is on youtube and that too for free
❤️ from India. I learn " linear algebra " from your videos .
God bless you , sir
this is art not a course . just wonderfull
5:36 :)) ''I ask..Ohh I 've just answered this question. Sorry!" lol
Such a funny and cute guy. :)) Wish you all the best boss
Hero is back! Watching during Covid-19. I like Linear Algebra!
Just listening to him makes my day
4:40 Can someone help me understand why Ax (All linear combinations of columns of matrix A) fill out a plane in 3D-space?
What I understand:
- Like Mr. Strang explains, if you have two crossing lines that meet at (0, 0, 0) and fill out the space between them, you get a plane.
What I don't understand:
- How do you get from "(1, 3, 2)^T*x1 + (4, 2, 1)^T*x2 + (5, 5, 3)^T*x3" where each vector obviously has 3 coordinates to the understanding that every linear combination of them resides on the same plane?
Just those 3 particular vectors are coplanar (e.g., via triple product, y × z is orthogonal to both y and z, but if x ∙ (y × z) = det(x, y, z) = 0, then x is orthogonal to y × z, hence it is coplanar with y and z). In general that isn't true, for example, for I₃ = diag(1, 1, 1) we have I₃ x = x, which gives us the whole ℝ³ back.
@@エリートコホモロジー thank you.
This is the issue with explaining LA in an hour heh
One equation with 3 variables means 2 degrees of freedom: one pivot variable, 2 free variables->2 spanning vectors after scalar extraction
Wow looking at the response, it feels like people go out of their way to make things waaay more difficult, or to sound smart. The prof didnt really explain it very well and I had trouble with it at first. Visualising it helps tremendously.
Imagine two vectors in space starting from zero, with a small angle between them, Like say the angle between your index and your middle finger. Now imagine a third vector that adds those two original vector, (it of course also starts from zero, its a simple addition). What youll find is that with these three vectors, you've lost a dimension (the third) and are operating on only two dimensions, i.e a plane. This is entirely because that third vector adds the first two.
I have another silly but probably effective example, that will help illustrate why the domain becomes a plane instead of remaining 3d. Let me know
The third column is a linear combination of the first and the second (C3 = C1 + C2), which means that the columns of the matrix are not linearly independent. The first two columns represent two vectors which define a plane in R3. Since the third column can be expressed in terms of these two column vectors, it is on that plane.
Thank you to Prof. Strang - a living legend!
Hope his eye might stay without getting so bad.
For those of you who miss understanding and rationale behind the very basics of matrices and vectors, I highly recommend "3 blue 1 brown" youtube channel. You won't regret it, the guy is a genius and explores the math the way it was explored by pioneers. You will like it.
Grant Sanderson rocks!
Thank you! He is so damn good at explaining. He shows the right way to learn Linear Algebra.
I support this message
A very helpful and condensed review series of linear algebra
Its a really intense and distilled video and I don't think I understood everything but I still love it! I am going to use it as a start, especially to get some basic ideas, for which it is amazing, go to the longer videos and then come back!
Thank you very much
thanks for your devotion professor Strang.
the best linear algebra prof ever !!!
Thank you so much for providing the courses!!!! Best linear algebra course ever!!!
I really like Prof. Strang's lecture. It is enlightening for beginners like me.
Maybe drawing on a blackboard will help to explain the concepts better than PowerPoint slides 😄
The legend returns
I like way you teaching. I have graduated and find job about mathematical software with your lessons.
Wow new way; l just finished this course yesterday. I used your blackboard lecture to study this part and I liked it. Thanks.
God bless Mr. Strang.
What a wonderful lecture. Thank you so much !
Feeling so alive so see a 85 years old teaching😢
A=CR is so brilliant that it can be used to build the simplest and shortest proof of *row rank = column rank*.
why is the linear combination from minute 5 only taking up a plane? I would think because it has 3 vectors of 3 components, it would fill r3... I am new to this study though so some clarification would be really helpful
ok he answered this! we don't need to evaluate the 3rd column because it is just the sum of the first two so it will be on the same plane.
@@microndiamondjenkins566 Are you still single because you are very pretty and I am single if is there any scope of .......you know what I am talking about
Thank you Dr. Strang!
Best way to spend a quarantine
Without advertisements 🙏👌😍
I still prefer Prof. Strang with chalk and board
I miss him working on the chalk board as well.
This video cured my depression
Thank YOU Professor Strang！🌹🌹
My full respect
Best linear algebra teacher
Thank you MIT
Strang is the GOAt
Thank you professor Strang 🙌
Excellent first lecture. That CR factorization was new to me.
There is just one detail I don't understand. Probably I am just a bit tired.
We have r independent rows in the R matrix. The row space of A lies in the span of those rows.
But how do we know the rows of R don't span a larger space than the row space of A?
6:04, why are there only 2 lines that meet not 3? Wouldn't each of the column vector represent a line, in which case the 3 vectors define the whole R^3?
did you get the answer to this? I have the same question.
@@harshitjuneja7768 I believe I understand it now. A single column vector represent a point not a line. A line has to be defined by 2 vectors, one indicating any point along the line, and the other the direction.
@@harshitjuneja7768
Imagine two vectors in space starting from zero, with a small angle between them, Like say the angle between your index and your middle finger. Now imagine a third vector that adds those two original vector, (it of course also starts from zero, its a simple addition). What youll find is that with these three vectors, you've lost a dimension (the third) and are operating on only two dimensions, i.e a plane. This is entirely because that third vector adds the first two.
I have another silly but probably effective example, that will help illustrate why the domain becomes a plane instead of remaining 3d. Let me know
Step by step video solutions for civil engineering questions
best teacher ever 😁
Hats off to u Sir......high respect to you
12:53 Do anyone have examples for ill-conditioned C and R?
Kind of confused
Same question! Don't know what 'difficult to deal with' means
Love it love it
I like that you're explaining column spaces first, but i tought it was a little confusing to jump into an example straight away on slide #4 since it's unclear which parts of what you're explaining apply to this matrix, matrices like it, or all matrices. i think it would have been better to have a quick summary of how different kinds of matrices result in the different kinds of column spaces, and then give examples of line, plane & R^3 to illustrate the differences.
Wonderful explanations, thank you!
Thanks the lengend comes back! Why is every lecture less than 15mins now? Is this because of no writing on backboard anymore?
i love this perspective
Thanks Professor!
Thank you so much for your help!
My distracted ass just wanting to hook up a PS4 to that massive television
Hail prof. Strang
@mitopencourseware, Thanks so much for providing us with a brand-new Lienar Algebra course! Really appreciate your effort to provide free, quality educational content during these difficult times. I have one question, though; is this course meant for first time LA students, or is this primarily for computer scientists who need refreshers on relevant topics? Thanks in advance.
my undergrad linear algebra course used his text. he was a cult figure among math majors.
in a just world, and a civilized society, Gilbert Strang would be a household name, and cardi b would be unheard of..
cardi who?
@@ronaldjensen2948 amen!!
tks to my hero
Hello, Can Some one point to me where is the PPT that Prof Strang is using in the videos???
It doesn't look like the PPT was made available. What we do have available is here: ocw.mit.edu/2020-vision. Best wishes on your studies!
Thank you very much
This is like yoga in linear algebra
I love this guy
I think it's need to be more edited For visualization.."
You can find unbeatable visualization here - th-cam.com/video/uQhTuRlWMxw/w-d-xo.html
Thank you Sir ! Best Regards
Ok I will come back after I finish calculus 1, 2, 3 & linear algebra class😢
You can just start to watch this series of linear algebra. You know he always makes it easy :)
정말 이렇게 늙고 싶다. 너무 멋지시다.
Kinda feels sad realizing that Prof. Strang has grown so old :/
I don't like how the video keeps switching between a version of the slides with Dr. Strang and a version without him. Very distracting :(
What's the point of switching the views around if we can already clearly see the material from the screen Dr. Strang is using?
MIT please minimize unnecessarily context switches in future videos. It's mentally taxing for some of us.
Thanks!
very nice !
2020 vision. Clear vision. Good joke. 20/20
Where can I find all your videos?
th-cam.com/users/MITsearch?query=Gilbert+Strang
@@mitocw Thanks
I do not understand slide five what so ever.
Thought this course was gonna be on factorising quadratics, expanding triple brackets, completing the square etc... then I read the channel name "MIT"... probably some university level stuff 0of
We have a full range of mathematics. Here is a list of what we have: ocw.mit.edu/courses/find-by-topic/#cat=mathematics. Best wishes on your studies!
8:09 ok
Ok this video is so helpful.
I did NOT know you can understand rref like that... Just wow.
Linear Algebra review day6
Legendary!!! Which presentation software does he use?
I’m sure he is just using Microsoft PowerPoint with a theme applied to the slides.
He is using this en.wikipedia.org/wiki/Beamer_(LaTeX)
LaTeX files were saved as PDF and placed into PowerPoint.
Please just give him a black board with a chalk and watch the magic
I miss him working on the chalk board as well.
LaTex!!!!!
Не, лекции Мануйлова лучше будут)
Respect. Chalk and board is better
Not a very clear explanation this time to know where everything starts and where everything ends.
Please, for the love of god, stop that jumping back and forth. Just show the slides! We know the prof. Thank you
Who disliked...?
Probably someone who is not precise with their cursor.
Gave up after five minutes. It was wandering and disjointed with no attempt to introduce concepts in order. I've done some linear algebra previously so I'm not a total newb, but this just didn't help.
ugh, not a single visual representation of the vectors, ever
linear algebra teaching just SUCKS everywhere, there's nothing revolutionary here
Yeah I think this video is really for those who have taken a full study of linear algebra already. He clearly presupposes knowledge of linear combinations, spaces, matrices, etc. I wouldn't recommend this if you're just starting out. My study of linear algebra bounces between abstract and concrete. My first introduction was from Lay/Lay's textbook and I just didn't gain much from it. I've been reading Sheldon Axler's textbook (far more abstract) and it's the first time I actually understood what a vector space was, what a linear map was, etc. He goes into what a matrix represents, too.
Really, a vector is just any member of a vector space. We can represent them geometrically (having magnitude/direction) or computationally (just a collection of values describing some feature), but they go far beyond that abstractly. What's helped me is not getting so much on what a vector "is" and the hardcore definitions of certain objects, but the relationships between objects. Who cares what a vector space is? It's anything built on a field that's closed under addition and multiplication and possess certain axioms. Then, whatever that object is, its elements are vectors.
This isn't a really rigorous explanation of any of this, but this has helped me out. I definitely share your frustration as it seems a LOT of resources out there presuppose a great deal of mathematical thinking and rigor before diving it and it leaves people thinking that can't understand the subject when instead there are often tons of unspoken prerequisites going in, even if they say otherwise.