15. Projections onto Subspaces

[@shubhayanghoshal5774]

0:15 "...Let's make this lecture immortal"

[@趙超]

NOW IS 2019

[@eduardosdelarosa5539]

He did

[@吴瀚宇]

2020 right now

[@quirkyquester]

Yeahhh, i like that i dea

[@martintoilet5887]

yeaheheahea

[@MikeJones-ue7ux]

I don't know if anyone else has noticed this, but the gradual shift into abstract concepts is done flawlessly by Professor Strang. In the first few lectures, really up until the lectures on bases, all of the methods Professor Strang uses to teach the subjects were concrete examples; very little was done with abstract definitions and arbitrary matrices. Over the last few lectures, Prof. Strang has made a gradual shift into abstractness which let's the mode of this sort of thinking sink in. Now, in this lecture, nothing is concrete and no numbers are ever used (until the end) and it feels effortless to understand! This man is a genius when it comes to math education. Thanks so much MIT and Prof. Strang for the amazing experience!

[@tarunkalluri1799]

The lectures are beyond fantastic no doubt, but I really feel he could have thrown some proofs in between, he mostly "says" things without ever proving them. Yes, we can get the proofs from anywhere else, but I want to hear what would his approach be.

[@damnit258]

yeah, it's weird yet fantastic!

[@NazriB]

Lies again? Postal Code

[@mingweifei7694]

This video clears up my questions and made this topic easy for everyone

[@Y2B123]

@@tarunkalluri1799 That would make the lectures much longer and means little to the general audience. This is just an introductory course much like entry-level calculus - it is doubtful most students would benefit from being taught real analysis as they are grappling with the idea of derivatives for the first time.

[@Al.Mo.]

after 9 years, it is immortal ,professor, it is immortal

[@bephrem]

I was thinking the same thing

[@ankitbabbar4883]

After 11 years too :)

[@9888565407]

after 15 years too prof !

[@tino139]

After 15 years

[@MrNiceFromUkraine]

@@tino139 after 20 years. The lectures were filmed in 2000

[@astrophilip]

For people who deal with statistics, this lecture is probably the most important in the entire series. This is so fucking good.

[@ramonmassoni9657]

a bit late but could you develop on that?

[@danielezeme3364]

@@ramonmassoni9657 he was referring to multivariate linear regression which is a huge topic in statistics.

[@randomdude354]

@@ramonmassoni9657 I'm a statistician, but the explanation about why the projection matrix is equal to its square is something I had never thought about. And yes, that property of the projection matrix is absolutely key in multivariate linear regression, which is a very important method in statistics.

[@RC-bm9mf]

This kinda thread itself also underpins and strengthens this body of knowledge about linear algebra set up by prof. Strang

[@KatherineRogers]

The combination of lecture 14 and 15 are works of art, things of beauty. Thank you MIT. Thank you professor Strang.

[@diysumit]

The thing I really like about these old timer Professors are the way they teach, he stops and ask questions as if he is also going through the problem he already knows everything but in the moment he is walking through us students and trying to look problems through our eyes, I would say this is the greatest method of teaching.

[@jessstuart7495]

This teaching method only works if the instructor has a rock solid understanding of the subject.

[@dalisabe62]

Indeed. The good teacher never stops wondering and postering. He pretends to not know to engage the student in the inquiry and even allows himself to make mistakes and get confused then tries to recover from the confusion exactly like a student who is in the heat of learning!

[@blablablerg]

Strang is a legend. He should get an ovation everytime!

[@RC-bm9mf]

I do, and I believe there’s at least one person who does for each lecture; I guess we collectively give him an ovation. :)

[@MatureFister]

i feel the same, in german unis there is this tradition to knock on the table as a form of applauding. I always feel like strang deserves this, too.

[@jeshwanthpilla5056]

You don't erase the board at MIT! You just raise it! :P

[@anshugarg5955]

I just can't thank you enough for showing such a clear picture of these concepts. I was struggling to understand 3D Computer Vision and how the linear algebra involved in it. Your lecture series has connected all the points which I have tried very hard enough to understand but I couldn't until I followed your lecture series. You have shown the whole picture by explaining the logic behind each and everything. You are an awesome professor. Thank You so so much. The way you explain is incredible, brilliant.

[@schen9580]

SO DO I !!!

[@alek282]

I rewatch this at least once a year and it never fails to amaze me, beautiful work professor

[@dalisabe62]

When Professor Strang follows the analogy between single vectors and matrices within the comparable formulas, he doesn’t use rigor to prove valid the analogy. He just arises the curiosity of the student as to how beautifully the math behaves in linear algebra as we move from one dimension to higher dimensions. The projection of one vector over another has the same logic and hence similar algebra as the projection onto a plane that could be defined by two vectors in the plane. Formulating a matrix from the vectors in the plane as the matrix columns creates a projection matrix that displays similar looking algebra (comparing little a with capital A). It is brilliant and provoking.

[@professorfernandohartwig]

This is the best explanation of the projection formula that I have ever seen. Prof. Strang not only provides a mathematical basis of the formula by deriving it, but also provides clear intuition and motivation. He also takes the time to go through the simpler case of projecting onto a line first, which allows building up the understanding of the content gradually. What a great lecture!

[@aravinds5920]

Somethings are so beautiful when comprehended a certain way. Prof. Strang's lecture are as much about the art of transfering such beautiful comprehension one might have experienced as it is about Linear Algebra. Many in youtube have the skill of making a difficult things easy to grasp. But what you also get from Prof. Strang is that sense of euphoria that comprehension of something beautiful invokes.

[@jessstuart7495]

The great thing about these recorded lectures, is that I can pause the video each time Dr. Strang asks a question, and spend several minutes thinking about it, or working it out on paper on my own, before proceeding. It takes me three times as long to get through the lectures, but I feel like it is time well spent.

[@chriscook25]

I can barely hesitate to say that these lectures are incomparable to the lectures I attend for Linear Algebra at university. It's uncanny how often when watching these videos I think, "I get it now, I get it!" Cudos to you Prof. Strang!

[@richsmith295]

Good for non mathematics

[@dwijdixit7810]

The Lecture has become immortal, sir! Thank you! (Ref - first sentence of Prof. Strang in the lecture)

[@skull8888888]

Now I finally understood the linear regression in its matrix form

[@LAnonHubbard]

This video is just incredible. I've been watching the Khan Academy LA videos and have covered this over there, but seeing this gives me yet another point of view. Coming at it from a slightly different angle is so good at solidifying the concepts. In this lecture I find Professor Strang keeps things high level and really shows the power of thinking at the high "linear algebra" level. Nice to see a video on projection without a cosine in sight :)

[@drsacoengg]

I asked to my prof " why do we minimize the square of the error in the LSM?" He replied "To reduce the distance" today I understood what does he mean by that after 3 years

[@go_rohit]

When you start asking questions yourself and find solutions of it hidden inside the teachings, you know how great the lecturer is to give you that sort of understanding. Thank You Mr. Strang for this immortal course.

[@zelousfoxtrot3390]

These kids rush out every time. They have no idea the blessing they were given. We never know how impressive a good instructor is until we are trying to learn these things on our own later in life. Thank you for making this available to all. (Yes, I rushed as a young college kid too, young people can be stupid.)

[@lucisetumbrae]

One of the crown jewel lectures of MIT OCW.

[@rushikeshshinde2325]

@@hyungjoonpark83 it is the most visited mit ocw course

[@SilverArro]

Rushikesh Shinde No surprise there. I’ve gone through a few of them, and this one is my favorite by far. Dr. Strang’s lectures are brilliant and intuitive, and he just has such a warm and likable personality. You really feel like he’s actually your professor.

[@kshitijsharma4624]

The subtle but crucial shift from row space to column space using matrices at the end was just phenomenal! This lecture has made my day

[@sudhanvasavyasachi2525]

This is perhaps "The Best Lecture on Projections" I have ever experienced. Wowwww

[@sanatanmeaning]

I m in love with how humbly he finishes his lecture.His lecture will be Immortal.

[@ChandrangsuBiswas]

Whenever I feel a little off and distracted I go through the comments, get motivated and excited all over again to learn from the legend himself !

[@rohit2761]

This lecture along with the entire course and you yourself will always remain immortal Prof Strang. Thank you from almost the entire world who have benefitted from these golden amazing insightful flawless transitioned lectures. You are a mathematical genius. Hope you read this comment someday and realise you have the biggest Indian Fan.

[@hurbig]

This is his first lecture that I watch about a topic that I didn't previously know. Now I can finally see how brilliant he is at explaining because I have never heard any of this stuff before and it still makes sense the first time I hear it. My brain almost exploded at the complex and yet simple beauty of what he's explaining. It's almost like he's doing a magic trick because we know all the laws of math and he can just take them and make something possible that I never would have thought possible.

[@raphaelambrosiuscosteau829]

When i connected dots in my head around 15 minutes when professor was talking about why we do this projection thing, it was one of the most mind fucking blown moments in my mind. I felt like going from top to bottom in the tiny brain - galaxy brain meme in a span of a second. Amazing course

[@allenjin4740]

I saw this video fours years ago when I was an undergraduate student having linear algebra class, and now I saw it again when I am a graduate student, He is just a legend!, No one teacher I have ever seen can be compared to him.

[@eugenelee3488]

Professor Strang is the BEST! Makes me motivated to enter MIT

[@roronoa_d_law1075]

Eugene Lee soo.. you entered MIT ?

[@allandogreat]

Did you graduate from MIT now?

[@renney77]

Have you started your career using your MIT degree?

[@ramonmassoni9657]

@@renney77 Are you an MIT faculty member?

[@Bl4cKeN1nG]

Note: Transpose(a)*(b-xa) is the dot product expressed in matrix multiply notation. Confused me for a moment.

[@趙超]

tks ,your reply helps me

[@theindianrover2007]

@@趙超 second you

[@shadownik2327]

I hope these mit videos stay the same for 20 more years. I want my kids to watch the same videos

[@AakarshNair]

Hard to appreciate how well presented this is in first go around. The more one learns LA the more one appreciates this style of presentation.

[@gauravbhasme7498]

I smiled when he casually introduced the concept of linear regression. Got different view on how this works using matrix.

[@EdwardG-c9p]

2020 now, he did make it immortal

[@ytberintaiwan]

Thanks to Dr.Strang, I do really found my passion in linear algebra.I have to say:It’s kinda fun. Solving problem step by step and also apply it on data science and there’s so many things i haven’t know and now i am really excited to explore more! But there’s still a pity for me is that i can’t take the lecture with professor since he’s now retired 😢.I am really looking forward to taking class in MIT with professor Strang.My parents should give me birth earlier…

[@SphereofTime]

42:00 Least squares fitting by line

[@kambizmerati1119]

My mind is about to get blown away. What appeared to be a dry, intangible, and futile branch of mathematics in the first few lectures turned out to answer one of the most gnawing questions, the answer to which was kept as a mystery for my entire life: How did people come up with the method of least squares? Why least squares? Says who? I love this guy!

[@Samurai_Jack__]

The most beautiful math lecture I have ever seen

[@VladimirDjokic]

This professor is a genius :)

[@clearloverr]

Thank you Professor, i am a medicine doctor. I just wanted to learn linear algebra out of curiosity. Even I, who had not dealt with mathematics for a long time, was able to understand the subject. Thank you very much indeed.

[@abhay9994]

Professor I am preparing for an entrance exam and your lectures are helping me most. Thank you professor Strang.

[@AntifachoOi]

I always thought econometrics taugh with sums was confusing. I finally found an intuitive way to understand least squares. The minimization of errors proof is cool, but the simplicity of your proof combining the left nullspace,is awesome; thank you once again professor Strang.

[@danieltrigo2928]

In all this time. this is by far the best explanation of least squares I've ever seen. Honestly thank you very much for the video.

[@wise1330]

I have watched lectures 14, 15, 16 over and over again, and gain new insights each time. Professors, including mine at the university, tend to drop the arrow sign above vectors while writing on the board. This causes great confusion for the uninitiated. For example, in this lecture there is an expression "p=ax" whether the writing "ax" means a (a scalar) times x (a vector) or a (a vector) times x (a scalar) is not that obvious. Admittedly, the confusion stems from my ignorance of the commutative property of scalar multiplication of vectors, which is, in fact, not mentioned explicitly in most textbooks (e.g., Lay et al, ; Larson & Hostetler). In retrospect, this "p=ax" notation is a prelude to the ingenious way of introducing projection matrix, which again, requires pre-knowledge of associative laws. Overall, these are great lectures that stimulate thinking. Thanks a lot.

[@stefan67367]

b-[(a×b/|a|×|b|)×|b|×a/|a|] a×b/|a|×|b| is a additionstheorem cos α cos β × sin α sin β = cos(α-β) b - (a×b)/|a|^2 × a a×a=|a|^2 Regards from Germany

[@nishantduttmishra9290]

This lectures makes the idea of linear regression crystal clear.

[@cupckae1]

My university didn't even teach me projections. After being introduced to PCA (A concept used in machine learning) I came to know this topic. Alas!

[@elGetero]

Finally I understood the origin of the least squares method... (i'm in 2nd year of aeronautics and this teacher is a thousand times better than my university ones. A few could learn how to teach additionally to the algebra lesson)

[@aayushkucheria2644]

You experience so many Ahha! moments in the lecture!

[@Mustafa-yc9zh]

"make this lecture immortal"

[@officialroshanb]

I am watching in 2020 and I must say it is fresh as mortal. Long Live Prof.

[@RePuLseHQKing]

e in N(At) is perp. to C(A) just killed me. 30:30 How smooth can someone explain the "big picture" step by step over weeks.

[@quirkyquester]

Thank you Professor Strang and MIT. This is fun and great! It boils everything down to the fundamentals. Best you can get!

[@ayushshaw1755]

The fun fact is that math is a magic, and there are a very few handful of magician like him who can perform this with so much effortlessly.

[@udayak6964]

I am binge watching this series .....at the end of each one I tell myself that I will watch just one more episode :) !!

[@madhurit.m4105]

Is Gilbert Strang the God of Linear algebra? I loved this lecture :)

[@moushumipardesi]

Watching this in 2020, legendary to date!

[@walterlevy5924]

This lecture is amazing! This is the deepest understanding of linear regression one could get. Thanks MIT for sharing it with the world ❤️

[@watchmanling]

Nobel Prize for teaching linear algebra.

[@ccdreamerc7710]

As a PhD guy, I can say if you master in this course, actually you master at LOTS of high-level subjects like econometrics, engineering, applied math, chemistry etc. Those "graduate" level courses sometimes just repeat the basic ideas from this course and make them hard to understand.

[@AlgebraParaTodos]

How come they did not applaude at the end? OMG you are awesome man!!

[@omarelabyad2010]

my mind is blown! not by linear algebra... but by Gilbert Strang's teaching

[@tzaidi2349]

I have no business bing here. Not in a math or engineering field. I never see linear algebra except at night, on long flights, or anytime I want to escape into a world of abstract spaces and subspaces with the immortal Professor Strang as my guide.

[@georgesadler7830]

Dr. Strang this lecture uses some abstract methods which help me understand projections onto subspaces much better.

[@El_inge_987]

At 4:43 , why does he divide x= (aT b)/(aT a) ? Those are matrices, and I always thought that it is not possible to divide a matrix by a matrix, that was the reason as to why we instead multiply by the inverse but not divide?

[@Oaklev]

+fyescas777 a^T a = || a ||^2 is a scalar!

[@rajupowers]

+fyescas777 a is a column vector. Listen @ 12:10

[@yakunli9111]

a^T is 1 by 2, and a is 2 by 1, so the result is a scalar.

[@tungdinh3664]

wow, this lecture has opened my eyes. Thank you, professor Strang!

[@eightminutes6842]

This is brilliant, go MIT!

[@donaldcaga5962]

I have so much difficulty coping with my Linear Models subject. Thank you prof for this! You're a blessing. Continue uploading your lectures. You are a great help to us!

[@mariansabianhh]

thanks professor Strang, you don't know how helpful this is for many of us!

[@harshvardhanbhatnagar3607]

This is not a lecture. It's a projection matrix that projects everything in the chapter onto the viewer's brain.

[@rickshawty]

Thanks to Gilbert Strang and MIT

[@mothershabubu5740]

I can't believe the students start packing their stuff, slamming desks around, etc. at 48:30 before he's even done talking. So rude. They've just received a lecture by a brilliant mathematician, yet they have no respect!

[@apocalypse2004]

Sometimes you have only 20 minutes in between classes to eat lunch and/or get to the other side of campus

[@wangryan336]

10 mins in my school, and sometimes you have to get to the other side of campus in 10 mins for an exam

[@mayue6195]

I guess they have more respect than we do, dude.

[@eccesignumrex4482]

it's called college, yo' -

[@danieldaza3412]

Mother Shabubu that's exactly what I thought

[@bboysil]

I got goosebumps the first time I saw this lecture. He is an incredible teacher.

[@martintoilet5887]

He made some nonintuitive things intuitive with just some simple example, love his videos

[@junkmana]

How can i just remember every bit of these lectures always? This is education in its true sense, sometimes i feel watching video is a better form of education than personal interaction, if you understand wat i mean

[@alanzhang7272]

17:57 did someone sneeze?

[@mekala]

If Plato was right about theory of forms I'm pretty sure that this is the IDEAL lecture.

[@tungdinh3664]

I just realized multivariable calculus is the prereq of this course. I understand most of the part but there was confusion sometimes. Now I need to take cal 3 so I can continue these awesome lectures from professor Strang.

[@tcveatch]

You missed the whole point of Strang’s gentle wise career which is that calculus before linear algebra is stupid and wrong, actually, linear algebra should precede calculus. We have been unnecessarily tortured and many of us our math careers destroyed, ended, by the unwise reverse decision. LA is much more comprehensible, useful, inspiring, and generally useful than calculus which I’ve never used after a PhD and 30 years more. I really wish my high school had offered linear algebra first. Then so many would have not dropped out, and would have been better served by their effortful learning.

[@FariborzGhavamian]

omg.. this is just amazing, thank you MIT thank you professor Strang

[@olouck2789]

I am lucky to have the teacher that explained us linear regression THAT VERY WAY! :D

[@yijiehu1200]

Damn, this is a great lecture.... somehow connect to the least square problem by the interpretation of pure linear algebra without using calculus. what an enlightenment.

[@lianghao8312]

It is immortal at 2019

[@amirakhatib7765]

This is best lecture everrr

[@eccesignumrex4482]

Gill gets savage on Trig 2:30

[@anilsarode6164]

With all my endurance I listen to this saga until this lecture just to understand Linear Regression

[@sarvasvarora]

ML students be going SMH

[@aceofshade]

other colleges should be envious that MIT has a great of a professor as Dr. Strang

[@lucasm4299]

Txafter and other countries

[@furkatsultonov9976]

Just to the point! Clear and Concise! He is my hero!

[@SphereofTime]

26:15 plane in linear algebra

[@yeslinsequeira4612]

12:50 why are we already talking about P & P transpose. For a 1D subspace, isnt aaT / aTa just a single value? Kind of confused me, maybe Im not getting something, can someone help please?

[@qinlun6859]

aaT is a symm matrix, and aTa is a number ,its dot prudoct of vector a

[@souravaich5379]

Professor Strang, You are immortal !! 🙏🙏🙏

[@ozzyfromspace]

Yeah, it really was that good of a lecture. Professor Strang's Immortal Lecture.

[@neehw]

33:44，A'A was not certainly has inverse (A'A)^-1

[@thelastcipher9135]

"Don't take the videotape quite so carefully"... Camera zooms in right after lmao

[@User-cv4ee]

I tried to reach same conclusion using the fact that p is perpendicular to (b-p) i.e. p(t) * (b-p) = 0 and then substitute p=Ax but failed to get similar result as @28:31. There is an x(t) hanging around. -> p(t) *(b-p) = 0 -> (Ax)(t) * (b-Ax)=0 -> x(t) * A(t) * (b-Ax)=0 -> x(t) * [ A(t) * b - A(t) * A * x ] = 0 --?? can I justify that the second term must equate to zero? any suggestions how to move forward? where " "(t) is " " transpose.

[@seanguo9366]

This is phenomenal.

[@Stkraf]

Legend... Thats one word match their capability :)

[@sreekanthpamuluru767]

At 8:39 How the Projection matrix P is derived.? I see that in the numerator it is dot product of a^T and b, which is a Real Number How could the associative law of matrices is applied here is applied here? Am I missing something here? Can someone enlighten me? If You have, Give me link of any supporting lectures on this so as to clarify this particular point.!

[@michaelltan1059]

Not sure if you've figured it out yet. But in this case you're describing, the projection matrix P is being defined as the matrix that acts on the vector b to create the projection of b onto the vector a. At 8:39, you'll notice that the formula for p = a(a^Tb / A^Ta). p = Pb is just a unique grouping of that formula. It might be helpful to look at 9:40 where Strang shows the grouping he is using.

[@arsingh8137]

can someone explain 32:37 how e perpendicular to C(A) resulted in At Ax = Atb

[@SAElwarwary]

This is art ! Thank you Prof. Strang