2. Linear Algebra

[@SeikoVanPaath]

Some notable Timestamps: 0:01:16 Matrices 0:05:22 Eigenvalues & Eigenvectors 0:11:40 Diagonalization 0:23:04 Singular Value Decomposition (SVD) 1:03:20 Perron-Frobenius Theorem

[@viktorkhan8518]

Upvoted.

[@user_20anjali]

...

[@kamilziemian995]

Thank you.

[@duyloc91]

The lecturer is now working as a portfolio manager at Millennium Management, one of the more successful investment firms out there.

[@No_BS_policy]

Nothing wrong with that. Just kinda sad. Many of our great mathematicians and physicists are working in the finance industry instead of using their talents for the greater benefits of humanity. It's all fvckin messed up.

[@optimizedpran1247]

Legit crazy, he's probably so rich now lmao.

[@daakudaddy5453]

@@No_BS_policy Working in finance, they would solve many more real world problems. Along with engineering, its the best application of their skills.

[@HaoqingWu-h3c]

@@No_BS_policy On the other hand, being a professional mathematician in a top 20 university is one of the most difficult jobs in the world. It's all due to supply and demand, there aren't that many jobs in academia. So if there are other places can hire all these smart people, why not?

[@JohnSmith-vy7ck]

@@No_BS_policyThe money that funds the research for all these projects for the future has to come from somewhere. This is where it’s generated. The Government isn’t going to pay for everything.

[@gilbertobatres-estrada5119]

I have a MSc in theoretical physics and a MSc in applied math. I can tell you this is a great lecture.

[@User-dn4vi]

No you don’t

[@amanpreetsinghgala2383]

I think this usually happens when you have thoughts and knowledge flowing at a greater speed than what you need to explain the class as a whole. This is what a scholar in action looks like. Mr. Lee , reminded me of my professor who used to brim with knowledge on a subject, and I pray if I could become half as good as him. Mr. Lee is a great instructor and its a review class, and not an elementary class of a school. So, e & oe. and we should be matured enough to read between the lines and be revision ready with him. If we are, then there should be no problem.

[@bowenzheng8580]

This is actually a very good lecture despite a few mistakes. I don’t have solid background in math or linear algebra but if you just carefully go through the lecture and pause to look up concepts that are not familiar, I am pretty certain you can fully understand it. A good place to get some visual understanding for basic concepts is 3Blues1Brown’s linear algebra series. It took me a whole day to go through this lecture but it’s very rewarding.

[@Ben-sk7ey]

Thank you for the referral to 3Blues1Brown. Excellent content.

[@111ark]

Great comment.

[@selfloveaffirmations1916]

That is exactly what I did!!

[@beastmode1647]

If it takes you 12 hours to understand a 1 hour lecture...then how good is that lecture really? Exceptions are warranted for high-level stuff like general relativity, but this is basic linear algebra The 3b1b linear algebra series is actually what’s good. If you have 0 linear algebra background, you’ll still learn a fair amount from 3b1b. Can’t say the same from this lecture

[@andressalinas4422]

The 3Blue1Brown Linear Algebra Series is exceptional, thank you for the referral, much appreciated.

[@thiagovasco7076]

You can watch as many MIT lectures as you want on the same topic and you will always learn something new.

[@danpalu2308]

Not me. I have learned everything.

[@Diana-yl1jo]

I haven't watch this video yet but I have to say that if you want to learn Linear Algebra go to see course called Linear Algebra in MIT or Khan. This is just one concentrated course for those who need to be remembered than apply it into finance.

[@iphgfqweio]

have you watched it by now?

[@fidelesteves6393]

Incredibile, professor. I have seen this content several times and now suddenly you gave me another way to understand some linear transformations.

[@tljstewart]

@59:00 answers the question "What does the eigen vector mean for stock price matrix...?"

[@nadekang8198]

This whole series of math in finance is more like a "Survey of Financial Mathematics". Only if you go through proofs and definitions like matrices, vector space, determinant, eigenvectors/eigenvalues and linear transformation, then go into probability theory(should be advanced probability theory as this is where they mention sigma-algebra and probability measure), statistics, regression (better with matrices), and then real analysis, advanced calculus, stochastic calculus (focus on Martingale), then you'll have an idea what they're really talking about.

[@edkure]

His handwriting on the board mesmerizes me! Such a talent 👍🏾

[@rezakian9829]

This guy is a great lecturer. Clear and concise.

[@vitordeholandajo156]

This professor is spectacular.

[@EasyGameEh]

the difference in view numbers between this and the intro video is hilarious. even wathcing mit lecture course people are saying that they still are only interested in most generic and shallow knowledge with some entertainment value.

[@EasyGameEh]

@Trails & Travel it seems like that's what i said more or less. one would think that mit in the title shall narrow interest enough but it widens it for the very intro part lol

[@Ibradiouf98]

For all the plaintiffs in this chat, remember that you came to watch this video because you do not fully understand it - So please stop pretend that the lecturer is not so good . At least he is lecturing at MIT and you 'so called better' are watching him. top the hate please. Ibra Diouf

[@AntonioLopez8888]

If you know the topic why to watch this.

[@beastmode1647]

Terrible argument. I haven’t seen any commenter claim they’re “better than” the professor. Obviously, I’d be significantly worse at teaching linear algebra than this professor But that doesn’t mean this professor taught this well. “Significantly better than beastmode1” does not necessarily mean “good teacher” This professor made numerous mistakes. He revised most of them ex post facto, but the sheer amount was still confusing. Furthermore, he left theorems unproved, which is just lazy teaching. If I didn’t have prior knowledge of linear algebra going into this, I would’ve been totally lost

[@owletkami8018]

So confusing when an instructor is walking you through something and as you are paying attention they go " oops made a mistake at the very beginning, let me just explain everything again "

[@tomctutor]

Most lecturers will sympathize with CL's dilemma of explaining a really complicated idea, as simple as he can without losing accuracy. If you think that he is in error you should also see Strang's MIT Lectures on Linear Algebra where he sweeps over difficult areas much the same way (and Strang is by far the expert in this field). I found this lecture very informative and inspiring, just wonder if he has any follow up vids here on MIT?

[@Grassmpl]

around 55:45, V should be n by m, not m by n, since you are multiplying by V^T

[@SphereofTime]

20:35 complex conjugate 34:31 Does AT A always a symmetric matrix?

[@suraj.panddey]

Guys all the value of eigen vectors are incorrect 51:09.

[@stepkurniawan]

The first 15 mins of the lecture have slapped me in the face. I know nothing about matrices.

[@jiteshbohra6164]

Gilbert Strang is great!

[@enisten]

So is Richard Feynman!

[@Aditya_katole02]

49:25, the matrix (3,3) place should be -17 instead of -13

[@tobiassugandi]

This is insanely good! love u Choongbum Lee

[@nikon800]

Very intuitive explanation of SVD. Thanks

[@deanflatt1370]

Did anyone else get -17 instead of -13 for the last entry in A^T*A - 25*I at 49:08?

[@WallaceRoseVincent]

Anyone interested in working through the course together?

[@smuksm]

Did you complete this course? Any advice for someone starting out?

[@gregorybattis9588]

Add me on linkedin. I am interested.

[@cssensei610]

I’ve just started. You need a bit of math though before you start. Would recommend 3Blue1Brown linear algebra series, before you start this lecture

[@adamyatripathi2743]

@@gregorybattis9588 Me too!

[@awsom82]

CS Sensei oh, thanks, I will try

[@nwgverified]

I was following for awhile but that got complicated

[@YoungEli9]

Financial math is easier to learn than just individually learning algebra, calculus, or differential equations for me because it’s basically applied math using the knowledge of finance, I mean honestly what can get better than that

[@STONE9523]

let go the instructor prepared or not, please! question guys: 1:10:49, "concluded | Av' | > | Av |, and this cannot happen". why? Many thanks for any help!

[@lemoi6462]

I'd say its because (with lambda the Eigenvalue of v' and lambda_0 the eigenvalue of v) |Av'|>|Av| => |lambda v'| > |lambda_0 v| => |lambda| > |lambda_ 0|, since |v|=|v'| and this is forbidden as he said earlier.

[@fawzibriedj4441]

@@lemoi6462 but v' is not necessarily an eigen vector

[@lemoi6462]

​@@fawzibriedj4441 Ok you are right, thank you. i guess its not that easy^^. I found the solution, it follows from the property of a symmetric matrix: max |Av| = max |lambda|, lambda being an eigenvector. You can find the theorem in the algebra script from the description.

[@fawzibriedj4441]

@@lemoi6462 Thanks for your answer I think this property is kind of strange, as in my opinion, there shouldn't be a "bigger" |A.v| If we suppose there exists a "u" such that |A.u| = max|A.v| I can always construct w=2u, thus: |A.w|= 2|A.u| > |A.u| What do you think ?

[@lemoi6462]

@@fawzibriedj4441 You are right. Actually i left something out, as i thought it would be too long, sorry^^: The max is over all v with |v|=1

[@engmostafa5]

good instructor

[@sigmaphiepsilonsociety5230]

He completely messed up the rigor and enthusiasm of students wanting to learn about the application of linear algebra applied to quantitative finance.

[@bigollameo]

Messed up the rigor? It's one class on application of linear algebra to treating acquired data sets, and generally in application courses we don't get into the deep rigorous elements. And the enthusiasm? Well, that's an issue of individual motivation. Problem lies with you, there. Not with the instructor. Maybe a little soul searching and navel gazing will help you. You're a little Stella, and you need to go and get your groove back.

[@Yonsei_econ]

1:09:10 I understand that there are infinitely many eigenvectors (scalar multiples) for a single eigenvalue, so why is it written as 'an eigenvector'?

[@Priyanshu-d6o1w]

Can you please tell whats the use of linear algebra in quant finance?? Also if I want to be a quant developer then also I have to study these things ?

[@Yonsei_econ]

@@Priyanshu-d6o1w I'm still an undergraduate myself, and I'm trying to integrate various disciplines in my studies, so I'm not sure if this is the right approach, but I'll share what I've learned so far based on my studies. If you want more professional advice, I recommend consulting with a professor at your university. * calculus and linear algebra are essential to study ai algorithm and studying additional subjects like analysis or mathematical statistics will be helpful. * I'm preparing for the CFA exam to gain a broad knowledge of finance. You might want to look into it as well. It could be helpful to you. * This video is a summary of linear algebra, so it might be difficult to understand. First, study the overall content with Professor Gilbert Strang's book and the free MIT lectures, and then try this course * I'm not sure if studying all the lectures of the 'topic in mathematics applications with finance' course is overdoing it or not enough. Still, I think it's good to be prepared to this extent. So, I'm making an effort to first study the necessary knowledge and related disciplines for each lecture, then attend the course to review and try to connect it with finance.

[@MedCreativityPlant]

Watch the instructor writing at 2x speed - lose grip on reality

[@Warrior1Spartan]

Question: at 44:34 for reference, the equation reads v1^T*Lambda2 v2/Sigma1. How does it go from that to Lambda2/Sigma1 * v1^T*v2? Or did I miss something?

[@yz9387]

lambda and sigma are scalars(numbers) that you can just take out.

[@Warrior1Spartan]

@@yz9387 Awesome! Thank you so much!!!

[@smferdous1]

Does not A of size mxn transforms a column vector in R^n to to another column vector in R^m? This lecture said it transforms from R^m to R^n.

[@SilverArro]

+S.M. Ferdous Yes, that's correct. For example, let's consider a 2x3 matrix (so m = 2, n = 3). Such a matrix will act on vectors of dimension 3x1 (so, vectors in R^3 which in our case is R^n), and will transform them into vectors of dimension 2x1 (so, vectors in R^2, which in our case is R^m). Therefore, this is a linear transformation from R^n -> R^m. Although watching the video now, it appears that he did correctly demonstrate that an mxn matrix represents a linear transformation from R^n to R^m. So yes, you were correct, but the professor was as well.

[@bongkem2723]

amazing course, wish i had this kinda course back in college :D

[@wallstreetwolf7681]

Holy Moley... it never stops!

[@ashutoshpanigrahy7326]

While calculating the Ui*A*Vi, it reduces to UiT*sigma_i*Ui which further reduces to sigma_i*L2norm(Ui)^2. How is the L2(norm) = 1?

[@theoqsoftware]

Am I the only one who is worryingly soothed by the electrical humming sound on the background?

[@njabulophiri4583]

Even the sound of the chalk on the board reminds one of those late noon lectures back in college.

[@기린-g2z]

17:31 bookmark

[@iodboi]

Q1: when we say det (A- λI) =0 has at least 1 solution, are we saying this will have 1 real solution at the minimum? Thanks for your help

[@turboleggy]

Eigenvalues have to be real but the vectors can be imaginary. In general there is no guarantee that we will get a real solution but the fundamental theorem of algebra says that there will always be a solution if we use complex numbers

[@noelplazo3404]

Can we get a certificate after finishing this? Greetings 🖖 🫂 from the Philippines 🇵🇭 🙏🏽💙💕

[@mitocw]

OCW does not offer any degree, credit, or certification. For MIT online courses with certificates of completion, visit: MITx Online/MITx on edX: openlearning.mit.edu/courses-programs/mitx-courses MITx MicroMasters: openlearning.mit.edu/courses-programs/micromasters-programs MIT xPRO: openlearning.mit.edu/courses-programs/mit-xpro MIT Bootcamps: bootcamps.mit.edu/ MIT Management / Sloan Executive Education: exec.mit.edu/s/ For questions about these programs, please contact them directly.

[@brian-kt1rc]

As with all lectures: Increase the speed of the video to make it flow better and not waste too much time

[@goulchat1]

No

[@bucketsniper5488]

did he mean a matrix A is ORTHOGONALLY diagonalizable if there exists an ORTHOGONAL (not orthonormal) matrix U such that A=UDU(inverse)

[@nickcimino829]

Just out of curiosity, if there is anyone reading this that has a career in finance, is any of what he is showing is applied in the job on a regular basis? Or is this more for bespoke applications? I am looking to get into finance (going into senior year of high school), and most if not all of this makes no sense to me.

[@nguyencongthienphu5362]

I still don't understand what the application of linear algebra is in stock market ☹️ Is there anyone could do me a favor and tell me how those kind of mathematics can be applied in stock market or in data analytics please? I would feel so so thankful for that 🙏

[@enisten]

To find an arbitrage portfolio or a replicating portfolio, you have to solve systems of linear equations (Ax = y). Check out the Part 2 of Andrew Lo's problems and solutions collection for his Finance Theory I class at MIT.

[@enisten]

I'm sure there are much more advanced uses of linear algebra in finance. I'd be happy to hear about them from someone else myself!

[@nguyencongthienphu5362]

@@enisten Oh thank you so much for giving me the suggestion 😍❤️ I really appreciate that 😊

[@Priyanshu-d6o1w]

Can anyone please tell whats the use of linear algebra in quant finance??

[@yorchais]

Anyone knows what is the point of normalizing the eigen vectors for Singular Value Decomposition? 52:14

[@yorchais]

nvm! it's just part of the process. V1 = [1 1 0]. Find the corresponding unitary vector which is V1 = [1/(2)^(1/2),1/(2)^(1/2),0].

[@fawzibriedj4441]

A bit late, but the point is to construct an "orthonormal" matrix (the real name would be orthogonal) You need normalised vectors

[@Semilla719]

46:55 comienza la accion!

[@melissachen1581]

Can someone explain why lambda 0 is the eigen value of multiplication of 1(@ Perron-Frobenius Theorem)??? What does that mean by “multiplication of 1"

[@银河搞钱指南]

means this lambda0 is unique. say when we try to solve the eigenvalue by (r-1)(r-3)^2=0, r=1 is of multiplicity of 1.

[@joyce1964]

why do some universities still use chalkboards? White boards are much easier to clean and much easier to see. Plus different types of coloured markers can be used.

[@luuukaain4133]

Well I'm guessing because they didn't want to spend or have the money for white boards and markers.. i always wonder the same thing though

[@Myrslokstok]

Well no reflexions, and markers dry out kind off Quick with cap off. Then luxyru chalk are good.

[@veraalex7446]

chalkboards are better

[@J58LRJ]

In a large lecture theatre chalkboards are actually a lot easier to read from a distance, partly due to lack of glare. Also a whiteboard marker becomes a lot harder to read as it runs out, something which is not true of chalk.

[@akkimaroine487]

It's teaching Maths tradition :).

[@zunelmhrz3040]

Just came here to read some comment after he said " 32:23 Just look like totally nonsense".. . feels like that :) *_*

[@diegososa9647]

Thank you very much. Great explanation

[@CoryMcaboy]

This guy is pretty good. Where did they get him from?

[@noras.9355]

made in china haha

[@ylou298]

No he's Korean

[@thetruth4865]

he studied at Seoul University which is extremely difficult to get into (well known for math, physics, etc...). there is a qualifying test that high school students take which determine which university you will attend.

[@Nick-jj6cl]

@@noras.9355 As a Chinese, I can tell that this guy is definitely from South Korean instead of China. There are subtle differences between Chinese, Japanese and Koreans which people from areas outside East Asia may not be able to tell.

[@edenchan61]

@@Nick-jj6cl right

[@miguelorozco936]

Es muy buen maestro 😃

[@realityiswhatyoumakeit8192]

hello i'm not sure if anyone will answer this but i wanted to ask, I'm a total beginner with these concepts of vectors/and matrix, or complex numbers, does anyone know which youtube video/ videos or channels i could watch to go to the basic to understand what is being taught her? Want to do some self studying , much help and luck and wealth for anyone that helps me out!

[@ishan605]

try khan academy

[@anshika7758]

Freecodecamp has condensed math lectures.

[@jacquesm1652]

what are some prereqs so that this lecture makes a little sense?

[@mitocw]

The prerequisites listed in the syllabus are: 18.01 Single Variable Calculus 18.02 Multivariable Calculus 18.03 Differential Equations 18.05 Introduction to Probability and Statistics or 18.440 Probability and Random Variables 18.06 Linear Algebra See the course on MIT OpenCourseWare for more information at ocw.mit.edu/18-S096F13

[@bighands69]

Start out with Matrices and then move to calculus.

[@chitharanjansarvesan5714]

Probably no one is checking this...but 18.05 and 18.440 lectures are not available on youtube...is there any alternate for them?!😅

[@jea_lee]

@@chitharanjansarvesan5714 go for 6.041 Fantastic professor

[@donaldbridgeman8424]

Classpad400 CAS for Algebra Need help on simple equation. 5X+2=10 ans x over 8/5 3x-2=10 ans x=4 How do you get ans on Classpad400 CAS

[@kara9567]

Donald Bridgeman x is not equal to 4, ans is 8/3...

[@gatsbylight4766]

+Ka Ra x=8/3? 3*8/3 = 12???

[@jackroettcher1549]

Is there a way to filter out that annoying background hum? Makes the video hard to watch

[@kikincream6962]

Can someone explain to me how to obtain the matrix 13 12 2 12 13 -2 2 -2 8 ?????

[@ladygood7048]

That’s matrix multiplication from A^T A 3x3+2x2=13 3x2+2x3=12 etc

[@spyinsecret0075]

This is all familiar to me. However seeing the use of different alphabet makes it new again😂 My mind was spinning trying to reconnect what I learn in hs and what was teach in the video😂 Ex: we use A instead of U. Thank you for the video!

[@SphereofTime]

15:17

[@AnkitSharma-ef7md]

Sir, I got your every point and I am sure that you are referring to the higher version of mathematics. A doubt has been plugged after looking at this lecture ( at 19.09), here it is; Symmetric matrices have real eigenvalues. I have a real symmetric matrix whose eigenvalues are imaginary. it is a 2x2 matrix with entries A11= -1 A12= 0 A21= 0 A22= -1 After solving we will arrive at a point (Lambda)^2 + (lambda)+1=0 This is nothing but the equation of complex cube root of unity whose values are imaginary. Any reader or viewer of this video would please solve and check it? Comment your answer in the comment box. Thanks.

[@MrMev2008]

Ankit sharma (-lambda-1)^2=lambda^2+2lambda+1= 0 since (a+b)^2 = a^2+2ab+b^2 so both the roots are -1

[@alaminhossain9803]

He unconsciously mistook to characterixe what's row and column of a matrix.

[@himjanand2034]

anybody wants to work with me through this course?

[@asdfasdfwae]

sure

[@estebanlopez1701]

How far have you gone?

[@RaylinRecords]

I do!!

[@mykytaradko5776]

Lmao at all these replies Nice "working through" this course

[@syndicat4847]

This instructor has managed to make linear algebra look like theoretical physics. I have a foundation in mathematics, physics and economics applications in calculus and I have no idea what he's talking about. Is it just me?

[@syndicat4847]

I guess that's why I would never qualify for MIT.

[@2904apoorv]

No, it is not just you. By the way, Prof. Gilbert Strang's lectures on Linear Algebra are just amazing.

[@snowy0110]

1:00:57

[@валерийсоколов-п4я]

===(2,2)(3,3)???

[@Chris-ez2sc]

29:46

[@baby-maegu28]

45:50

[@AntonioLopez8888]

Sense as if they don't know the application of their eigen stuff for stocks. Their answer wasn't convincing.

[@SphereofTime]

4:30

[@stuartcampbell7854]

When I try and download the course direct from KZbin the audio is very static sounding ! You are being jammed MIT !

[@neerajtatikola4797]

why's he using sidewalk chalk

[@madam_im_adam]

I'm still no clearer of the applications in finance

[@bird175]

Good mathematician, poor teacher, teaching is a skill where you need to overcome the curse of knowledge, how do you explain to someone something you take for granted. You only develop this skill through the experience of finding out what students get stuck on and refining your definitions until you present it clearly. This is more of a 'read from the text' style and just regurgitates what he's been trained in. Though I understand this is just a review lecture and the facility are throwing him a bone.

[@chico6854]

love it!

[@mitenmehta79]

can some one further simplify this lecture may be can break into 4 more videos on EVD, SVD, COR and then actual stock price example. Also things like Norm etc. need to be explained.

[@TheTruthSeeker756]

6 minutes in and my head hurts😂

[@Mathskylive]

Không gian vec tơ ❤

[@bucketsniper5488]

v can not be the zero vector, for eigenvalue and eigenvector, he forgot that

[@89gattomatto]

Nice lecture ^^

[@astaragmohapatra9]

Why people are all in praise for this video, as if MIT will enroll from the comment section of a KZbin video. The content was not entirely helpful and was very difficult to grasp.

[@gilbertobatres-estrada5119]

I guess people appreciate the mathematical intuition you can get from it

[@Myrslokstok]

Very good!

[@валерийсоколов-п4я]

The teacher was not prepared a lot of mistakes

[@BULLSHXTYT]

Sad that I cannot follow the lecture in less than 10 minutes.

[@AvirupChakraborty-ex9bf]

I am Indian. Just go through RB2

[@darthmilo1833]

first year, first semester engineering math

[@thetruth4865]

Koreans love math...

[@muhammadshakil4410]

How?

[@ruemignon]

They also write the number 7 in a cute way...

[@justinscarpello8709]

Lol, any questions so far? I have Absolutely no idea what this guys is talking about. I thought linear algebra meant y=mx+b. Yikes

[@professionalprocrastinator8103]

That's cause you're stupid lol

[@kvnsrinu]

Yeah you would be clueless if you didn’t take Dr Strangs MIT videos here on Linear Algebra. This is very high level summary

[@beastfromeast-w2d]

may I just say, my undegrad degree ~was~ is worth shit?

[@hjon9119]

please gift him with a better board eraser

[@stewiegriffin8384]

this is dime

[@malekmawaheb6285]

c'est excellent

[@neerajaggarwal4689]

3Blue1Brown is great

[@валерийсоколов-п4я]

incorrect decomposition

[@lokeshkumarsingh9636]

he should learn linear algebra himself more with more clarity in concept. otherwise he will spoil the name of MIT

[@williamhughes2837]

The sweltering math theoretically provide because force cosmetically stir alongside a wooden pump. redundant, magical radio

[@bakhodirjonkakhkharov344]

As an engineer(Dipl.Ing) I can confirm this is a good lecture)

[@liangyumin9405]

讲的一般啊。。

[@melissachen1581]

同感