This man single handedly saved my university algebra course, my teacher was just reading notes, he's actually expalining in a very clear manner.

Good professor with good old blackboard and white chalk teaching method. This is way better than all the fancy powerpoints that many teachers use now a days.

@9888565407

4 жыл бұрын

hey did ya benefit from these lectures

@yusufkavcakar8744

3 жыл бұрын

@@9888565407 yeah ı did

@jiayuanwang1498

2 жыл бұрын

I can't agree more!!!

Thanks to him! I passed Linear Algebra.. I watched his videos for 4 days before final exam and I got 74 from final.. If I couldnt watch Dr.Strang's lectures, I would probably fail...

@snoefnone9647

7 ай бұрын

For some reason i thought you were saying Dr. Strange's lecture!

this is a bazillion times more straightforward and clear than the lectures i pay for at my university. :( I appreciate this being online

@bsmichael9570

8 ай бұрын

He tells it like a story. It’s like he’s taking us all on a journey. You can’t wait to see the next episode.

I think Strang leaves out a key point in the difference equation example, which is that the n unique eigenvectors form a basis for R^n, which is why u0 can be expressed as a linear combination of the eigenvectors.

@alessapiolin

7 жыл бұрын

thanks!

@wontbenice

6 жыл бұрын

I was totally confused until you chimed in. Thx!

@seanmcqueen8498

6 жыл бұрын

Thank you for this comment!

@arsenron

6 жыл бұрын

in my opinion it is so obvious that it is not worth stopping on it

@dexterod

5 жыл бұрын

I think Strang assumed that A has n independent eigenvectors since most matrices do not have repeated eigenvalues.

I agree. > Just one small correction at 32:30: It should have been S * LAMBDA^100 * c instead of LAMBDA^100 * S * c.

@Slogan6418

4 жыл бұрын

thank you

@ozzyfromspace

4 жыл бұрын

The sad thing was, a few moments later he was struggling to explain things because even though he hadn't pinned down the error, he someone knew that something wasn't quite right. But he obviously had the core idea nailed

@alexandresoaresdasilva1966

4 жыл бұрын

thank you so much, was about to post asking about this.

@user-px8iy5jj8q

4 жыл бұрын

I stuck on this for like 10 mins, until I saw the comments here...

@maitreyverma2996

4 жыл бұрын

Perfect. I was about to write the same.

Math surprises you everytime...🤔 Never thought that connections between rate of growth in system dynamics, fibonacci Series and diagnalization of an INDEPENDENT vectors will finally boil down INTO GOLDEN RATIO OF EIGENVALUES at END! 😳

This is magnificiant!! I have no words to express how thankful I am towards the exposure of this video

From this latest lecture , I am learning more about eigenvalues and eigenvectors in relation to diagonalization of a matrix. DR. Strang continues to increase my knowledge of linear algebra with these amazing lectures.

Thank you a lot Prof. Strang you really made this course clear to understand, the way you teach these topics are superior, what really special about it is you put yourself in the position of your students answering their questions without them even asking, a true skill acquired not only by decades of teaching but actually having the mindset of a true teacher. Mr. Strang, words can't describe how good these lectures are its a true form of art. And thanks a lot for MIT OpenCourseWare for providing these lectures in good quality and free of charge, hopefully one day to I will pursue a master in my degree in electrical engineering in MIT.

The explanations of this professor of all those abstract theorems and blind methodologies are simply briliant

It's getting more and more interesting when differential equations are involved!

Really happy this is online! Thank you Professor :)

Absolutely amazing! This lecture really helped me to understand better the ideas about Linear Algebra I've already had.

You know you’re in for some shenanigans when they pull out the “little trick” Professor Gilbert is an incredible teacher; I struggled with Eigenvalues and vectors in a previous course and this series of lectures has really helped understand it better Love your work Professor Gilbert

This teacher made fall in love with linear algebra thankyou ❤️

Thank you for existing MITOCW and Prof. Gilbert Strang.

This is so beautiful!

through single matrix transformation, the whole subspace will expand or shrink with the rate of eigenvalues in the direction of its eigenvectors, suppose you can decompose a vector in this subspace into the linear combination of its eigenvectors, so after many times of the same transformation, the random vector will ultimately land on one of its eigenvectors with the largest eigenvalue.

@MsAlarman

2 жыл бұрын

Mindbending

Who would have guessed, when this guy explains it, it almost sounds easy! You, dear dr. Strang, are a master at what you do...

Hard lecture to get through personally but does illustrate some of the cool machinery for applying eigenvectors

For the curious: F_100 = (a^99 - b^99) * b/sqrt(5) + a^99 , where a = (1 + sqrt(5))/2 and b = (1 - sqrt(5))/2 are the two eigenvalues of our system of difference equations. Numerically, F_100 = ~3.542248482 * 10^20 ... it's a very large number that grows like ~1.618^k 😲 Overall, great lecture Professor Strang! Thank you for posting, MIT OCW ☺️

I have learned about the Fibonacci sequence in my high school, and it is so good to have a new perspective on the magical sequence.I think the significane of learning lies in the collection of new perspectives.😀

Thank you for posting this. These videos will allow me to pass my class!

44:00 well that's an outstanding move

A great lecture showing us the wonderful secret behind linear algebra

a big thanks tothis prof for his efforts to give us cours about linear algebra

This is one of the best in the series

bruh the fibonacci example just blew my mind. crazy how linear algebra just works like that!!

At 28:07, uk = (A^k)uo, can also be written as uk = S*(Lambda)^k*(S^-1)*uo. Also, we can write uo = S*c as explained at 30:00. therefore, uk = S*(Lambda)^k*(S^-1)*S*c=S*(Lambda)^k*c

@jeanpierre-st7rl

4 ай бұрын

Hi @ 29:46 Uo = C1X1 + C2X2 +C3X3... Is U0 a vector? If so, How can split this U0 in to a combination of eigen vectors? What is Ci ? If you have any info pleases let me know. Thanks.

Bravo!!! Very much the best and premium stuff.

Thank you, Prof. Thank you, MIT.

What a brilliant lecture !!!

the lecture and the teacher of my life!

@serden8804

4 жыл бұрын

bro yasiyor musun

Thanks for uploading.!

I love professor Strang's great lectures. Just one small correction at 32:30: It should have been S * LAMBDA^100 * c instead of LAMBDA^100 * S * c.

@starriet

2 жыл бұрын

Nice catch!

@jeffery777

2 жыл бұрын

haha I think so

@eyuptarkengin816

6 ай бұрын

yeah, i though of the same thing and scrolled down the comments for a approval. Thanks mate :D

33:40 Correction: Eigenvalue matrix be multiplied to S from the right. That has been made in the book. Probably, it slipped off Prof. Strang in the flow.

There is a small writing mistake at 32:30 by Prof Strang. He writes (eigenvalue matrix)^100 multiplying (eigenvector matrix) multiplying c's (constants). It ought to be (eigenvector matrix) multiplying (eigenvalue matrix)^100 multiplying c's. At the end of the lecture Professor Strang does narrate the correct formula but it is easier to miss.

@clutterbrainx

10 ай бұрын

Yeah I was confused for a very long time there

I got 70 out of 75 in my final linear algebra exam thanks MIT...

The best math teacher ever.

Now I get it, so its like breaking the thing ( vector or matrix or system really) we want to transform into little parts and then transforming them individually cz thats easier as the parts get transformed in the same direction and then adding up all those pieces. E vectors tell us how to make the pieces and e values how to make the transformation with the given matrix or system. Wow thanks ! It’s like something fit in in my mind and became very simple. Basically this is like finding the easiest way to transform. Thanks to @MIT and Professor Strang for making this available online for free.

What a great professor!!!

He is my linear algebra super hero!🙂

I had to pause to figure out how he got the eigenvectors at the end. Plugging in Phi works but it wasn’t until I watched again that I noticed he was pointing to the lambda^2-lambda-1=0 relationship to reveal the vector.

That was amazing and awe-inspiring. :)

A*S=S*Lambda (Using the linear combination view (Ax1=b1 column part) of Matrix Multiplication), That is Brilliant and Clear! Thanks!

@neoneo1503

3 жыл бұрын

Also expressing the state u_0 to u_k as linear combination of eigenvectors (at 30:00 and 50:00)

@fragileomniscience7647

3 жыл бұрын

Well, it's - if you interpret it that way - just a basis transformation from the standard base (up to isomorphism, then just additionally multiply the transforms of the alternate basis) onto the eigenvector basis. Provided of course, that either the characteristic polynomial factors distinctly, or that geometric and algebraic multiplicity match (because then the eigenspaces distinctly span the vector space up to isomorphism; if they weren't, you'd just have a subspace as a generating system). For anyone who wanted one more run-through.

@neoneo1503

3 жыл бұрын

@@fragileomniscience7647 Thanks! =)

well i got impressed at the begining, but when he stated the second eigenvalue i realized it is just the golden ratio... That does not demerits him, he's great!

Did we ever prove that if the set of eigenvalues are distinct, the set of eigenvectors are linearly independent? I ask because at ~ 32:00 taking u_o = c1*x1 + c2*x2 + ... + cn*xn requires the eigenvectors to form a basis for an n-dimensional vector space (i.e. span the column space of an invertible matrix). It feels right but I have no solid background for how to think about it

@roshinis9986

8 ай бұрын

The idea is easy for 2d. If you have two distinct eigenvalues and their corresponding eigenvectors, you don't just have one eigenvector per eigenvalue, the whole span of that vector (its multiples forming a line) are also the eigenvectors associated with that eigenvalue. If the original eigenvectors were to be dependent, they would lie in the same line making it impossible for them to scale by a factor of two distinct eigenvalues simultaneously. I haven't yet been able to extend this intuition to 3 or higher dimensions though as now dependence need not mean lying in the same line.

@jeanpierre-st7rl

4 ай бұрын

@@roshinis9986 Hi @ 29:46 Uo = C1X1 + C2X2 +C3X3... Is U0 a vector? If so, How can split this U0 in to a combination of eigen vectors? What is Ci ? If you have any info pleases let me know. Thanks.

agree! that's what I plan to use in my teacing

Master Yoda passed on what he has learnt by fibonacci and 1.618.

@cuinuc I think they are actually the same, because LAMBDA is a diagonal matrix, you can have a try.

I read something on SVD without even knowing about eigenvalues and eigenvectors, then watch a youtube video, explaining that V is actually the eigenvector decomposition of A^TA. Which is extremely insane when I got to see this video oh my godness. Now even haven't watched your SVD lecture, I can even tell the precise concept of it. Oh my godness Math is so perfect!!

I love you professor !!!

Beautiful lecture. Thanks

@PhilOrzechowski he does it to make first order difference equations system out of second order

why are we representing mu(k) vector as the combination of eigen vectors in the problem of Fibonacci sequence?

MIT, thanks you!

Great lecture !

Fantastic! Thanks for uploading.

Great stuff. I was able to do my homework with this lecture. I will definitely be getting Strang's book.

The golden ratio arose from the Fibonacci sequence and has nothing to do with eigenvectors or eigenvalues. The beauty of using the eigenvectors and eigenvalue of a matrix though is limiting the effect of the transformation to the change in magnitude only, which reduces dynamics systems such as population growth that is a function of several variables to be encoded in a matrix computation without worrying about the effect of direction or rotation typically associated with matrix transformation. Since eigenvectors and eigenvalues change the magnitude of the parameter vector only, the idea of employing the Eigen transformation concept is quite genius. The same technique could be used in any dynamic system that could be modeled as a matrix transformation but one that produces a change in magnitude only.

@Arycke

9 ай бұрын

Hence the title of his *example* as "Fibonacci Example." Nowhere was it stated explicitly sthat the golden ratio didn't arise from the Fibonacci sequence, so I don't see where you got that from. The example has a lot to do with eigenvalues and eigenvectors by design, and is using a simple recurrence relation to show a use case. The Fibonacci sequence isn't unique anyway.

Fibonacci numbers being solved for as an algebraic equation with linear algebra was pretty cool.

43:36 that moment when your professor's computational abilities goes far beyond standart human capabilities

@BalerionFyre

8 жыл бұрын

Yeah wtf? How did he do that in his head?? lol

@BalerionFyre

8 жыл бұрын

Wait a minute! He didn't do anything special. 1.618... is the golden ratio! He just knew the first 4 digits. Damn that's a little anticlimactic. Bummer.

@AdrianVrabie

8 жыл бұрын

+Stephen Lovejoy Damn! :D Wow! AWESOME! I have no words! Nice spot! I actually checked it in Octave and I was amazed the prof could do it in his head. But I guess he knew the Fibonacci is related to the golden ratio.

@IJOHN84

5 жыл бұрын

All students should know the solution to that golden quadratic by heart.

@ozzyfromspace

4 жыл бұрын

Fun fact since we're all talking about the golden ratio. The Fibonacci sequence isn't that special. Any sequence F_(k+2) = F_(k+1) + F_k for any seeds F_0 = a and F_1 = b != -a generate a sequence that grows at the rate (1+sqrt(5))/2 .. your golden ratio. Another fun way to check this: take the limit of the ratio of numbers in your arbitrary sequence with your preferred software :) edit: that's a great excuse to write a bit of code lol

absolutely a brilliant example for how to apply eigenvalues to real world problem

@gianlucacococcia2384

4 жыл бұрын

Can I ask you why A1 x x1 is lambda x1?

@gianlucacococcia2384

4 жыл бұрын

Zhixun He

thank you :)) you are amazing

You can hear the students chuckling as they recognized the Golden Ratio. Didn't quite recognize it as (1 + root 5)/2.

There is a MISTAKE on the formula of the minute 32:31. It must be S(Λ^100)c in order to work as it is supposed. However it is an excellent lecture, thanks a lot. :)

@YaguangLi

9 жыл бұрын

Yes, I am also confused by this mistake.

@sammao8478

9 жыл бұрын

Yaguang Li agree with you.

@AdrianVrabie

8 жыл бұрын

+Bryan Astudillo Carpio why not S(Λ^100)S^{-1}c ???

@apocalypse2004

7 жыл бұрын

u0 is Sc, so S inverse cancels out with the S

@daiz9109

7 жыл бұрын

You're right... it confused me too...

@PhilOrzechowski , he says that he just adds it to create a system of equations.

Gilbert strang a great math teacher............

46:08 It should be F_100 is similar to c_1 * lambda1 * x_1. The professor missed x_1 here. But if you assume x_1 is 1 (which is the case here), then this is correct.

@starriet

2 жыл бұрын

Not actually. F_100 is a number and x_1 is a vector.

I'd say if you play this video at speed 1.5, it's even more awesome!

@abdulbasithashraf5480

5 жыл бұрын

Trueee

@ozzyfromspace

4 жыл бұрын

1x all the way. I savor the learning ☺️

something wrong with this lecture, 32:39, A^{100}u_0=SM^100c. Here I use M to substitute the eigenvalue diagonal matrix. The professor said A^{100}u_0=M^100Sc which is not correct.

awesome!! Greetings from Peru

so it seems like the professor emphasized the importance of the eigenvalue here, that's nice. but is the eigenvector of any importance? what's a good example of eigenvectors?

Thank you.

very nice discussion about Fibonacci ... great !

Here comes a question, How U0 is equal to c1x1+c2x2...+cnxn. at 29:50, confused, could anyone explain it to me?

Can't you also use the determinant to figure out that Aᵏ → 0 as K → ∞ ? i.e. if det A < 1 then Aᵏ → 0

U0 == "you know it" First time I've heard his boston accent c:

So, the second component of u(k+1) is useless, right? The actual value is given by the first component.

@thovinh5386

5 жыл бұрын

Yep, you can use u(k) = u(k) and it still works.

If any one ever asks you about why the Fibonacci and the golden ratio phi is connected , point him/her to this video. Thank you Dr. Strang

19:21 I would sure like to see the proof that if there are no repeated eigenvalues, then there are certain to be n linearly independent eigenvectors

Is there an error at 32:30? Shouldn't S be multiplied before (lamda matrix)^100?

Iam from india I love your teaching

Great professor

i'm still a little confused, what is u naught supposed to represent? Are those just the linear combination of the eigenvectors?

@lucasm4299

6 жыл бұрын

Nicole O Yes. They form a basis

thank you

At around 32:45, Prof. Strang writes Lambda^100*S*c. Notation wise, shouldn't this be S*Lambda^100*c?

but will it be easy to find the C vector in c1x1+c2x2+...cnxn where xi are eigen vectors and n is very big.

@jeanpierre-st7rl

4 ай бұрын

Hi @ 29:46 Uo = C1X1 + C2X2 +C3X3... Is U0 a vector? If so, How can split this U0 in to a combination of eigen vectors? What is Ci ? If you have any info pleases let me know. Thanks.

Is there a practical use to matrix diagonalization, or is it just a cool trick like 3 balls Mill's mess or sword swallowing ?

@gianlucacococcia2384

4 жыл бұрын

Can I ask you why A1 x x1 is lambda x1?

32:41 There is a slight error here. The result Λ^100 * S * C may be wrong. I think it should be S * Λ^100 * C.

I didn't get , how Au =ΛSc is right. Shouldn't Λ be at right of S. Is there anyone who can help ?

Do all matrix have eigenvalue and eigenvector? (including complex eigenvalue )

@rolandheinze7182

5 жыл бұрын

Yes square matrices but not always real eigenvalues

the best of the best

15:02 interested information inside matrix - eigenvalues

Just wondering...what keeps us from calling the eigenvector matrix E instead of S ? Is E already used for something else ?

@abdulrhmanmadi7392

3 жыл бұрын

Yes, it is used for elimination matrix.

@lolololort 1/2(1 + sqrt(5)) is also the golden ratio! Math is amazing =] I'm sure the professor knew the answer and didn't calculate it in his head on the spot.

beautifully simple how that Fibonacci worked out

I have a doubt in Difference equations part. He writes u_0 as a combination of eigen-vectors of A. Why should this be true?

@olfchandan

7 жыл бұрын

eigen vectors span the entire space (Remeber - S is square invertible matrix). So, U0 will be a linear combination of eigen vectors.

@suziiemusic

7 жыл бұрын

A set of n independent eigenvectors, each one with n components, is a basis for Rn, and therefore any vector in Rn (including u0) can be written as a linear combination of these n eigenvectors. We could choose any other set of n independent vectors as a basis and do the same thing. The "standard" basis would be the columns of the identity matrix, which in 3 dimensions correspond to the x,y and z axes.

Why the skew-symmetric matrix have zero or imaginary eigenvalue?

@sidaliu8989

5 жыл бұрын

By the definition of the skew-symmetric matrix (A^T=-A), all entries in the diagonal of the matrix must be 0. So when we come up with the characteristic equation, it will be lambda^n+b^2=0 (since the trace is zero and the determinant is some square), and this will give us pure imaginary solutions if b^2>0.