In the middle of my PhD, with all the stress, Dr. Strang's lectures are the only relaxing time I have. Nothing else really feels so great. What a legend. I wish I could attend his lectures physically.
You should.. uh.. get a guitar or maybe go on a hike or learn to cook a new dish from scratch. Glad to hear Gilbert's lectures are peaceful for you, but the stress of academia can be monumental and debilitating over time. Hobbies outside of academia can save your life. Fishing is a great hobby--lots of time to think about science or whatever you want, but you're exercising, and spending time in nature.. best of luck on your degree!!!
does it make sense that during the middle of my PhD I put everything on hold and binge followed lectures+bookchapters+exercises to once and for all fill the gap in the knowledge in crucial areas in wireless and DL. It is taking time but It feels like time well spent
I've finished this course a while ago. still, going through Markov chains in probability was confusing till I came back and watched this, and again, Mr Strang came to the rescue. I don't know how much I need to thank you for your online courses for it to be enough, as simply saying thank you doesn't do you justice. I just want to say that you, Mr Strang is one of a minority of people who indeed make this world a better place. Thank you, from some corner of this Earth.
This lecture is amazing. Pro.Strang gives an intuitive perspective Fourier Series, and how it is related to orthogonal vector. I know Fourier transform pretty well but now have a deep understanding of that.
my proffesor in signals and systems explained in a single class all the the prequisites for linear algebra leading to fourier series, i can gurantee you not a single person unless he's already mastered linear algebra understood her. This guy is most amazing proffesor ive ever seen, makes complex things really simple@@yuchujian8837
My god this is so beautiful. So much insight making things fall into place. "Finding coefficienes in a Fourrier series is exactly like an expansion in an orthonormal basis". Now I get it!
Prof. Strang's lectures are legendary 😭I've been following this course for a while and it has been a delight to see the concepts unfold in such an elegant and coherent way
This is the moment in time where I have to say thank you! In my opinion this lecture connects everything that happend until now beautifully and builds the foundation for solving most advanced engineering problems.
Professor Strang's introduction to projections using Fourier Series as an example generalizes to other orthogonal functions (Legrendre Polynimials, Bessel Functions, etc.). This is pretty cool, because you'll see this stuff ad nauseum in electrodynamics, quantum mechanics and statistical mechanics.
Yes, the collapse of wavefunction is a central concept in Quantum Theory. Measurement can be understood as projection of the wavefunction onto one of the orthonormal basis states i.e., taking the dot product between the wavefunction and a basis state.
My goodness, this is beautiful. The insights given by prof. Strang, especially on Fourier series, are out of this world! Thank you very much four your intuition on these profound topics, they are very valuable.
Professor Strang can almost always explain a concept from an angle other professors rarely touch upon. Excitingly, that angle seems always to be the right angle most appropriate to understand that concept.
Prof. Strang, you are truly amazing for even a Ph.D, in terms of your lucid explanation and causal but very deep explanation which encourages thinking more than number crunching! It is really an honor to listen to your lecture!
The link between Orthonormal vectors and Trigonometric functions by the example of Fourier Series! Great example with Great connection between Algebra and Function! at 44:00. Thanks!
This is certainly God level! Things he is saying at times, not only predicting the answers to himself but also to the audience, just to assure them that they can do it too. Also, not being there predicting and waiting for more than a few more secs., for others to still follow!
It is worth pointing out that A in the the notation N(A) (16:00) is a generic reference to matrix, not the Markov matrix used as a concrete example. Clarify this confusion, and we could better understand why the eigenvector of eigenvalue 1 is in the null space of ‘A’.
The eigenvector calculation around minute 33 confused me until I saw how the zero in row three gives an extra degree of freedom in choosing the vector. The check is to multiply by row 2. A nice trick.
I’m taking 18.06 now but Gilbert Strang resigned just before I took the class. Very very sad. His lectures here are perfect though so I can just watch these in my dorm instead of walking all the way to lecture.
In the Fourier series explanation, I think (1, cos, sin, ...) are not unit vectors, cause inner product of cos and cos is pi, not 1. So, I guess, to be orthonormal basis, we need normalization with 1/pi with cos vector. So, In my understanding, just 'orthogonal basis' is correct term for functions (1, cos, sin, ...) as a basis for set of all possible Fourier series' (a vector space).
Hawkings radiation compels us to go in for Blackhole as singularity becomes unsteady flow for a vaporisation in a way supports Einstein's refusal to accept Blackhole as singularity.
I didn't know about Markov matrices, they are very interesting. So all n x n matrices with each entry equal to 1/n are Markov matrices, with eigenvalues 0 and 1 (1 because they are Markov matrices, 0 because their rank is equal to 1).
me too haha. I guess because its somehow related to calculus? or differential equation, if you are not quite familiar with those topics, i guess it might add difficulties to us to see what the lecture is actually trying to prove, and the actual use of these matrix. However, you might probably remember these stuff, when you actually need them in the future. then you can pick up these knowledge again and solve the puzzle hopefully
Ah man, I know that feeling. You get to the end of a lecture and realise that you haven't understood anything for several lectures. It can really be worth it to rewatch carefully.
The invention and development of Google is due to Markov Matrices. The things that we see and use everyday in our lives is built on mathematical theory and concepts.
Prof. Strang's lectures are legendary 😭I've been following this course for a while and it has been a delight to see the concepts unfold in such an elegant and coherent way
![fellow fellow - พรุ่งนี้ไม่มีใครรู้ feat. INK WARUNTORN [OFFICIAL MV]](http://i.ytimg.com/vi/QP6JyjYx_W0/mqdefault.jpg)
Audio channels fixed!
It would be nice to fix other audio channels as well :D
Thank you!
In the middle of my PhD, with all the stress, Dr. Strang's lectures are the only relaxing time I have. Nothing else really feels so great. What a legend. I wish I could attend his lectures physically.
You should.. uh.. get a guitar or maybe go on a hike or learn to cook a new dish from scratch. Glad to hear Gilbert's lectures are peaceful for you, but the stress of academia can be monumental and debilitating over time. Hobbies outside of academia can save your life. Fishing is a great hobby--lots of time to think about science or whatever you want, but you're exercising, and spending time in nature.. best of luck on your degree!!!
@@mississippijohnfahey7175 great!
Yes. You are right
does it make sense that during the middle of my PhD I put everything on hold and binge followed lectures+bookchapters+exercises to once and for all fill the gap in the knowledge in crucial areas in wireless and DL. It is taking time but It feels like time well spent
@@euler12i am with you brother, Me too in middle of PhD, I just can’t really learn something until I realize how useful that thing is😮
I've finished this course a while ago. still, going through Markov chains in probability was confusing till I came back and watched this, and again, Mr Strang came to the rescue.
I don't know how much I need to thank you for your online courses for it to be enough, as simply saying thank you doesn't do you justice. I just want to say that you, Mr Strang is one of a minority of people who indeed make this world a better place.
Thank you, from some corner of this Earth.
same boat
This lecture is amazing. Pro.Strang gives an intuitive perspective Fourier Series, and how it is related to orthogonal vector. I know Fourier transform pretty well but now have a deep understanding of that.
I really appreciate the way he related the Fourier series to orthogonal vectors. Before watching this video all I knew was memorizing the formula
my proffesor in signals and systems explained in a single class all the the prequisites for linear algebra leading to fourier series, i can gurantee you not a single person unless he's already mastered linear algebra understood her. This guy is most amazing proffesor ive ever seen, makes complex things really simple@@yuchujian8837
My god this is so beautiful. So much insight making things fall into place. "Finding coefficienes in a Fourrier series is exactly like an expansion in an orthonormal basis". Now I get it!
Prof. Strang's lectures are legendary 😭I've been following this course for a while and it has been a delight to see the concepts unfold in such an elegant and coherent way
Thank you Professor Strang !
but I think 14:43 is in the nullspace of (A-I)^T rather than nullspace of A^T isn"t it ?
I can not agree more. A litter mistake
This is the moment in time where I have to say thank you! In my opinion this lecture connects everything that happend until now beautifully and builds the foundation for solving most advanced engineering problems.
Professor Strang's introduction to projections using Fourier Series as an example generalizes to other orthogonal functions (Legrendre Polynimials, Bessel Functions, etc.).
This is pretty cool, because you'll see this stuff ad nauseum in electrodynamics, quantum mechanics and statistical mechanics.
Taking a measurement on a Quantum System (COLLAPSING THE WAVEFUNCTION!!) is a physical dot product.
Which of those topics topics do you think are useful for quantum computing? (applied quantum mechanics)
Yes, the collapse of wavefunction is a central concept in Quantum Theory. Measurement can be understood as projection of the wavefunction onto one of the orthonormal basis states i.e., taking the dot product between the wavefunction and a basis state.
My goodness, this is beautiful. The insights given by prof. Strang, especially on Fourier series, are out of this world! Thank you very much four your intuition on these profound topics, they are very valuable.
Professor Strang: You have the amazing ability to bring out new rabbits out of old hats! Hats off!
Professor Strang can almost always explain a concept from an angle other professors rarely touch upon. Excitingly, that angle seems always to be the right angle most appropriate to understand that concept.
Prof. Strang, you are truly amazing for even a Ph.D, in terms of your lucid explanation and causal but very deep explanation which encourages thinking more than number crunching! It is really an honor to listen to your lecture!
I completely understood in math terms the concept of wavefunction collapse by measurement (projection) after watching this brilliant lecture...
Those lectures are pure gold. gorgeous!
Strang: That is the first time in the history of linear algebra in which a eingenvalue has a component 3300.
LA: *Oh, not today, no!*
The link between Orthonormal vectors and Trigonometric functions by the example of Fourier Series! Great example with Great connection between Algebra and Function! at 44:00. Thanks!
This is certainly God level!
Things he is saying at times, not only predicting the answers to himself but also to the audience, just to assure them that they can do it too. Also, not being there predicting and waiting for more than a few more secs., for others to still follow!
20:28 when you're about to say "Oh f***!" and you remember the camera is rolling! Haha gotta love Prof. Strang.
Great video great lecture great professor.
It is worth pointing out that A in the the notation N(A) (16:00) is a generic reference to matrix, not the Markov matrix used as a concrete example. Clarify this confusion, and we could better understand why the eigenvector of eigenvalue 1 is in the null space of ‘A’.
You are emancipation of Algebra thank you sir.
Great analogy of infinite basis for functions
The eigenvector calculation around minute 33 confused me until I saw how the zero in row three gives an extra degree of freedom in choosing the vector. The check is to multiply by row 2. A nice trick.
I’m taking 18.06 now but Gilbert Strang resigned just before I took the class. Very very sad. His lectures here are perfect though so I can just watch these in my dorm instead of walking all the way to lecture.
Lifechanging lecture, really
Great lectures! I just wanted to comment because I am learning a lot from all his lectures! And I already have a degree in physics!
Cool! Good to know that, how encouraging! Did you got PhD or BS?
In the Fourier series explanation, I think (1, cos, sin, ...) are not unit vectors, cause inner product of cos and cos is pi, not 1. So, I guess, to be orthonormal basis, we need normalization with 1/pi with cos vector. So, In my understanding, just 'orthogonal basis' is correct term for functions (1, cos, sin, ...) as a basis for set of all possible Fourier series' (a vector space).
I just came for the first 5 min then I watched it till the end! So engrossing!
Prof Strang the G.O.A.T!!!
35:17 Fourier Series
amazing lecture
Hawkings radiation compels us to go in for Blackhole as singularity becomes unsteady flow for a vaporisation in a way supports Einstein's refusal to accept Blackhole as singularity.
Thank you MIT
He is a legend!
I didn't know about Markov matrices, they are very interesting. So all n x n matrices with each entry equal to 1/n are Markov matrices, with eigenvalues 0 and 1 (1 because they are Markov matrices, 0 because their rank is equal to 1).
hey Yuxuan Wang! congratulations for reaching here :)
Just beautiful
Little notation mistake at around 16 min Null space of (A-I)^T
14:38, shouldn't that (1,1,1) be in the left nullspace of (A - I) instead of left nullspace of A as demonstrated in the tape？
Yeh I have this problem too,😂 he may was wrong
yes, he might be wrong
Yes
thx bro😉
Mind is officially blown.
45:08 .... and what a "transpose" of a funcion would be? Nevertheless, nice to listen to lessons with the voice of Frank Sinatra!! :)
Camera: I see you're leraning about Morkov Matric- (3:43)
Heyy nich shirt ma dude
I am completely lost in the last two lectures
me too haha. I guess because its somehow related to calculus? or differential equation, if you are not quite familiar with those topics, i guess it might add difficulties to us to see what the lecture is actually trying to prove, and the actual use of these matrix. However, you might probably remember these stuff, when you actually need them in the future. then you can pick up these knowledge again and solve the puzzle hopefully
Ah man, I know that feeling. You get to the end of a lecture and realise that you haven't understood anything for several lectures. It can really be worth it to rewatch carefully.
here is the question, so why did you back to this lecture and leave the comment lol ,
I never understood Fourier series more clearly
44:25 Isn't vectors right hand side weird? I think it must be V^T*W = W1*V1^(T)+... so it becomes Dot product
The notation is just kinda sloppy there. v1, v2,... are actually elements of the vector v
03:15, 3:45
Brilliant!
free enlightenment
15:42 Shouldn't X1 be in N(A-I) instead of X1 in N(A)?
He came.
Put hands in his pockets
Explained Fourier series using projections
He left
🛐 Chad Professor
16:02 is it in N(A-I) instead of N(A)?
Yes.
He's calling A to both original A and A-I , that's why he says (1,1,1) is in the left nullspace.
yes.
Yes, I figured the same thing.
@@mauriciobarda no. (1,1,1) is in the N(A^T). x1 is in N(A)
Excellent!
Can someone say u-not better than Gilbert Strang?
20:23
The invention and development of Google is due to Markov Matrices. The things that we see and use everyday in our lives is built on mathematical theory and concepts.
Some smart people moved from CA to MA, and some stayed in CA and made a billion. If only the probabilities had gone more my way.
PLEASE DO NOT CODE A.I TO USE ALL THESE FREE INFORMATION ESPECIALLY MIT LECTURES ON TH-cam😭😭🤣I HOPE THE AVERAGE JOE TAKES NOTE AND START STUDYING ASAP
AI now can solve university level math problems. But I doubt it can truly understand even the basic arithmetics.
dddd :D here i am
Prof. Strang's lectures are legendary 😭I've been following this course for a while and it has been a delight to see the concepts unfold in such an elegant and coherent way