@@Parkertannerz we haven't really seen the messiah, but we have seen this man teach with all the passion in the world. Professor Strang's lectures are nothing short of blessings
You can see how passionate he is to the idea of making everyone see linear algebra as intuitive as he sees it. This is what being a professor should be all about. Absolutely fantastic.
Thank you professor Gilbert Strang and MiT for this absolutely fabulous series on linear algebra, for the benefit of students all over the world. This, for me, is USA at its very best. Greetings from Trondheim , Norway.
I"ve watched the 18.06 from lecture 1 to this section, suddently realized that all contents were in Prof. Strang's brain and he didn't utilize any auxiliary materials/tools to control the rhythm, so amazing... Thank you Sir for your great teaching!
i am working real hard on this course carefully reading the book and doing all the exercises hopefully i can become a real linear algebra professional thanks to professor Gilbert Strang
Prof.Strang is very very cute . I love how he teaches . I feel a warmth in my heart seeing how he moves around the classroom trying to teach the students .It often feels like he is trying to teach himself . I can connent with him easily . It's so AWESOME!
You have to watch these lectures a bunch of times. There is so much he is giving you here just in the little asides that are so helpful to understanding this stuff on a deeper level. You can't get it all in in the first Pass.
He is working magic here in his lectures. I am experiencing one of those "math is beautiful" moments with these concepts and ways of thinking about matrices.
Reviewing these in grad school since I've let my linear algebra get rusty. I picked up the book from the library. I wish my linear algebra class was this good. I just purchased his "Linear Algebra and Learning from Data" as soon as I saw that there's also lectures
Thank you Dr Strang for the amazing videos. I love the way you explain things so simply. Most of my teachers at University fail to explain simply things. They go into big terms while the basics is tough. With your videos I barely attend my classes but I am able to nail my exams. Thank you again!!
this lecture really puts together the concepts of pivot ranks, null space and special solutions, and complete solutions to Ax=b. I think we've also covered row reduction to get the augmented identity matrix to find an inverse and transpose.
This video was recorded on a Saturday and I am watching it on a Saturday. The Professor caught me off guard when at the end, he said, "have a great weekend. See you on Monday"
Dr. Strang thank for helping me to relearn linear algebra and it's important concepts. When I took this class at the University of Maryland Baltimore County the professor did not care if I learn it or not.
I'm proud to have found out that he had made a mistake before Mr Strang. It means I've actually undertood the previous lecture (which I had to watch several times to grasp)
He is able to be so wonderfully passionate because he cares that his students understand. So he is able to be in a one-on-one framework while standing and talking to an entire class.
There is an important piece of information missing in the lecture; The fact that any particular solution p_i is reachable from another particular solution p_j + some combination of nullspace. Now one can claim that all solutions are included in x_p+x_null
@@yt.abhibhav Yes. That's what professor said in 11:53. You can set whatever you want in free variable, but setting them all to 0 makes the following computation easier.
? Ax = b has only one ONE particular solution given all free variables are set to zero. The particular solution graphically is the vector joining the (0,0,0...) to the point where Ax = b is satisfied but all all free variable components are zero The null space is then defined graphically by finding the ratio of each free variable with every pivot variables and finding the line which has all the same ratios with respect to variable magnitude In this way we find n-r lines for n-r free variables the linear combinations of all those lines give us the whole "plane" where Ax = b The particular solution isn't unique; it is the vector found in one step of this method to find the complete solution to Ax = b all other solutions p_j like you say are indeed x_p + x_n
I do not know how to say thank you. I started loving and understanding linear algebra from when i started watching your lectures. Thank you so much. I really appreciate it. wish you the best.
Best way to learn -> Understand and understand how precious education opportunity we all here get just by generous decision by MIT and prof Strang. We never pay any(even ads), we even can do repeat as much as we want and stop for a while to think or write.
It's so weird these videos are from 1999.. I was in elementary school learning the multiplication table in 1999.. and now I'm using these lectures to help my pass my University midterms.. woah
@@ahmadabdullah262 Wow, I have no memory of those exams. I ended up dropping out of engineering after a couple years - it wasn't for me. I did some travelling, and went into architecture. Ten years later I just graduated with a master of architecture degree, and I'm transitioning from architecture into product design. Funny where life takes you... I have NO IDEA what is going on in this video
Every time I see one of Professor Strang’s lectures I run and get my old Finite Math with Calculus book just to see how much I’ve forgotten 😉 He’s awesome to listen too🙏
He always asks permission from some unseen student in the sky that only he can hear: “can I give an example here? OK, can I try this?” It’s a pleasure to take his course.
I actually passed my 3 Applied Linear Algebra courses with flying colors by just remembering all these results because the lecturer never tried to explaine what's happening behind the scene. So after almost 20 years I'm still here trying to figure out what's what. I shoulda gone to MIT.
"ok.... diurghurghh uhg" - Gilbert Strang, 46:37 jokes aside, i love these classes, he is the best math teacher I have ever seen by a mile. Makes me really passionate about learning linear algebra compared to my uni classes
I teach some electronics courses at a local community college, and I am watching these as a refresher on linear algebra. I am taking notes on Dr. Strang's teaching style!
21:47 those blackboards are very nice actually. I also watched the Stanford physics lectures (by professor Leonard Susskind), and they also have very nice blackboards (whiteboards actually). I found that top universities always have excellent blackboards, because those good professors always prefer to do the demonstrations and calculations by themselves on the board. My uni was very mediocre and so were the teachers. after a whole semester they seldom wrote anything on the board, all they do is repeating what's on the textbook and playing the powerpoint with most calculations and results already written on it, One exception was my Mathematical Physics professor, he was a brilliant guy and he always do the demonstrations on the board, and during classes we can often hear him complain about the size of the board and the poor quality of the chalk :)
"...giving other self-learners in the world opportunities of learning from the best." True enough. Of course, it would also be nice if they would release a few more lecture series from other subjects. I'm spoiled now... it is so frustrating walking into a course without already knowing it forward and back. ;)
Conclude the lecture: The solution will exist only when rank(A)=rank(A|b), which would be x=x(particular)+x(nullspace), x(nullspace) exists only when rank(A)
Here's something they didn't quite spell out to me in my Linear Algebra class: Let E be the product of row operation matrices. Let F be its inverse, FE = I. Obviously if Ax = b then EAx = Eb (because f(x) = f(y) for all f when x = y). But also importantly and relevantly if EAx = Eb then Ax = b: If EAx = Eb then Ax = IAx = (FE)Ax = F(EAx) = F(Eb) = (FE)b = Ib = b. That is, divide by E (i.e. multiply by F) on both sides to recover Ax = b. This is the justification for the algorithm: Compute E * [A b] where E is chosen such that solutions x can be read off easily. Solving E*[A b] means x solves EAx = Eb, but then Ax = b. So the outputs of the algorithm are in fact solutions to Ax = b.
At 43:32, shouldn't Professor Strang have put *infinite* as opposed to *1* solutions to Ax = b. When r = m = n, all n (or m) - dimensional vectors can be obtained from linear combinations of A's columns because the column space is also n-dimensional.
On 20:30, Professor Strang mentions that X-particular is unique and can’t be multiplied by a constant (it is not a line space). So I went back to step one of X-particular where he said (12:00) - that setting all free variables to zero is one way of finding X-Particular. Why is X-p one guy if there are many ways to find different xp? thank you
You can specify the exact same set of "solution space" (i.e. set of vectors that solve the system) just by adding a vector from the null space to the Xp. The solution space may now look different on the surface, however rest assured that all possible Xp's are "reachable" just by adding a vector from the null space.
I think the main point of this lecture is not to find the solution at all, but to know when you can find it. So you can watch in perspective and answer when you can and when you cannot to find it. That's the reason for what he did'nt begun by putting numbers as results of the equations, but with literals, meaning they can be any number and the answer to that system just can be found if those numbers meet the condition of beeing a mix of every column in the matrix.
This information gives solution of pulling up and pulling down gravity operated on two planets orbiting the orbit bt its mass difference operated on 60 degrees in between them when the mass difference is much higher side. Now the matrics giving more informations by column vector by elemination.
I’m confused about how he explained adding a x particular to nullspace will not give a subspace. I think it this way. Nullspace should always contain the origin, so in this case it is a plane going through the origin, since x particular is a vector not in that plane, adding them together will never give a zero, so it is not a subspace.
