@@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
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 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.
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!
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.
@@3rddegreeyt144 It's been so long that i forgot what the joke was but I think it's about Strang drawing another line on the blackboard to represent the 4th dimension, which obviously no uni does. The joke could be that the blackboard is so good it can represent any number of dimensions.
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
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
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.
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.
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!!
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.
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"
"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 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.
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
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
"...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. ;)
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.
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)
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.
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 :)
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.
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.
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!
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.
Well, this is a lecture to people who have no idea what a vector, Matrix or Eigenvalue is! They have to learn first before using the PC to solve problems. If not, it is GIGO!
At 45:20, if we are talking about null space then indeed there would be an infinite number of solutions, but if we are talking about Ax=b then there needs to be a particular solution in the first place for the infinite solutions to be existing. Am I right?
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.
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.
Can’t I also choose other values for x2 and x4 to find other particular solutions? Would they not also shift the null space to another spot and cover regions which were not already defined? Or does the particular solution + null space contain the set of all possible solutions for Rx=b?
i believe it contains all solutions since if Ax = b => Ax + A(any vector in null space) = b + 0 = b , and since the null space contains all such vectors, any other solution you obtain would be this x + a vector in the null space.
I can never draw a 4-dimensional space on a 2-dimensional blackboard. It would look wrong no matter how I do it. (There is a vector perpendicular to an entire 3-D universe?)
At 28:11 ,How can we assume Nullspace to be 0,if free variables are 0,in Reduced echleon eqn=[-F I] to find nullspace F=0 as free variable is 0,But I can still write 1 in place of I to get Nullspace non zero???
Ayush Kumar I wondered the same, however I realised you are choosing the free variable to be the I. If you don't have a free variable then you can't choose that I in the first place. I would suggest only follow that template if you have free variables.
Xparticular is calculated by setting all the free variables to zero. But in case, when rank=n, there wont be any free variables. How to find the Xparticular in this case?
The number of free variables gives you the number of dimensions of the solution. No free variables is a zero dimensional solution which is a point. Just for anyone reading this really.
When he plots the solutions x in R^4 at minute 23-24, he plots Xp, Xn and Xcomplete=X. Shouldn't Xn be drawn through the origin and Xcomplete be drawn away from the origin and at the 'plane' without going through the origin?
+Hiten Chojer The plane that he drew on the board wasn't Xn, it was Xcomplete, which is Xp+Xn. It's like Xn that has been shifted away from the origin by Xp, just like he said..
+Adam Levin You are right. I had the same thought as Chojer initially. But the man eventually made it clear that the plane is Xcomplete which is the Nullspace being shifted to go through Xp.
oh.my.god. he is just TOO PRECIOUS!!!!!
every time he starts with his way too excited "OK!" I actually get excited too :)
Thanks Dr. Strang!
Excellent quality of teaching.. really appreciate the initiative taken by MIT to make these lectures available to everyone across the world for free
abhishek G.Y
MIT!! 🇺🇸🏆❤️
Prof Strang is a blessing to humanity!
@@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
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 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.
Nice description 👍
I don't think I'd pass linear algebra without your lectures
i got so invested in the lecture i was replying whenever he asked the class a question
Anxiously Waiting at the end for him to switch the r=m
Haha same, I get to the point where I'm like, "you heard me, right?" 😂😂 These lectures are just too good!
so did I
good one, made me laugh ^_^
I love his blinking single eye, literally. this man is great.
Xiang Zhang double the speed, double the blinks.
"lemme give an example, okay BRILLIANT example" That "brilliant" always gets me
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!
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.
to be honest, the blackboard is better than my uni
Mine either !!
Well can your uni represent four dimensions?
It’s better than mine too
@@theblinkingbrownie4654 how can a uni represent 4 dimension
@@3rddegreeyt144 It's been so long that i forgot what the joke was but I think it's about Strang drawing another line on the blackboard to represent the 4th dimension, which obviously no uni does.
The joke could be that the blackboard is so good it can represent any number of dimensions.
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
all the best Sir!!
A true informative lecture by a brilliant professor! Can't say how much I enjoy the way he explains every concept; so simple, so elegant!
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.
One of my favorite MIT professor. May Allah Almighty bless with long life.
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
How was that class? I got the 4th Ed to follow this class.
I`m learning English watching these incredible classes, thank you so much Teacher Strang and MIT
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's a great professor. very rarely you see such a talented teacher in math.
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.
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!!
Thank you so much, what an amazing lecture! 23:50 - 24:12 is definitely going in my notes word by word!
Watching these in junior-year ME is incredible. He essentially condensed the past three years of studying into one simple mental picture.
@@brandonlawrence5851 sameee
These lectures are addictive.
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.
The taste is different with MIT people! You "feel" the science with them! I envy the ones who had the opportunity to sit in these class rooms!
omg that resonates with me so deep!
all points of the nullspace are shifted by the vector x-particular
this lecture NEVER GETS OLD.
랄뚜기가 왜 여기에..
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.
Its amazing how the previous recitation perfectly syncs with the lecture at 16:17
Thank you Prof. Strang & thank you MIT.
Brilliant lectures, much appreciated.
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"
Gilbert Strang - a legend in the making...
Guess he's already made as a legend...
thank you MIT, all these linear algebra lectures are very helpful, and Prof. Gilbert Strang teach very well
thank you very much!!:))
"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 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.
thank you so much professor Gilbert Strang. You literally make my day.
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
And here I am 10 years later trying to pass my finals. Btw how did they go?
@@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
@@LukieRawr I changed from architecture to computer science. Guess you never know where life takes you huh
@@ahmadabdullah262 So how its going for you mate?
Amazing. Think about it, xparticular plus the nullspace gives final x in r4, is the same as y=a*x+b in r2! This really blows your mind. Great 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
So basically, professor here assumed a scenario of just an unique particular solution? Please confirm!!
@@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
45:00
in the case r = m < n, then
rref (A) = [ Id. F ] , if we allow permutation of columns.
This part of the lecture is fantastic when he overwrites I and F!!
"...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.
;)
Bless this man and his family, he should be an idol for teachers
The summary is brief and billiant!, Ax=b, cool!
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🙏
Thanks a lot Dr. Strang, i'm really excited for the next lecture.
My Dog! His lectures are a work of art! Incredible
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.
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)
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.
These videos finally make me realize that I wasn't stupid, my undergrad professor was!
So I have to draw a 4-D picture on this MIT cheap balckboard - Einstein could do it..:D
hypnoticpoisons I read this comment just as he said that!
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 :)
46:37
I just record the timestamp in case I forget the conclusion of this great course in the future.
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.
23:57 What a genius. I love this lessons.
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.
I really like his teaching. Perfect.
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!
I have fallen for Dr. Strang. He is actually Dr. Strange. 💗 I don't know how he gives me some feeling through these videos. Only kudos....👏.
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.
Thanks (2 years after) because I didn't understand it. Now it's obvious :D !
"What about the solution to Ax=b, what's the story on that one?" hahahah he's adorable
Dr. strang is too energetic!!!
sorry....but really not sorry 38:32. Thank you Gilbert Strang!
"Me and the guys in the nullspace"
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
Same doubt
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 should have been in his lectures instead of being 8 years old. 😞
35:17 The lovely sound of science
oh god, prof. strang is so funny. he is acting natural. like him so much.
25/11/2024.
iNDIA won Perth Test. Jassy B Man Of The Match. Virat 100* and Yashasvi 161.
thankyou professor!! thanku MIT!!
I nod my head to the screen like listening to rock music, while I am not.
"Einstein can do it." I love it when he said that :)
Great videos! the only thing that i'm missing is "to rise my hand and ask him some questions"..
Language of column space , OMGGG
I LOVE YOU MAN
Thank You, Prof Strang.
Well, this is a lecture to people who have no idea what a vector, Matrix or Eigenvalue is!
They have to learn first before using the PC to solve problems.
If not, it is GIGO!
Einstein couldn't do it, but... Gilbert can!
Thank you MIT!!
At 45:20, if we are talking about null space then indeed there would be an infinite number of solutions, but if we are talking about Ax=b then there needs to be a particular solution in the first place for the infinite solutions to be existing.
Am I right?
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.
He got me when he said: "can I just right [ IF]"
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.
When he says *OK* you know it's about to get good 😂😂
Did he choose the b vector (1,5,6) at random, based on 0 = b3-b2-b1? EDIT: yes
Thanks a million times!
33:07 dude, there's something moving under the professor's desk.
I believe it’s vector trying to escape from the desk space.
Can’t I also choose other values for x2 and x4 to find other particular solutions? Would they not also shift the null space to another spot and cover regions which were not already defined? Or does the particular solution + null space contain the set of all possible solutions for Rx=b?
i believe it contains all solutions since if Ax = b => Ax + A(any vector in null space) = b + 0 = b , and since the null space contains all such vectors, any other solution you obtain would be this x + a vector in the null space.
I can never draw a 4-dimensional space on a 2-dimensional blackboard. It would look wrong no matter how I do it. (There is a vector perpendicular to an entire 3-D universe?)
Time.
At 28:11 ,How can we assume Nullspace to be 0,if free variables are 0,in Reduced echleon eqn=[-F I] to find nullspace F=0 as free variable is 0,But I can still write 1 in place of I to get Nullspace non zero???
Ayush Kumar I wondered the same, however I realised you are choosing the free variable to be the I. If you don't have a free variable then you can't choose that I in the first place. I would suggest only follow that template if you have free variables.
21:55 My brain stopped working when he drew ℝ4 on 2 dim board! Woah!!
i'm wondering that when he draw the picture of the complete solution , why doesn't the Null space pass the origin ?
At t = 11:00, finding the complete solution to Ax = b.
such a great professor, Salute you
Xparticular is calculated by setting all the free variables to zero. But in case, when rank=n, there wont be any free variables. How to find the Xparticular in this case?
+adarsh k murthy I guess that when rank=n, there will be either one solution or NO solutions. In the case of one solution, Xcomplete = Xparticular.
+Sam Mao thank you very much
+adarsh k murthy
yes either 1 solution or nothing,then we go for solutions with errors-as in next lectures
The number of free variables gives you the number of dimensions of the solution. No free variables is a zero dimensional solution which is a point. Just for anyone reading this really.
when r< m, n, i.e. the last case in the summary board, can it have exactly one solution?
When he plots the solutions x in R^4 at minute 23-24, he plots Xp, Xn and Xcomplete=X. Shouldn't Xn be drawn through the origin and Xcomplete be drawn away from the origin and at the 'plane' without going through the origin?
+Hiten Chojer The plane that he drew on the board wasn't Xn, it was Xcomplete, which is Xp+Xn. It's like Xn that has been shifted away from the origin by Xp, just like he said..
+Adam Levin You are right. I had the same thought as Chojer initially. But the man eventually made it clear that the plane is Xcomplete which is the Nullspace being shifted to go through Xp.