Wow. I was initially skeptical when I started watching thinking this is just another video on this topic. I already knew this material but now I know it much better. Very clear explanation. Thank you.
Ax = b in general has no solution here, since b is in an n-dimensional space and Ax is in a 2-dimensional (at most 2) subspace of that space, since Ax is a linear combination of 2 n-dimensional vectors. Just for anyone else that got stuck there for a moment.
I think what makes this topic confusing for beginners is that the x vector in Ax=b does not represent the "x's" but rather represents the betas. And the b in Ax=b does not represent the betas but rather the y's. It is an unfortunate accident of the nomenclature.
At 7:17, you said "if AX equals B is overdetermined, that means B is not in the column space of A, there's no solution to X". Where does that come from ?
it's not necessary that an overdetermined system has no solutions, it is unlikely in a real world scenario but it's not necessarily the case. If the columns of an overdetermined matrix are dependant for example in the 2D case then they lie on a line. my point is that the fact that there is no solution is not because the system is overdetermined but because b happens to not lie in the column space
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This is a great overview of the problem! A teacher of mine went over the steps without all the justification, so this was very helpful!
Wow. I was initially skeptical when I started watching thinking this is just another video on this topic. I already knew this material but now I know it much better. Very clear explanation. Thank you.
Ax = b in general has no solution here, since b is in an n-dimensional space and Ax is in a 2-dimensional (at most 2) subspace of that space, since Ax is a linear combination of 2 n-dimensional vectors.
Just for anyone else that got stuck there for a moment.
What an incredibly intuitive explanation!
I think what makes this topic confusing for beginners is that the x vector in Ax=b does not represent the "x's" but rather represents the betas. And the b in Ax=b does not represent the betas but rather the y's. It is an unfortunate accident of the nomenclature.
That would be a beginning beginner.
@@ProfJeffreyChasnov me is a beginning beginner ^o^ ty for the great lectures
Sir, you are amazing! This is by far the best at least according to me a best explanation for least squares optimization problem.
Great video to quickly review the concepts. Thanks Jeffrey.
At 7:17, you said "if AX equals B is overdetermined, that means B is not in the column space of A, there's no solution to X". Where does that come from ?
great video , finally understand clearly the topic
Is this dude really out here writing backwards
or maybe some tech has been made that can flip a video (omg what has the world come to)...
I think that he is recording himself behind a transparent wall, and then he reflects that recording around the vertical axis.
or he's writing on a mirror and the video might be everything in the mirror
@@cathyfeng3385 writing on glass and then flipping video horizontally
TENET
it's not necessary that an overdetermined system has no solutions, it is unlikely in a real world scenario but it's not necessarily the case. If the columns of an overdetermined matrix are dependant for example in the 2D case then they lie on a line. my point is that the fact that there is no solution is not because the system is overdetermined but because b happens to not lie in the column space
In 7:15. If Ax=b it is not true that this implies that b does not lie in the collumnspace of A. It may be overdetermined and have a solution.
We are doing the least-squares problem here.
you are the best!!!
You are a legend 👍
Very helpful, thank you
thank you
Nice sir
great!
you lost me at Ax =b. . you have x under B and B1, A under X and b under the y's.
If y=Ax +b, isn't what you wrote confusing?