Least Squares

In this video, I solve linear systems equations that are unsolvable (whoa!!!). More precisely, I show how to solve the systems using the least-squares method, first by doing it the Peyam way using orthogonal projections, and then by using the classical way (by multiplying by A^T). Finally, I show why the two methods are equivalent. Along the way, I also talk about the QR decomposition of a matrix. Enjoy!

[drpeyam]

Note: At 4:30, part of the video was accidentally cut out. That part is simply the calculation of b^, which is on the right-hand side of the whiteboard.

Also at around 5:00, I switched b and b^ but I correct that mistake a couple of minutes later in the video.

[gastonsolaril.]

You've just clarified so many things about a subject that I've never been able to totally understand throughout my 9 years of graduate and post-graduate engineering courses...
Hail and long live Dr Peyam, First of His Name, Ruler of the Seven Kingdoms of Mathematos, Lord Paramount of Integrals, Series, Transforms, Eigenvectors and Stochastics!

[taubone]

The awesome Peyam method is the best method.

Also, bless you (14:11).

[JLConawayII]

Vastly superior to the most-squares method.

[loganreina]

For my final calc II project, I gave a derivation and a few examples of the least squares method in use as both an introduction to partials and a practical application of them. I was pretty proud of the presentation.

[tomatrix]

Wow great video Peyam! Happy New years!

[shareefsyed]

Thank you so much for making this! It really helps!

[codebits]

Am watching one of your old vids😂😂😂
You've really upgraded🙌🔥
Good teach though✔💪

[justcarcrazy]

Why is (Q_t)(Q)=(I)? Isn't (Q^-1)(Q)=(I)? Is this a definition, a happy coincidence, or a necessary condition?

[joebaldwin]

18:55 How is, for Q orthogonal, QQ^T not the identity but Q^TQ is? Im confused since Q^-1 = Q^T and therefore Q^Q-1 = I and Q^-1Q = I

[dhunt]

sneezing from all the white board dust?

[AndDiracisHisProphet]

I have never seen any of those methods.

[arthurribaudet]

Your first method only works if the matrix is already orthogonal?...

[JorgetePanete]

So, do you mean that it's like **I WANT A FUCKING SOLUTION, YOU FUCKING FUNCTION!!**
?

[mitchkovacs]

Was this cut incorrectly or am I missing something? There seems to be a jump at 4:30

[thatdude_]

can you prove that the integral from -inf to inf of exp(-x²) = sqrt(pi), but using the wallis product and the gamma function (gauss' representation of the gamma function specifically)? i haven't seen a video about this here on youtube, and i think it's a lot more interesting that the ole "polar coordinate substitution" method XD

[michalbotor]

frankly, i think that if Ax=b doesn't have a solution, then calling x-hat its least squares solution is, forgive me, preposterous; even if it's already established in the literature, i believe, that we should insist on calling it what it actually is -- least squares approximated solution.