Part 3: Orthogonal Vectors

A Vision of Linear Algebra
Instructor: Gilbert Strang

Professor Strang describes in detail orthogonal vectors and matrices and subspaces. He explains Gram-Schmidt orthogonalization, as well as the Least Squares method for line fitting and non-square matrices.

License: Creative Commons BY-NC-SA

[sarvasvarora]

LMAOO he never forgets dissing Schmidt

[inigom]

Graham has the idea, I don't know how Schmidt got into it... 18:06 Nostalgia

[ramonmassoni]

It's 2020 and Schmidt is still getting wrecked lmao

[TheWitness]

I feel like watching Messi in his waning days when I am watching these videos of Strang. I used to watch his MIT videos back in 2010. We are blessed to still have you Pr. Strang.

[HuyDo-wlsu]

I've just found out he's 86 this year. Such a legend.

[maxwellsdaemon]

This is the fourth video of the series I am watching. While I think I am familiar and comfortable with LA, I already learned some useful and interesting concepts and points of view from these lectures. BTW, this is how I imagine Mr. Rogers would teach LA, if Mr. Rogers were a math teacher.

[freeeagle]

10:07 In mind, "Let me write something. OMG! Where is my blackboard?" Miss your moving pieces of blackboard up and down many times in one session of the lecture, dear Professor.

[taylorlorenztransormation]

Unbelievable commentary. I was watching the videos out of order and thinking to myself “this guy is unusually good.” Then I heard him say his name.

[herambpatilofficial]

I just love MIT OCW,
I request you, people, to make a playlist of organic chemistry...

[maxwellsdaemon]

Note that there is a subtle "mistake" on the slide titled "Orthogonal matrix" (around 6:10). An example was presented where Q is the 2D rotation matrix, with eigenvalues exp(+ or - i \theta). The argument presented showing that |\lambda^2|^2 = 1 is true if the complex conjugate is taken at the same time as the transpose (^T) operation and if ||Qx||^2 and ||x||^2 are non-zero, so that these can be canceled from both sides. For the eigenvalues of the 2D matrix Q, obviously \lambda^2|^2 = 1. But if the complex conjugate is not taken with the transpose operation, ||Qx||^2 and ||x||^2 CANNOT be canceled from both sides since ||x||^2 = 0 for the 2D matrix Q.

[echolee]

The geometric illustration of least square is soooo cool👍👍

[milleskov]

Dear Gilbert. I adore you so much! Even when it seems i ought to use admire or to be infatuated by you, i stand by baby doll adoration for your straight out simple wonderful math explanations ❤️ ILU for teaching us. THANK YOU!

[DerekWoolverton]

The length of a vector (-1, 2, 2) at 4:20 would be the same as the length of a vector (1, 2, 2) which would be sqrt(11), not sqrt(9).

[thomasfisherson]

This dude's the OG of linear algebra.

[videofountain]

Animated, Paced and Clear presentation.

[Tiaraz]

Can we please get a detailed answer key to some extent for (odd or even questions) for the new book expected in 2021. Even if it’s for purchase I would be willing to buy it. Detailed solutions help out tremendously especially in showing exactly why a question is done in such a way. Thank you!

[brandomiranda]

I love prof Strang, but I think the proof he presented is a great example of why (unfortunately) ppl don't like proofs. Yes it shows why things are true but I'd assume for most ppl it comes out of thin air. There is little motivation to how it would occur to someone how to do it. e.g. no motivation of how you'd even know in the first that orthogonal is a property to expect from symmetric. Even if we did expect it, how in a proof real ppl doing them think hmm I have this property transpose and I need to use it as a key step in the my proof...etc. It is just too magical without any motivation from how the result would be expect or the proof method itself would be expect. Just a random set of facts that worked. This is what I think has to change for ppl to stop hating proofs. Make them appealing to humans, rather than to computers. Where did they come from? Where did the technique used work? etc.

It's not a math journal (even then I think the above should apply), it's a class room.

[pavanchandra]

Professor Strang, Can you explain the limitations of least squares? When will the error vector be significantly large enough that it would nullify the method itself?