my calculus 3 professor taught us, think of a saddle point as a point on a "pringles chip" a lot of people know exactly what it looks like.

After viewing the 3blue1brown video on Duality, I am just seeing xT*A*x as the results of applying a linear transformation to a vector x and then projecting that new vector back onto x; if the vectors still point "in the same direction" i.e. the projection is positive, then A is positive definite

This lecture is definitely a positive effect on my grasp of the matrix, and this lecture plays a pivotal role in the whole series. Thank you prof. Strang. Nobody has explained those concepts so clearly and coherently -- a whole new world is ahead of me. This is a must-see. A genuine human heritage.

It's a .... superbowl

This lecture is especially beautiful..

Is that video available or not? I'm following the course on you tube but I can't visualize this particular lecture (n° 27).

I watched this once, taking all notes. Then watched it again (with break) without taking notes. Then I read my notes (after a break). The lecture's quite good! Btw for those that thought the second derivatives thing came out of left field, Du is the directional derivative operator, so you want Du( Du( f(x1,x2,...,xn) ) ), which gives the expressions for x^T*A*x for a connected f and A. This tells us that x^T*A*x is like a second order operation for the derivative of a function when A is the Hessian (matrix of second derivatives) of said function f(). This wasn't clear to me initially so I was kinda lost. Best!

I believe the graph of 2x^{2}+12xy+20y^{2}=0 do not exist

Quite possibly the hardest lecture, so far.

Like watching someone like Aristotle teach

This is maybe the best lecture in the entire course.

The funny thing is, I'm also doing a general relativity series and got to covectors. I understand what they are computationally, but couldn't "visualize" them, beyond the typical "stacks". Then I stumbled on the interpretation of covectors as linear maps, which served as a connection to linear algebra. I tried so many things to built a geometric interpretation all night but nothing formed in my head properly, so I was like, "meh, I'm 80% done with Professor Strangs lectures, might as well do another one". He led with x^H*A*x > 0 and for some reason, everything just started to click (x^H*A is an implicit map, so I have ideas about how to analyze things in my other GR class). Funny how his lecture was what I needed to get my head turning again. Thank you, and awesome lecture! ☺️🙌🏽 Stay safe during

If I had teachers like Gilbert Strang, I likely would've had a Ph.D in Math by now. No kidding! I love this guy! He made me fall in LOVE with Mathematics.

31:08

This lecture is just amazing, what a beautiful thing, all coming together...

fancy tie!!!!

Never Seen someone like this.... Amazing!!!!

From this lecture, I really understand Positive Definite Matrices and Minima thanks to Dr. Gilbert Strang. The examples really help me to fully comprehend this important subject.

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At 41:54, Strang gets the eigenvalues correct from memory. I'm impressed!