New linear algebra video! Cramer's rule (plus some bonus material)

Thanks! If only I could edit videos in place...

3blue1brown

Very many thanks for this excellent series, looking forward to more! Just one issue that must have tripped up others as well: in the eigenvector question at the end of chapter 13, the x and y coordinates of v1 and v2 are the wrong way round. They're correct in the exercise writeup. BTW: well-chosen exercise, with that beautiful discovery at the end!

Thanks for the video! What about covering covariant and contravariant representation?

Dude I’m so happy with the direction this channel is going.

They should just be scaled versions, are they not? Maybe I forgot to mention that in the writeup.

3blue1brown

I'm glad you liked it, and thanks for the feedback! I'll take that into account.

3blue1brown

Hi Grant, Thanks so much for all this material. One question about the eigenvectors exercise writeup. The Eigenvectors in the writeup don't seem to match the two given at the end of the Eigenvectors video. Am I missing something obvious?

Hello, Grant! You've made a wonderful video yet again. Thank you for the effort you and, now, your team are making to inspire the world's learners. As for feedback, you could make the red text that fills the screen (as in "Not the most computationally efficient") a little smaller or perhaps shrink into a box like the equations. Other than that, which is inconsequential, nothing else stands out. Particularly on the scale of the lesson in its entirety, your design sense remains impeccable. Thank you.

Debbie Newhouse

Excellent video! I have my Linear Algebra Final in two days so this was nice timing. At 1:47 when you use the coefficients in the linear system to make the matrix the 1 turns to a -1. Not sure if that's a mistake, maybe I'm just missing something. Anyway thanks for the hard-work and the deeper understanding.

Great point!

3blue1brown

I think that it's very likely we'll cover this at some point, either on the physics side, the information series side, or on the connection between the two.

3blue1brown

Hey! I know it's a little bit out of the scope of what your videos usually cover. But do you think you could eventually do a video explaining entropy? It's something I've been trying to understand recently and I'm having a hard time understanding it intuitively. Thanks so much for all you and your team do!

When you talk about Ortho Normal (ON) transformations you only mention (and animate) rotations. You should point out that reflections are ON transformations too, otherwise people might think that rotations are the only ON transformations. You could even mention something about how rotation preserves orientation while reflections do not, and how this affects the sign of the determinant.

Maybe subconsciously, but I can't say any such consistency has been deliberate.

3blue1brown

I like the pi creatures expressions at 3:22. Do you give each one a consistent personality across the videos?

Gabe

I use Cramer's Rule all the time at work to quickly solve 2x2 systems, but I never really understood why it works. Thank you for this very clear explanation!

Vincent Zalzal

Thanks, Grant. I'll write my math finals tomorrow so I hope I can use this there :) Good that I have early access.

Good note at 1:00, thanks. Can you elaborate on what you mean with the second suggestion? Was the volume too low, or do you mean something less literal?

3blue1brown

Keen eye, thanks!

3blue1brown

I'm so glad *you* exist.

3blue1brown

Grant, so much thanks. I'm in my first linear algebra class right now and went through the series to prep, so the fact that you're adding on to the series now is an unexpected stroke of good luck!

Alex Loftus

This was great! Thank you so much for this addition. I don't recall learning about Cramer's rule last year, but our professors seemed very keen on Gaussian elimination, so maybe that's why. In any case, giving us any intuition for what was going on geometrically was never their priority.

A ridiculously simple derivation! For the past 7 years, I have learnt Cramer's rule several times; but never the derivation. Each time I tried to think up a proof; but it couldn't come to me. This video is so satisfying to me @1.00, when you say Cramer's is not efficient, those two lines weren't so clear and were mixing with the background graphics Also, the part where you reveal that you've derived Cramer's rule was too silent for me.

Magnasium

Great stuff as usual, Grant. There may be a phantom minus sign at ~11:07. Not really important, but thought you might like to know. Keep up the fantastic work!

Thanks Grant!

I am so glad you exist.

Jonathan Fuzaro Alencar