Inverse matrices, column space and null space | Chapter 7, Essence of linear algebra

How to think about linear systems of equations geometrically.
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Future series like this are funded by the community, through Patreon, where supporters get early access as the series is being produced.

Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld

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3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted about new videos, subscribe, and click the bell to receive notifications (if you're into that).

Various social media stuffs:

[GustavoMerchan]

These should be STANDARD pre-course material for any linear algebra student ... Imagine going to those lectures having watched them first :)

[ChumX]

These videos have the opposite effect on me than a regular lecture, they put me in a relaxed meditative state, concentrating on deep intuitions instead of a hectic state lost in calculations

[xat]

God bless you even just for that Nullspace visualization.

[austinsimpson]

Holy crap dude, you've just synchronized an entire semester of lukewarm understanding into one single video that connects all the dots, absolutely wild.

[nirmalaliyanaarachchi]

This whole series has been a collection of "holy sh*t", dispersed at different time stamps.

[markoruzic]

I have to stop every 5 minutes just so i can pull myself together, this is so mind blowing that they are not showing this in all classes to all the students

[playitothebeat]

the vibe that every one is feeling in the comments section is priceless

[doctoridk]

words can't describe how mind-clearing this series is

[mazisilas]

Leave me alone. I'm not crying because I'm sad, I'm crying because I get it.

[thekillermuffin]

Can you please do a similar course on probability? Love these!

[RunstarHomer]

There will never be a channel that tops 3Blue1Brown. After months of studying, this video made all the difference in the world in 12 minutes.

[disappointedbutnotsurprised]

This is mind blowing. I can't believe a lecture on maths (something I've been running from all my life) of all things has managed to seize my attention so much that my dopamine addicted brain is actually "studying" instead of watching shorts or anime.

[CoasterMagicX]

In high school math, we recently learned this way to use matrices to calculate linear equations. But, without the intuition that you teach here and *WITHOUT LEARNING WHAT MATRICES ARE*. It felt like magic how we plugged in numbers into our calculators and watched it give us the answers. Math should never feel like magic. Thanks to you, I now know where it comes from, and it doesn't feel like magic anymore. Keep making these videos!

[Titurel]

I eat popcorn when I watch these...It's THAT entertaining...

[ffggddss]

I must tell you that I'm very impressed with your treatment of lin-alg in this series of videos. You've packed a solid chunk of a good LA course into a handful of 12-min videos, at a seemingly leisurely pace, with loads of intuition-provoking effects that help make the key concepts stick. It reinforces for me, the reason I subscribed to your channel.

I'd also like to add – around the 1-minute mark in this chapter, where you're talking about the importance of matrices – a suggestion of another very strong reason:
• the way they allow characterizing continuous, non-linear transformations as being locally approximated by linear ones.

This really is something that makes them universally useful. It takes us into the concept of tangent spaces, and, ultimately, tensor calculus, which is the natural language of general relativity and other applications of curved manifolds, by relating nearby tangent spaces. Of course, this more full-blown explanation is well beyond the level of your series here, but I'm confident that you could drop the hints in a most understandable way about this, without going into unnecessary detail...Or maybe in a followup series that delves deeper.

Meanwhile, I'm sitting back, enjoying the show! [Now where's that microwave popcorn...?]

Fred

[kjekelle]

0:00 intro
0:28 what about computations?
0:51 system of equations
2:38 visual
4:15 inverse and identity transformations
5:39 link with the determinant
8:02 “rank”
8:54 "column space"
9:38 "null space"
10:56 in sum

[Cerealbox]

I'm already sad that this series is finite. ):

[fossilfighters]

3Blue1Brown, as a student in pre-calc right now, I find these a perfect accompaniment, both helping me intiutionalize (I'm making it a word) vectors and matrices and adding on extra, more complex things as icing on the cake.
Thank you, sir.

[julijanailic]

I am probably one of the lucky students that discovered this series during my linear algebra 1 course, and I have to say thank you from the bottom of my heart. Not only have you awaken my passion for math through your other videos and make me want to study it at university but you have also helped me intuitively understand these key concepts. Often times professors give us definition that don't make sense in your head until you see an exemple. And you series is exactly that illustration I need to fully understand. Keep up the amazing content Grant!

[cutcc]

Very interesting how systems of equations is in the 7th episode of this series whereas in school, it's the first thing you learn. Really shows the difference between institutionalized learning and conceptual/intuitive learning.