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

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

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

God bless you even just for that Nullspace visualization.

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

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

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

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

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

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

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

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

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

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

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!

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.

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

This is one of the most enlightening video series i've ever seen.

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.

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!

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.

To understand non full rank transformations, I like to think about it in the language that: "It loses information".

I remember in college as a math major having the intuition squished out of me by the professors bent on formal rigor. I was never any good at abstract math after that. I am happy to see that intuition is making a comeback. I think that all breakthrough mathematics is a result of intuition. I really like what this guy is doing for the intuitives.

I will die a happy man if you make a similar set of videos for Probability and Statistics. Having watched these videos, I will never be able to look at Matrices the same old-boring-cryptic way. Tight Hugs!

Weeping, literally weeping over how familiar concepts finally make sense. I'm so glad I discovered you in my first year of engineering. Thank you so much.

"... Plus, in practice, we usually get software to compute this stuff anyway." Not for me, my professor(s) have essentially outlawed the use of calculators. :'(

I really wish that I had watched this before I learned Linear Algebra

This guy is epic. A great indication of how well someone understands a particular topic is how elegant and/or imaginative their explanations are. I've always found a lot of math a bit hard work. Not hard work in that I couldn't get through it, but hard work in that I always felt I was remembering arbitrary systems for solving arbitrary problems. This guy truly brings math to life and inspires me to learn more. Absolutely nobody, ever, not in anything I've ever read/heard/watched has managed that. Amazing.

7:55 And all the cases for Cramer's rule for the case when the determinant is zero suddenly makes sense. You're beyond brilliant! Wish I saw these videos before.

My comment will be another in a sea of many, but I wanted to express my gratitude for your existence. Your goal of giving maths a more intuitive approach really helps people like me who don't like to accept rules and formulas as they are and just use them. Please keep the good work up, it is invaluable.

Our maths professor explained everything mentioned in this playlist almost only numerically. Watching this got me more mind blows than ever before in such a short amount of time! PS: You truly saved my semester, dear sir

i feel like i have opened my third eye

I'll definitely support you on Patreon after getting a job!

Hi. Thank you. A) thank you for your honesty stating that you are focused on the intuition and not the calculation. B) thank you for actually focusing on the intuition rather than the calculation. You make linear algebra so much easier to understand. C) as a side note - thanks for the awesome graphics.

I absolutely love this. I learned Linear Algebra in a very good setting, but I never quite understood the linear transformation side of things until this series. this is one of the best series on youtube. bravo.

This series is incredible - I use linear algebra every day but the deep level of intuition I'm gaining is astounding (and I'm only half way through!)

I am in my first year of bachelor in physics and astrophysics and this playlist helps me so much! It kind of reconnects me with reality and what everything actually means instead of what we see at uni. There we just see all the math part, computing and the (seemingly very important) proofs. Grant, I think you're doing an amazing job of explaining quite complex math while keeping your and our feet on the ground! I really love your way of explaining things. Thank you

When I got a notification for this at work, I immediately went to the break area, popped in headphones, and watched it. You're doing a great thing, my friend

I took linear algebra 2.5 years ago and ever since I always wondered what is the meaning of a determinant and this series blew my mind. I wish they'd mentioned your channel to new students. This is pure gold right there!

I wish these videos were around when I was in school. I'm going back for a master's and these are the most informative refresher I could have asked for. Knowing the intuition behind the concepts, I remember the equations from school without issue.

With the help of this great video and a quick review of my old linear algebra textbook, I was finally able to understand many things intuitively, for example why the dimension of any vector space is the sum of the dimensions of the kernel and the image of any linear transformation whose domain is that vector space, and what the transformation does such that it holds. It's just amazing. Thank you, 3Blue1Brown!

I think I spotted an error. At 8:56 where you define Column Space... shouldn't it be ... Set of all possible outputs Ax?

It's unbelievable how these animations make it so intuitive and easy to understand while dry text from a book has the opposite effect.

Dude, you haven't just given me solid insight and understanding of these concepts but also injected a desire for me to practice math. Thank you!

I watched these videos almost 2 years ago in absolute desparation of understanding linear algebra and it already was an eye-opener back then. But coming back to rewatch this after having taken other math classes it finally feels like it's all clicking together. What you're teaching with these animations could never be accomplished on some lecture slides. Thank you!

holy. shit. this was whats 'inverse' was all about

Wow! Simply superb! I was used to sit endless time trying to grasp in depth a few concepts like these. Now, barely 12 minutes of watching a video and everything seems soooo clear and sooo easy, and in such an amusing way, that it feels almost like cheating! Thank you very much for all these interesting videos!

It's amazing how many properties of rank, nullity, and column space instantly began popping into my head watching this. In my abstract linear algebra class we were so focused on pumping out proofs about n-dimensional vector spaces that I never really got these. It makes sense that nullity and rank add up the way they do

I took linear algebra in college (it wasn’t even required for my engineering degree, I just wanted to learn more about it) and despite being taught how to CALCULATE all of these values, I had zero intuition for what they were. I didn’t even know that the kernel was the set of vectors that get mapped onto the zero vector by a transformation. They just told us HOW to calculate the kernel, not what it was. Same with the determinant. I finally realize why a matrix doesn’t have an inverse when the determinant is zero, because you lose information about where those original vectors were when you lose rank, because it becomes no longer possible or to reverse the transformation. I passed that class with an A and had no idea of any of this intuition, only how to get the answers I needed for the exams.

Rank-nullity theorem: Rank(A) + Nullity(A) = dim(A)

I just want you to know I feel a sense of joy watching these videos I really can't describe especially after how professors teach us. Your videos are the highlight of my day

I love this channel! I can't even describe how enlightened I feel now. I've been totally lost in math classes during my master's and PhD. Now I feel I can actually understand the concepts I've been trying to use all those years.

10:08 This is an awesome illustration of the Rank's formula! Thank you, you made me understand a lot.

I find my self thanking you every few videos.. You help enlighten us with the very missing concepts that no lecturer of ours (university), as it seems, dares to delve in to while constantly and rapidly firing at us computational babble! I was laughing out of hopeless frustration during the lecture today (I would cry if else) and your unique time invested videos really contribute in restoring emotional balance in upset students like myself..

Whoever you are, you're a godsend. Intuition and motivation are simply indispensable to appreciating analytical expressions, and you are unparalleled in providing both through these productions.

Compared to ALL the other math teachers and professors I have had over the years, you NAILED it.

All of these videos from the series just keeps blowing my mind.

It's saturday night my girlfriend go out with her friends while I'm watching these videos that I LOVE!!!

pls make the whole measure theory or topology. that would be of great value since there are few lectures about those topics online

I've just discovered your channel, and I appreciate this series! In my university class, I am learning how to do linear algebra computations and memorizing all of the processes to derive answers. However, your videos help me understand what actually occurs with those computations. Understanding the reasoning for linear algebra will help me as a math teacher much more than memorizing a series of formulas/processes.

Thank you very much for this series! I was able to solve questions of linear algebra before but never really understood the meaning behind it. You're a gift to education!

I saw those images only in my imagination, I am very glad that now I can see them as a real-life video and I will definitely recommend this set of videos to my friends who don't understand algebra

I want to hug this guy he is a gift from god

These lectures are awesome, thank you so so much! In uni I never really grasped the visualization of the calculations we were doing and what they acutally mean. We only spoke about definitions and theorems of these defintions. Now these videos just make it click into place and even inspire!

I can't express enough gratitude towards you. We have been taught matrices since high school but I never actually understood the intuition behind them or their properties. Thank you for creating such content which actually helps us visualize them. As all of them are animations, I understand that they must be very tough to create. Nevertheless, thank you again for such an amazing job. May God bless you.

great job. I in my late 40s and wanted to learn linear algebra for my thesis. And nothing could have been better than these series of videos to start with. Gob bless you.

I honestly cannot comprehend the artistry of your teaching ❤️ it's absolutely flawless and it inspires me beyond explicable words...

Unlike other videos who just explain how everything is as it is, you really dive deep into why it is right. You are the reason why I finally understand all of these concepts, by just watching the video once.

It's really incredible ... during my studies of linear algebra in Computer Science degree - almost 20 years ago - no one explained in this way and everything was very abstract and to be memorized without understanding what it was !! Compliments, I'll follow also your Calculus set for Analysis

You have added a great deal of value for me. And from the looks of it, for a lot of other people as well. Thank you so much.

In a time series econometrics class right now. These videos have helped immensely. I will end up with an A in the class. Thank you so much. Beautiful.

You nailed it! Couldn't imagine math could be this easy , when you explain things my mind has no choice other than understanding it, thanks for the great explanation and visualization you brought to this world. I really mean it , and say it from the bottom of my heart.

I'm really glad that you've been posting such a great course about liner algebra. Thank you!

i love you, seriously this is amazing i mean you revealed the simplicity and the beauty of linear algebra. When i studied it i was sure it was more logical but the book was extremely (lame) and now i am wandering in the internet to get it the RIGHT way like here.

Take notes professors, this is how teaching is done.

This series is pure gold. I can't get enough of it!

The Nullspace visualization just eluminated so much, it linked to the topic of Eigen values and gave me the biggest "Ohhh" moment. God bless you.

Holy sh*t! I'm finally understanding it! I have been trying to self-study linear algebra for about 3 days now, and nothing was really making sense intuitively, till I watched these videos. Thank you so much!

I am reborn after watching this playlist.

I got a solid A in my Linear Algebra course, but I have taken away more intuition about the subject in these videos than I did in that course. It isn't a slight at my old instructor either, you are legitimately an amazing instructor. Thank you so much for giving us these gifts.

Best tutorials ever. Most practically explained linear algebra videos . Can't find such material anywhere.

I've finally understood it.

For some reason, I was so excited for this video to come out. This was the highlight of my day. I live a sad life :(

These series of videos are brilliant. From the moment each video starts till the end it invites all to think and understand. You're great.

These videos are so amazing because they connect all the dots and show us the big picture of a subject that we are learning. Thanks!

I would nail my linear algebra exam if I could find this series earlier

I know that you have a lot of videos planned after the conclusion of this invaluable series, and I'm indescribably excited for whatever it is that you decide to cover - but I personally feel wrapped up in, even lost in the world of Cantor; I feel as if I'm skirting the edges of a real intuition for the realm, and something tells me that a video or even series of videos by you on the world of infinite ordinals, inaccessible cardinals and other awe-inspiring entities that inhabit the endless would help me scale the wall. In any case, thank you immensely for all your effort and your tremendous efficacy. You've helped myself and surely many others break through more completely into the wider world of mathematics. Time to start counting down the days until the next one!

This series makes math seem beautiful and exciting and I am here for it

Wow what a brilliant way to explain the inverse and identity matrix! This world needs more teachers like him.

Waiting for the next one with bated breath!

this is oddly relaxing. like i could see my mom enjoying this

The nullspace visualization BLEW MY MIND BRO. I was SOOOOOO lost. I literally feel like my mind has been transformed to R^n

I wish that this visualization was presented to all college students when they learned linear algebra. Back in 2003, 2004 I did this in my university without learning the essence of this work. Thank you so much for producing these videos. I wish that I can watch a video about geodesic as well. I was so determined to learn about it back then. Now I have a hope that there is someone out here to give visual interpretation to it.

I was sold , the moment you said "some squishes are squishier"

My professor started the lecture by showing this video.

I really appreciate and love how he emphasizes more on INTUITION part than COMPUTATION...!!! Thanks!!

This series paired with professor Strang's lecture series is the single greatest resource for mastering basic linear algebra.. no doubt about it

Holly shit, after 2 years in math engineering you blew my mind at 2:47, now it all makes sense!

you re a god man. this series is so great.

IT ALL MAKES SENSE NOW !!!! Finally, the essence. I'd been doing the calculations in my class, and I'm good at it - but I can't tell what is happening - things made sense when I tried to map every linear algebra thing I learned onto Cartesian Plan - Coordinate system. Which stopped making sense after a while. But, IT ALL MAKES SENSE NOW !!!! Thanks a tonnnnnn.

I had an epiphany at 4:35 watching the numerical representation of the rightward and leftward sheer (which numbers in the matrix were multiplied by -1) which finally gave me an understanding of why the identity matrix is a diagonal of 1s from top left to bottom right, which as a kid learning random facts about higher math with no context seemed so bizarre. The n by n identity matrix is a representation of the basis vectors in n-dimensional space that have had no transformation applied to them, and as a result are their own inverse. Being its own inverse then uniquely defines it as the identity matrix. Being able to come to this conclusion on my own feels intuitive and invaluable. (I paused the video to type all this up) I kept watching before posting this comment because I figured it would be your next point. My linear algebra class starts tomorrow. Thank you Grant for these incredible and intuitive explanations and animations :)