Inverse matrices, column space and null space | Chapter 7, Essence of linear algebra
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- เผยแพร่เมื่อ 14 ส.ค. 2016
- How to think about linear systems of equations geometrically.
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These should be STANDARD pre-course material for any linear algebra student ... Imagine going to those lectures having watched them first :)
In my Linear Algebra class we never discussed what the determinant, null space, or rank actually meant. We did discuss the Algebraic theory behind it very well but never touched the Geometric concepts. These videos are filling a lot of holes in my knowledge.
+Gustavo Merchan I am going to have this pleasure! And I can picture that I will recommend this videos to a bunch of people!
Can someone help me understand the intuition behind 5:33? Why does the vector x move when the basevectors change but vector v is just lying there untouched? Shouldn't the vector v also move to a different position? Why is only the vector x repositioning? I don't understand the logic behind it
+iFoxx16 the intuition behind the x=A^-1*v transformation is quite simple. at first you're given the Ax=v transformation, that is the transformation A that takes the x vector and *transforms* it to the v vector, so going by pure math when you multiply the equation by A's inverse, you get the corresponding value of the x vector (which in the case of a real life problem you'd be looking for) and diagrammatically the inverse function of A takes the v vector and transforms it into the x vector.
*So summarized A turns x into v so intuitively A inverse turns v into x*
(Sorry for the long post hope it helped :))
+Abdul Adeshina Ahh, thanks bro. The video misguided me. I thought the video showed the base vectors changing in a linear transformation. But the intuition behind linear transformation is that you multiply it with a transformation matrice and your vector can change it's direction and length, it isn't the base vectors changing, I get it now.
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
I sorta have that as well
Wow, can't believe you managed to put that into words
You’re right. Hahaha
That is exactly what math is at its core.
literally this.
God bless you even just for that Nullspace visualization.
you understood it? I got nothing!
@@123gregery Watch it again and again and again.. You guys really give up fast! I've watched this for like the 5th time and now everything seems natural since I practiced the computations too. The explanations are nearly perfect, and thee commenters help extend that.
For the first time I understood that, and makes sense in my mind. If you go by the teacher explanation you may never get to fully understand the concept.
I was mind blown by that Nullspace visualization, so that was what Nullspace meaning is
My problem with understanding was when he suddenly takes A*v (it has been A*x until then) and calls that column space.
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 know right. It’s such a weird sensation for some reason.
all the dots and crosses * :D
Leave me alone. I'm not crying because I'm sad, I'm crying because I get it.
I was in a series of shocks after each word he said, so I paused the video each time and felt like I wanna cry
Every word here is so correct
That's exactly what's happening to me right now
Yes，He says fantastic good
A strange feeling indeed
This whole series has been a collection of "holy sh*t", dispersed at different time stamps.
Why?
@@dineshdange5883 Because it's full of moments where you come to realize things that you possibly didn't understand during formal education. These videos focus on the _why_ , not the _how_ , if you get what I mean.
@@dineshdange5883 Cuz in every video when I learn something I feel like saying "holy sh*t"
The biggest holy shit moment for me on this vid was 10:34
I have holy shit-ed so many times troughout this series. I have spent numerous hours feeling retarded while watching the lectures provided by my school, only to grasp it all right now within the span of minutes.
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
SAME
the vibe that every one is feeling in the comments section is priceless
This is what happens when education is done right. Imagine if all teachers and professors taught this way. I think many students would actually start to enjoy school. More than halfway through my linear algebra course and I only _now_ finally understand what everything means thanks to this playlist. It really is just mind-blowing. Before, I was just memorizing a bunch of steps to find an answer that had no meaning to me. Like, great, I found the determinant, not sure wtf that is but here's my answer for it, now give me my grade.
I'm already sad that this series is finite. ):
Ya... but infinite videos would be... really, really short. I mean, Aleph Null length videos would be nothing compared to w^2 long videos. ( ͡° ͜ʖ ͡°)
+Baumgarten Ralph Quite Exactly. 😅
vsauce? ( ͡° ͜ʖ ͡°)
linear algebra stil researched and infinite
Go see the series of Maththebeautiful's channel to complete your knowledge of linear algebra.
Can you please do a similar course on probability? Love these!
YES PLEASE!
thekillermuffin yes!
Please
It would be wonderful if you could do it.
You mean... statistics, right? :p
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.
words can't describe how mind-clearing this series is
I eat popcorn when I watch these...It's THAT entertaining...
Same XD It was a relaxing lesson!
Jajaj
Same here
guess you were eating kernels while learning about kernels...I'll show myself out
I just take out a notebook and start making notes lol
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!
Math still feels like magic but the mathematicians feel like wizards
I dont like the way you are using 'magic' in a negative sense. The intuitive part is what is so magical about math
@@assholable he just means something unknown dude
@@rowdysrohan5641 yeah but you gotta admit, "math shouldn't feel like magic" sounds bad even if you force yourself to not hear it that way, like something an evil, antagonistic cartoon show maths teacher would say
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 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
@@downsonjerome7905 Congratulations!
The other Fred
And other resource(books or lectures) equivalent to these brilliant lectures to learn this stuff?
Well said, @ffggddss
This is one of the most enlightening video series i've ever seen.
To understand non full rank transformations, I like to think about it in the language that: "It loses information".
Would be interesting to see your coverage of tensors as well by the way. Your videos are great fun :)
Since non full rank matrices are not invertible, it does seem to lose information after the transformation.
I just want you to know that Dr. Susskind was insulted by this comment.
This is exactly how I think about it
Dimensions, to be more precise.
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!
This is exactly the problem I also have with the lectures. I quickly lost the understanding how all the things belong together. I‘s a pity I only see those videos now.
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.
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.
+fossfighters101 "intuit" is a nice word for "intuitionalize".
I prefer "intuinternalize"
Thank you.
:D
+goohz essence and meaning... you mean the intuition? That is, the intuitionalization?
"... 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. :'(
That is correct, you need to know how to calculate them to rellay learn the meaning. Of course in real life you will not calculate a simple 3x3 matrix and you'll use a calculator, but you still need to understand what the calculator or the software is doing. That's why in all algebra and calculus exam calculators are forbidden but the exam itself only require very simple calculations, if you know the method of course
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.
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!
I really wish that I had watched this before I learned Linear Algebra
i haven't taken linear algebra yet and im watching and understanding them just fine. 3b1b did a great job
I'm watching this in *this number has been removed for security reasons*th grade, and I am loving it!
@TheBlueRaven This. I agree that it's better to have a rough idea of the concept before getting into those videos...
I watch Essence of Calc before Calc BC and I'm not doing so well.
Maybe I should've went Essence of Calc -> Khan Academy's Calc BC -> Actual class, where a grade matters
@@NoorquackerInd If you really want to learn math, you must do problems. Learning math and physics is not the same as knowledge based subjects like biology. You must build problem solving skills and a working knowledge. That's why you can understand this stuff intuitively and still fail your exams. Math tests are tests of skill not knowledge. I recommend Professor Leonard's TH-cam lectures and solving problems in a standard textbook or pausing his videos and solving along with him to really do well in your classes.
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.
wait till you find out that 1 = 0.99999... or the infinity of the size of the reals is bigger than the size of the integers
@@rickroller1566 That's quite intuitive, it's just not trivial intuition.
I'm sure those things were obvious for great mathematicians of the past.
Intuition is way more powerfull than analysis.
@@vazn4143 I always thought of it as intuition solves problems more creatively and can solve problems you don't already know better than rigorous analysis, but is prone to error. The best approach I've found is to search for an intuitive solution to a problem and then back it up with rigorous symbolic logic to make sure my intuition makes sense. That said I'm a physics and engineering student so maybe pure math runs on different rules.
@@taylorward7576 I think from my experience i could say there's two types of math person.
There's the guy that so much into the formal "side" of math that he can just juggle with the symbols and obtain proofs without even asking himselfwhat is the meaning of the things he juggle with.
And there's people like me, who rely on the meaning of things and symbols to solve problems. It's doesn't mean that you lose abstraction because your symbols could have an abstract meaning in whatever mathematical structure you're working in. But its necessary to me. And then my intuition arise from the meaning of those ideas.
It could be geometric intuition, algebraic or anything.
But i would not be able to do any math at all (and i would not like math anyways) it i couldn't develop this intuition so yeah its pretty important in my opinion
@@rickroller1566 Thats pretty intuitive, like 0.999999... is trying to be as close to 1 as possible, may as well be equal to 1.
Same goes for infinity the size of reals being bigger, since we know there are infinitely many reals between any two integers, you would intuitively think there are more reals.
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.
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 you. Luckily, my calculus professor often provides geometric intuition for the lessons he teaches. That is what most professors miss in my opinion, they focus on us remembering the equations as opposed to having an intuitive understanding of what the equations mean, and their implications for other concepts in math. For example, I understood the concept of the derivative mostly through what it meant when applied in physics. This applied concept helped me understand the fundamental math behind it even more. Every class becomes a lot more beneficial when you focus less on remembering equations and focus more on solving problems.
I think I spotted an error. At 8:56 where you define Column Space... shouldn't it be ... Set of all possible outputs Ax?
Oh! Good catch. I guess technically it doesn't matter, since for that sentence there's no reason the symbol v can't stand for the input. But certainly for clarity's sake it would have been better to have used an x.
@@3blue1brown This comment really helped me! I was really confused about the whole idea of "output for your matrix" until I saw this, even though I've watched the rest of the series up until this point.
So basically, 8:59 :)
@@WeirdAlSuperFan same here too ..i was als9 confused and hence went to check the comments..finally satisfied
phew. thanks for the catch. i was kinda confused as well! thank you grant for this wonderful video!
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.
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.
I'll definitely support you on Patreon after getting a job!
How was your GATE result?
@@dddiyamirza I'm going to give my 1st attempt in 2021
vamshi goud Oh great! I am also preparing for 2021!
@@dddiyamirza cool!
@@dddiyamirza branch?
i feel like i have opened my third eye
Eye of agamotto!!
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
Thanks!
how did your course went?
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!
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
Rank-nullity theorem:
Rank(A) + Nullity(A) = dim(A)
That don't make a damn bit of sense
@@High_Priest_Jonko It sure does! Observe that the rank of a matrix A is the number of dimensions in the output of the matrix transformation and the nullity (dimension of the null space) is simply the number of dimension in whatever span of vectors that gets 'squished' to the origin. If we look at a matrix transformation that squishes 3D space into 2D space, we observe that a whole line (lines are 1 dimensional) of vectors gets 'squished' to the origin which in turn means that the nullity is 1. The rank of the transformation, we defined to be the number of dimensions in the output, which we know to be 2D. Now we see that rank(A)+nullity(A)=2+1=3 which should give us the dimension of our matrix A. Well, because we looked at a matrix that squishes 3D space this has to be true!
Preben That was beautifully articulated, my friend. Well done!
1
@@preben3453 beautifully explained! Thanks
I want to hug this guy he is a gift from god
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!)
holy. shit. this was whats 'inverse' was all about
haha i feel you bro
It's unbelievable how these animations make it so intuitive and easy to understand while dry text from a book has the opposite effect.
It's saturday night my girlfriend go out with her friends while I'm watching these videos that I LOVE!!!
Full rank winning.
Right choice
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.
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 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.
All of these videos from the series just keeps blowing my mind.
Take notes professors, this is how teaching is done.
pls make the whole measure theory or topology. that would be of great value since there are few lectures about those topics online
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
By the way, do you have a patreon account? I think you are doing a great job and this job definitely has to be rewarded, animations are beautiful, the quality of the sound is nice, and overall impression after watching your videos is great
Thanks for the offer! These videos come to you courtesy Khan Academy.
Compared to ALL the other math teachers and professors I have had over the years, you NAILED it.
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!
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!
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
10:08 This is an awesome illustration of the Rank's formula! Thank you, you made me understand a lot.
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!
I've finally understood it.
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.
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.
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.
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..
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 :(
me too, I get even more hyped now that they are published every week rather than everyday
That or his videos are that good!
These videos are pretty cool, they get to touch the core of mathematics and life. I don't think one such video being the highlight of your day necessarily means you live a sad life.
The only thing sad about your life is that you think that's a bad thing
Aeroscience but dont lie, seeing in your subrcriptions a video of this channel really hypes you up
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
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 would nail my linear algebra exam if I could find this series earlier
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.
This series is pure gold. I can't get enough of it!
I honestly cannot comprehend the artistry of your teaching ❤️ it's absolutely flawless and it inspires me beyond explicable words...
I am reborn after watching this playlist.
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.
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'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.
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.
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!
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.
This series makes math seem beautiful and exciting and I am here for it
"this isn't meant to teach you anything"
proceeds to explain everything i didn't understand
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.
Wow... From this video I got some intuition of what is an eigenvector. All I knew is this is a thing and you have to calculate it. Damn this visualization is good. I was even able to formulate a problem, "what should be the eigenvector of rotation matrix?" Never have I ever thought I could get an intuition of eigenvectors let alone create a problem! Thank you very much.
Wow what a brilliant way to explain the inverse and identity matrix! This world needs more teachers like him.
this is oddly relaxing. like i could see my mom enjoying this
Waiting for the next one with bated breath!
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.
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.
Holly shit, after 2 years in math engineering you blew my mind at 2:47, now it all makes sense!
@Pedro Takahashi Gunderson apparently something that doesn't help you learn linear algebra in any reasonable time scale.
you re a god man. this series is so great.
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 :)
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 was sold , the moment you said "some squishes are squishier"
I hope you see this comment , sir! I love your videos, they've literally changed my view on Linear Algebra, and math in general. It would be awesome if the quality of these videos was 60fps, for more crisp visuals :D
Best tutorials ever. Most practically explained linear
algebra videos . Can't find such material anywhere.
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.
11:18 What do you mean by "the idea of a null space helps us to understand what the set of all possible solutions look like"? I can understand that that's the case where the v vector is the zero vector, but otherwise, how does the null space help?
+ViolaBuddy Remember: linear transformations preserve linear combinations. So, let's say that x is a solution to Ax = v, and let's say that u is any element of the nullspace. Then
A(x + u) = Ax + Au = v + 0 = v
On the other hand, if x and y are two solutions so that Ax = Ay = v, then
Ay - Ax = A(y - x) = 0
which means that y-x is some element of the nullspace, which is to say that y = x + u for some u in the nullspace.
The point is this: there is an exact correspondence between the solutions to Ax = 0 (the nullspace) and the solutions to Ax = v. If x is one solution to Ax = v, then every other solution to Ax = v can be written in the form x + u for some u in the nullspace. Geometrically, the solution to Ax = v is the nullspace *shifted over* by x (which gives us a space "parallel" to the nullspace).
Oh, that makes so much sense, actually! Thanks! This probably leads into orthogonality and the dot product that he's going to talk about next time, since it seems that the null space is necessarily perpendicular to the column space (and geometric perpendicularity corresponds with linear algebra orthogonality).
+ViolaBuddy Your hypothesis that the nullspace is necessarily perpendicular to the column space is interesting in that it is *generally false*, but true for a very important subset of cases. For example, if you take the matrix
0..1
0...0
You'll find that the column space and nullspace not only fail to be perpendicular, but are in fact exactly the same! It is true, however, that if A is a *symmetric* matrix, then the column space and null space will be perpendicular. More generally, it is the *row space* of A which is necessarily orthogonal to its column space, which I expect 3blue1brown will touch on when he gets to dot-products.
Geometrically, symmetric matrices (unlike general linear transformations) correspond to stretches, squishes, and flips along perpendicular axes.
bengski68
Ah wait, I see where my mental image breaks down. Gah, the accursed human intuition that never seems to match up with actual math! It seems that null space is more about the input vectors x whereas column space is more about the output vectors v, and so there's of course no relationship between the two in the general case (please correct me again if I'm wrong!). Though saying it that way, that should've been obvious by definition: null space is the collection of all x's such that Ax = 0 for a given A, whereas column space is the collection of all v's such that there exists an x such that Ax = v.
...I think I'll have to stare at examples a bit more to fully grasp this, especially the part about row space and symmetric matrices - or I may just wait until (hopefully) 3blue1brown covers it. In any case, thanks again.
No problem!
You have a lot of the right ideas; your guesses about input vs output vectors (which are mostly correct) become a lot clearer when you start thinking about non-square matrices. In general, the column space lives in the "output space" which is of one dimension (i.e. number of dimensions), and the null space lives in the "input space", which might be of a different dimension. The row space, on the other hand, is there in the "input space" along with the null space.
My professor started the lecture by showing this video.
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 really appreciate and love how he emphasizes more on INTUITION part than COMPUTATION...!!! Thanks!!
By the fourth minute of this video you implicitly explained in just *one sentence*, why a linear system of equations A*x=b with n equations has exactly one solution if Rank(A)=Rank([A b])=n, multiple solutions if Rank(A)=Rank([A b])
Notes for my future revision.
8:55
.Column of matrix
= where basis vector lands.
.A column space of a matrix
= All possible output of matrix of all columns
= The space (across the dimensions) the transformed basis vectors.
It's not where the basis vectors landed, but the resulting *dimension*.
In short,
Column space of a matrix
= Column span
= Span of (all) columns
= Span of transformed basis vectors
.Span of transformed basis vectors (can be lower dimension or same dimensions as the original vector basis)
= The space with all column spaces included
= All possible output (from all basis vectors) after the matrix transformation
= The space (within the new dimension) where the vectors could possibly be transformed into
---
Rank (of a matrix)
= Number of dimensions of the column space
= Number of dimensions after basis vectors being transformed by the matrix
Full rank matrix
= When the rank is as high as it can be
= When the rank equals to number of dimensions
= When the rank equals to the number of columns
Not full rank matrix
= Matrix that squishes input to a lower dimension
---
09:38
Zero vector
= [0, 0]
= Origin
= Always included in column space
= Always present after any transformation
In a full rank transformation, the original span is maintained, so the only vector that lands at the origin is [0, 0].
For a non-full rank transformation, many vectors collapse to the origin.
For example, when a 2D span (a plane) became 1D span (a line), there is a line in the 2D span that collapses to [0,0].
---
Null space
= Kernel
= The space (of set of vectors) that collapse into the origin/null/zero vector
---
10:47
When the v happens to be the zero vector, the null space gives all the possible solution to the equation.
Ax = v
If v = zero vector, the values of x will be within the null space.
---
In a system of equation, ax + by + cz = ... or Ax=v, the solution refers to the values of x,y,z or vector x.
Column space helps to understand when a solution even exists (i.e. when the resulting vector v lies within column space of matrix A).
Null space helps to understand what the set of possible solution can look like.
Thanks 🤍
You are a beautiful human being
the hero we need but didn't deserve
thanks !
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.
I'm really glad that you've been posting such a great course about liner algebra. Thank you!