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.
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
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 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.
@@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 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
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!
@@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
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 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.
@@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.
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.
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.
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
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.
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 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.
@@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.
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.
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.
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.
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.
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.
"... 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
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 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
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!
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.
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.
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!
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 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.
6:15 as long as det != 0, there is a A-1; 6:43 when det == 0, there is no inverse (A-1); 7:42 ~ 8:01; 8:22 Rank 1, 2 & 3; 8:54 what's called "the Column space of a matrix"; 10:33 ～ 10:45 null space; 10:45 ~ 10:55 null space usage; 11:12 ~ 11:23 column space & null space's high level usage;
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 :)
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 RUclips lectures and solving problems in a standard textbook or pausing his videos and solving along with him to really do well in your classes.
@@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!
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!
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 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 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 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
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
The part at 10:33 made so many things make sense to me. I'm currently studying for my LA final, and this is the first time I've been able to properly visualize so many of these concepts. Thank you Grant.
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
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.
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.
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!
0:28 what about computations? 0:51 systems of equations 1:21 "linear system of equations" 2:04 into a single vector equation 2:38 visual 3:14 simple examples 4:15 inverse transformation 4:59 the identity transformation 5:39 considering the determinants 7:26 special case with det(A) = 0 7:43 "rank" (squishes) 8:54 "column space" 9:38 "null space" 10:56 in sum
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.
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.
I watched this course a long time ago (about 6 years ago) when I started my undergraduate studies, and now I am a PhD candidate, and I am reviewing some of your videos. I like how you geometrically explain why a matrix isn't invertible when the determinant is 0, as it's more than just seeing what happens in the formula for the inverse of a 2x2 and/or 3x3 matrix. since you get a 1/0 occurrence. Excellent video as an introduction to the world of invertibility. Concise, clear, and cohesive, and the course as a whole is perfect as a jumpstart for a journey into linear algebra.
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!
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!
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])
There's no words for how thankful I am. Not even for the algebra, but for finally feeling like I'm good enough. So many years as a child and adolescent being made to feel like I was of average intelligence because I couldn't get math or science. I now realize that math is easy, it's the schools and teachers who make it hard to understand.
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.
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'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 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
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
I want to say though, that there is a good intuitive understanding of how gaussian elimination works. The standard way this is taught, when you have Ax = b, you stick A and b together into an augmented matrix, and perform row operations on that until you get to RREF. Knowing how multiplication works make it pretty obvious why you get the solution to the system from that form, but it's not entirely clear what it means intuitively to stick A and b together and then do row operations on that matrix. To understand this, you need to realize that row operations can actually be represented as linear transformations i.e. they can be represented as matrix multiplication. So, if you perform some sequence of row operations to get to RREF, that sequence can be represented as a single linear transformation, which can then be represented as a single matrix multiplication. If we call that matrix R, then what you're really doing is R*(Ax) = R*(b). From there, it makes sense why you can stick b and A together and do the same operations, because you're really multiplying them by the same transformation matrix R. It also makes sense why the row operations don't "change" the system - because they're all just linear transformations. So geometrically, Ax = b is asking "what vector x, when linearly transformed by A, results in b"? and applying the same linear transformation to both sides of that doesn't change the relationship. Thinking about gaussian elimination as a sequence of row operations that can actually be represented as a linear transformation makes a lot of the other calculations, like finding inverses, make a lot more intuitive sense
The hilbert curve, you could in theory map each vector in the line to a point of the plane. You don't need something that fancy anyway, say your vector is (x, y), you take the length squared of the vector L = x^2 + y^2, and then you would construct the output vector the following way: every even digit in the decimal expansion of L goes to the x coordinate of the out vector, and every odd digit goes to the y coordinate of the out vector. (Than actually only covers the quadrant I, but could can surely fix it)
I whole-heartedly agree that this is a fascinating topic, which is why I made a video about the Hilbert Curve :). As it is, I'm struggling to get all I want about linear algebra into the scope of this series, which has so far required a ruthless exclusion of the many fascinating things that are not quite in the "essence".
Why do you start with (x,y), don't you want to start with only one number, since you look for a function f: *R* --> *R*²? If you're looking at an input value x, you could of course say x=n.d_1 d_2 d_3 d_4 d_5 ..., with n being natural and d_m a digit for every natural m. Even only looking at the unit square [0,1]x[0,1], though, there is no way you could hit anywhere near (0.45, 0.45), or any other number far away from numbers with only even / odd digits. There is a reason the hilbert curve is so famous in mathematics. It's not trivial to come up with such a function. Also, many, many numbers are mapped onto the same point, 1.12, 2.11 and 1.21 for example all go to (1.1,2), or (.11,.2) if you want to add them all after the comma. My first guess for hitting as much space as possible (with relatively easy functions and no recursion or limits) would be x |--> (tan(x),floor(x/pi)), so every time the tan jumps, it goes "one step up", hitting at least *RxN*, which is still far, far less than *R*², but I'm lazy.
+Franz Luggin The transformation I wan to create is from R×R to R×R, but I want it to have the propety that the null space of the function he mentions (a line through the origin) maps to the whole plane, to do that I think of that line passing through the origin as a numberline, and then do the trick with the digits. Of house I'd have to fix it for the sign but it's possible. Having said that, the way I defined the function many values map to the same place, but that doesn't mean it's not a function, it's still valid (any linear transformation of rank 0 maps all points to the same value, and it's still valid). Definig a function that does what the Hilbert curve does is not hard, the thing is that the Hilbert curve is more "nice mathematically"
iam in the middle of taking linear algebra in uni and learnt some before while doing graphics programming, when doing gp i was like : wow linear algebra is so cool and usefull i need to learn that! then uni: what is even goin on why do i calculate random abstract things with no meaning at all?? this series: wow ok now everything clicks and makes sense!!!! you sir are the literal glue between abstract uni concepts and the actual usefulness of the nonsense it seems to be. without it i would've probably finished the course and forgot all of this but now iam absolutely hooked. THANK YOU!!
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.
Loving this series of videos. I didn't have linear algebra in school, and I'm finding it fascinating. But I don't understand what the difference is between the definition given for "span" near the beginning of this playlist and that of "column space" given in this video. From what I understood, the span of a pair or trio of vectors is the plane or space defined by all of their linear combinations. What is the difference in meaning between that and the span of the column vectors in a matrix/transformation? What am I missing? I tried googling for an answer, and the best I could find was that column space is "a very special case of span," but it wasn't more specific than that.
Column space is the span of the vectors forming the columns of the matrix. So span is more general because it could refer to any set of vectors, it does not matter if you have them together in a matrix or not.
9:16 column space: the span of the column vectors; rank: the dimension of the column space; full rank: rank equal to the number of the columns; 9:16 null space/kernell: for none-full rank matrix, the set of vectors that land onto origin(0,0)
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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.
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
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.
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.
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.
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
fr lmao. Uni lectures be useless at this point
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
@@eterty8335 i think a better word for this feeling is "maths shouldnt be a black box"
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
words can't describe how mind-clearing this series is
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
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
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
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.
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.
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?
This is one of the most enlightening video series i've ever seen.
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
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.
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 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.
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.
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.
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.
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.
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!
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.
"... 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
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 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?
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 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!
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 !
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.
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 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!
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 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.
6:15 as long as det != 0, there is a A-1; 6:43 when det == 0, there is no inverse (A-1); 7:42 ~ 8:01; 8:22 Rank 1, 2 & 3; 8:54 what's called "the Column space of a matrix"; 10:33 ～ 10:45 null space; 10:45 ~ 10:55 null space usage; 11:12 ~ 11:23 column space & null space's high level usage;
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 :)
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 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 RUclips 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'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?
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.
Amazing video again. Linear algebra becomes interesting and beautiful, and not just a bunch of boring abstract definitions. Continue to watching.
pls make the whole measure theory or topology. that would be of great value since there are few lectures about those topics online
It's unbelievable how these animations make it so intuitive and easy to understand while dry text from a book has the opposite effect.
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
This is seriously the future of education. Boring academic books kill all motivation whereas this is exciting and engaging.
10:08 This is an awesome illustration of the Rank's formula! Thank you, you made me understand a lot.
The nullspace visualization BLEW MY MIND BRO. I was SOOOOOO lost. I literally feel like my mind has been transformed to R^n
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!
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 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 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.
All of these videos from the series just keeps blowing my mind.
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.
I want to hug this guy he is a gift from god
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
The part at 10:33 made so many things make sense to me. I'm currently studying for my LA final, and this is the first time I've been able to properly visualize so many of these concepts. Thank you Grant.
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
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.
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.
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!
0:28 what about computations?
0:51 systems of equations
1:21 "linear system of equations"
2:04 into a single vector equation
2:38 visual
3:14 simple examples
4:15 inverse transformation
4:59 the identity transformation
5:39 considering the determinants
7:26 special case with det(A) = 0
7:43 "rank" (squishes)
8:54 "column space"
9:38 "null space"
10:56 in sum
holy. shit. this was whats 'inverse' was all about
haha i feel you bro
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 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.
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.
I watched this course a long time ago (about 6 years ago) when I started my undergraduate studies, and now I am a PhD candidate, and I am reviewing some of your videos. I like how you geometrically explain why a matrix isn't invertible when the determinant is 0, as it's more than just seeing what happens in the formula for the inverse of a 2x2 and/or 3x3 matrix. since you get a 1/0 occurrence. Excellent video as an introduction to the world of invertibility. Concise, clear, and cohesive, and the course as a whole is perfect as a jumpstart for a journey into linear algebra.
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!
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!
Take notes professors, this is how teaching is done.
Thanks!
The null space explanation is probably the single best example of what this channel is about.
I've finally understood it.
I honestly cannot comprehend the artistry of your teaching ❤️ it's absolutely flawless and it inspires me beyond explicable words...
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])
Wow. This lecture is just mind blowing. EVERY Linear Algebra course should introduce this series at the beginning of their course.
There's no words for how thankful I am. Not even for the algebra, but for finally feeling like I'm good enough. So many years as a child and adolescent being made to feel like I was of average intelligence because I couldn't get math or science.
I now realize that math is easy, it's the schools and teachers who make it hard to understand.
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
i feel like i have opened my third eye
Eye of agamotto!!
I would nail my linear algebra exam if I could find this series earlier
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'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.
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.
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
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.
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
I want to say though, that there is a good intuitive understanding of how gaussian elimination works.
The standard way this is taught, when you have Ax = b, you stick A and b together into an augmented matrix, and perform row operations on that until you get to RREF. Knowing how multiplication works make it pretty obvious why you get the solution to the system from that form, but it's not entirely clear what it means intuitively to stick A and b together and then do row operations on that matrix.
To understand this, you need to realize that row operations can actually be represented as linear transformations i.e. they can be represented as matrix multiplication. So, if you perform some sequence of row operations to get to RREF, that sequence can be represented as a single linear transformation, which can then be represented as a single matrix multiplication. If we call that matrix R, then what you're really doing is R*(Ax) = R*(b). From there, it makes sense why you can stick b and A together and do the same operations, because you're really multiplying them by the same transformation matrix R. It also makes sense why the row operations don't "change" the system - because they're all just linear transformations. So geometrically, Ax = b is asking "what vector x, when linearly transformed by A, results in b"? and applying the same linear transformation to both sides of that doesn't change the relationship.
Thinking about gaussian elimination as a sequence of row operations that can actually be represented as a linear transformation makes a lot of the other calculations, like finding inverses, make a lot more intuitive sense
this is oddly relaxing. like i could see my mom enjoying this
I am reborn after watching this playlist.
in 7:09, isn't there a function that maps a line onto the plane? Of course it's non-linear, but I think it's worth a mention
You mean one from the number line to 2d space? Or are you referring to something space-filling like a Hilbert curve?
The hilbert curve, you could in theory map each vector in the line to a point of the plane.
You don't need something that fancy anyway, say your vector is (x, y), you take the length squared of the vector L = x^2 + y^2, and then you would construct the output vector the following way: every even digit in the decimal expansion of L goes to the x coordinate of the out vector, and every odd digit goes to the y coordinate of the out vector. (Than actually only covers the quadrant I, but could can surely fix it)
I whole-heartedly agree that this is a fascinating topic, which is why I made a video about the Hilbert Curve :). As it is, I'm struggling to get all I want about linear algebra into the scope of this series, which has so far required a ruthless exclusion of the many fascinating things that are not quite in the "essence".
Why do you start with (x,y), don't you want to start with only one number, since you look for a function f: *R* --> *R*²?
If you're looking at an input value x, you could of course say x=n.d_1 d_2 d_3 d_4 d_5 ..., with n being natural and d_m a digit for every natural m. Even only looking at the unit square [0,1]x[0,1], though, there is no way you could hit anywhere near (0.45, 0.45), or any other number far away from numbers with only even / odd digits. There is a reason the hilbert curve is so famous in mathematics. It's not trivial to come up with such a function.
Also, many, many numbers are mapped onto the same point, 1.12, 2.11 and 1.21 for example all go to (1.1,2), or (.11,.2) if you want to add them all after the comma.
My first guess for hitting as much space as possible (with relatively easy functions and no recursion or limits) would be x |--> (tan(x),floor(x/pi)), so every time the tan jumps, it goes "one step up", hitting at least *RxN*, which is still far, far less than *R*², but I'm lazy.
+Franz Luggin The transformation I wan to create is from R×R to R×R, but I want it to have the propety that the null space of the function he mentions (a line through the origin) maps to the whole plane, to do that I think of that line passing through the origin as a numberline, and then do the trick with the digits. Of house I'd have to fix it for the sign but it's possible. Having said that, the way I defined the function many values map to the same place, but that doesn't mean it's not a function, it's still valid (any linear transformation of rank 0 maps all points to the same value, and it's still valid).
Definig a function that does what the Hilbert curve does is not hard, the thing is that the Hilbert curve is more "nice mathematically"
iam in the middle of taking linear algebra in uni and learnt some before while doing graphics programming,
when doing gp i was like : wow linear algebra is so cool and usefull i need to learn that!
then uni: what is even goin on why do i calculate random abstract things with no meaning at all??
this series: wow ok now everything clicks and makes sense!!!!
you sir are the literal glue between abstract uni concepts and the actual usefulness of the nonsense it seems to be.
without it i would've probably finished the course and forgot all of this but now iam absolutely hooked. THANK YOU!!
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.
"this isn't meant to teach you anything"
proceeds to explain everything i didn't understand
Loving this series of videos. I didn't have linear algebra in school, and I'm finding it fascinating. But I don't understand what the difference is between the definition given for "span" near the beginning of this playlist and that of "column space" given in this video. From what I understood, the span of a pair or trio of vectors is the plane or space defined by all of their linear combinations. What is the difference in meaning between that and the span of the column vectors in a matrix/transformation? What am I missing? I tried googling for an answer, and the best I could find was that column space is "a very special case of span," but it wasn't more specific than that.
The Columns of a matrix don't just represent any vector. They represent the "Basis Vectors" after the transformation..i hope you catch the idea now
Column space is the span of the vectors forming the columns of the matrix. So span is more general because it could refer to any set of vectors, it does not matter if you have them together in a matrix or not.
Waiting for the next one with bated breath!
10:15 Beautifully explained. Couldn't need anything more to understand the nullspace.
9:16 column space: the span of the column vectors;
rank: the dimension of the column space;
full rank: rank equal to the number of the columns;
9:16 null space/kernell: for none-full rank matrix, the set of vectors that land onto origin(0,0)