Dear linear algebra students, This is what matrices (and matrix manipulation) really look like
2024 ж. 12 Мам.
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I wish someone would explain why linear algebra instructors never motivate the math techniques (re: algorithms) that they teach. Linear algebra is always presented as a set of "recipes" to follow. But students never know whether they're baking a pie, batch of cookies or a cake. This video has provided me with more insight than the semester-long course on Vectors and Matrices that I took in university. It's a shame that linear algebra is taught so poorly. It's such an important topic.
Most of us are just burning water. XD
You're right. My teacher says he isn't supposed to prove stuff in that class. Ok but then I don't understand
I thought the same when I was in university. I needed some justification “why I need this”. Ironically, my job is strongly related to mostly all topics from uni’s course and even much more. And now I really think that university course was fine. There was really little ways teachers can tell you why it is useful and great before we become more or less fluent with basic math topics. Only after that mathematicians can properly show you the great applications. Once you have accumulated knowledge (in a dumbest way) above some threshold - something clicks and you begin to feel bigger picture. Before that any attempt to clarify will be either too vague or strongly misleading and really impractical
KZhead math videos blow away traditional math classes at University/College, I'm learning math I never thought possible
@@mission3479 watching a video and understand it's content doesn't mean you've understood the entire topic. This is just a fraction of college level linear algebra and if every topic was explained like this you'd be in uni forever, not to mention how much work this is to make. If you believe you've learnt something new and useful that's great but youtube videos can only be a compliment to your coursework, not a replacement.
Adding this visual element is a great idea for helping students to grasp the more intense abstract concepts of mathematics. I felt like I had an okay understanding of linear algebra after taking a class in it but this really helps to solidify my understanding.
cosmicafroninja Totally agree with you. Had a professor in class of Linear Algebra, who wasn’t the most educational lecturer. Literally said, this video has given more visual understanding, than 40 hours linear algebra in class.
This isnt really intense abstract concept IMHO. But i agree with you. I really don't know how can people learn linear algebra without geometric intuition built properly. Ive seen lecturer describing linear dependency just by writing the equation without commenting anything ..:(
Graphic, animation are the best, most effective teaching techniques whenever possible, math, physics, ... being the most appropriate subjects.
My god, thank you. It always seemed SO DAMNED ODD when we learned matrices, because in fact, without explaining their contextual utility, it's like teaching about nouns, without telling someone that ... YOU'RE NOT WORKING ON SENTENCES right now -- just nouns. So don't be surprised when nothing sensible occurs from the concept -- because we're not thinking a complete thought ... but rather, accepting that things which look like this can be manipulated in basic basic ways ... in which we'll learn more relevant rules to LATER.
Yeah, it makes me wonder if math would be better taught with two teachers. One who is a pure mathematician or math academic and one who is uses math practically in some field. Thankfully we have the internet where we can find great videos and content creators like this in order to supplement which ever complement you are lacking. And the more I think about it, having 2 teachers for any class would be a good idea (outside fiscal issues obviously). Receiving at least two different perspectives on the concept can allow you to easier pin down the concept.
This just made my entire semester of Linear Algebra make a whole lot more sense.
Watching this on the evening before my linear algebra midterm has replenished my motivation!
Good luck!
Amazing!! I just learned linear algebra at the University and yet I learned a few things from the video
That part about circuits was very interesting to me, seems quite useful(especially for Electric Engineering)
@0:26 Think of an mxn matrix as a set of m row vectors (each with n elements) and a set of n column vectors (each with m elements). @0:59 Matrix multiplication by a vector can be thought of as adding scaled column vectors together. The elements of the input vector tell how much to scale each column vector -- the first element tells how much to scale the first column vector, etc. The result of the multiplication is the vector you get having added the scaled column vectors together. @1:17-1:33 A system of linear equations (which can be rewritten as matrix multiplication) can also be thought of as an intersection of planes. The output vector (the result of the matrix multiplication) determines where the planes of the equations lie. The point of intersection of those planes rerepresent the input vector. Intersection need not be a point, it can be a line or plane etc. @2:06 Recap: A system of equations, which can be represented as matrix multiplication, can be thought of as intersecting planes or the sum of scaled column vectors. Intersecting planes help you solve for the input vector space (i.e., the set of all input vectors that makes the system equations equal). Sum of scaled column vectors help you visualize the image of the linear transformation (i.e., a mapping from the set of input vectors, the domain, to the image in the codomain). @2:22 The set of all input vectors in the domain that map to the zero vector is called the nullspace (aka. kernel) of the linear transformation. It usually includes (0,0,0), the origin but the kernel could be a line, a plane, etc. @3:12 Gaussian elimination algorithm simplifies the system of equations to give you the kernel. @3:23 In gaussian elimination algorithm, when you multiply an equation by a constant, the plane changes shape but the part of the plane that is also a part of the kernel of the system of equations does not change. @4:13 If any two equations can be 'rotated onto' one another (forming a single indistinguishable plane), there is a 'free variable' which means the kernel space has moved up a dimension (i.e., a point to a line, a line to a plane, etc.). @4:55 The number of dependent variables are called pivots, the number of free variables indicate the dimension of the kernel (e.g., 1 free variable means the kernel has 1 dimension or in other words it has the shape of a line). @5:26 A system of equations can be thought of as taking the inner product (i.e., dot product) of the input vector and each row vector. @5:53 when looking for the kernel (i.e., where the output vector is the zero vector), each equation in the system of equations is a constraint that says the kernel vector is perpendicular to the row vector. @6:10-6:44 The row vectors of the multiplication matrix span (i.e., the image of all linear combinations of the row vectors) a subspace that is perpendicular to the kernel. @6:45-7:10 Each vector in the kernel contains elements that are scalars for the column vectors (of the transformation matrix) such that the scaled column vectors sum to the zero vector (i.e., the sum of the scaled vectors, put end to end, points back to the origin). @7:10-7:25 Linearly dependent vectors. @7:25-7:51 Column space. @8:11-8:22 The column space and the row space are always the same dimension. @8:55-end Applications of matrix multiplication.
Amazing! Another video directly related to what I'm studying right now! Keep it up and maybe I won't have to study all semester. Thanks :)
I am amazed at how quickly this got complicated, yet it stayed digestible. The visual graphics complement the numbers and the vocals exquisitely. Great video!
Thank you so much. I am literally taking linear algebra and was very confused by the null space. This video really helped especially the visualization
the one thing that amazed me is when we scale a linear equation in Gauss Jordan elimination the point of intersection still remains the same. Just wow!!
Well, think of it this way: If you have an equation ax+by+cz=d, then r(ax+by+cz)=r(d)=rd, and if r is not 0, you can also do it the other way around. Thinking of it in form of a matrix doesn't help in this case, but as an equation, it isn't that hard
Amazing work! I admire your passion! Your videos really inspire us. This period of time I am studying about directional derivatives and gradients and I have to admit that they are difficult to understand. I know that this math section is absolutely essential for my other subjects. Could you please make a video about grads and directional derivatives because I want to learn the reasons why those things are so important and about their implementation in real life? Thanks again about your help cause your videos are really helpful! I really appreciate it!
It’s because of you that my interest in math continues to grow daily
Great video! I wish this was introduced in my linear algebra class. It would have solidified the notion of "why" we were even doing Gaussian elimination in the first place as well as understanding the effect of what row reduced echelon form looks like. Keep it going!
You need to start a school. Everyone would sign up
Even though that probably won't happen, if it would, I would sign up.
Then you should’ve watched past 15:03 for the genie in the bottle.
I'd sign up too. Count me in.
Because I watched 3blue1brown I think I know why at 2:50 the last plane intersects the other planes over a line and not over a single point. One thing that Zack didn't mention is that determinant of this matrix is zero, but that just means one of those row vectors or column vectors is linearly dependent on other vectors. Determinant is 0 when you lose some degree of freedom. If you look at 3 by 3 matrix as 3 unit vectors these 3 unit vectors can usually cover the entire 3d space. You can reach any point in 3d space by adding those unit vectors. However, if a unit vector is a linear combination of another two unit vectors, that means that that unit vector "doesn't add anything to the table" . You can always use the linear combination of the other two unit vectors instead of the third vector. Which means that 2 unit vectors can cover as much space as 3 unit vectors can. The 3rd vector is kind of useless in that sense. Every point in space that you can reach with that unit vector, you can reach without that unit vector as well. So, instead of being able to cover 3d space, matrix only covers 2d space, because one of the unit vectors is useless at covering space (the other two unit vectors are just as good at covering space without the third vector. Because, if a vector is a linear combination of the other two unit vectors, you can always use the linear combination of the other unit vectors instead of that third unit vectors. The third unit vector is useless). Anyways, I was going to leave a comment about why the intersection at 2:50 is a line and not a point looking at the problem from another angle. If one of the rows or columns is linearly dependent on the other two, it means that you can eventually get rid of one row or a column which will leave you with 2 equations and 3 unknown variables. 2 equations will help you get rid of one variable, so you are left with 1 equation and 2 unknown variables. And that's a graph of a line. Also, you have 1 degree of freedom you can set one variable to bewhatever you want but then the value for the other variable is fixed.
2 unknown variables can represent a plane, not a line. Each equation gets rid of one variable, so if you have 2 equations and 3 variables you will end up with just one variable
@@sergeirachmaninoff6397 2 mutually independent unknown variables represent a plane, 1 independent unknown variable represents a line. The number of mutually independent unknown variables = degree of freedom = number of dimensions. I haven't said anything contradictory to this in my comment.
As one gets further into mathematics and its applications, most problems boil-down to "Find the inverse of the matrix A." or "Compute the eigenvalues of the matrix A.", etc.
Very true for things like control technology
@@albingreen344 And differential equations when you solve for a particular and/or an initial value problem.
But do you know why? I’ve noticed that when you press a particular pedal on the floor of all cars, the thing will start moving or at least make a revving sound. Is that pedal in flying cars? In trains? In drones? It’s definitely on bikes but you have to use both feet. That’s what your statement sounds like to me.
@@oidbio2565 Lmao! You reply to an innocent comment written three years ago to say that? You must be pretty needy, or at least that's what your statement sounds like to me. You don't have to thank me for the attention, though. It's free.
@@oidbio2565Bad example
I'm taking Linear Algebra right now, and this has really helped me visualize everything I've learned so far (up to Eigenvalues, Eigenvectors, and Matrix Diagonalization), so thank you so much!
Math can be so simple yet complicated at the same time. Once you visualize it, all makes perfect sense and you wonder why you didn’t grasp it sooner. Looking in your textbooks without these visual insights can be a really terrifying experience!
You did a great job with this video! I liked watching the planes rotating about the null space line.
Putting out really good stuff bud! Keep up the good work and the world thanks you!🍻
This is beautiful to watch. I do wish I've seen it back when I was doing Linear Algebra, maybe my Quantum computing class would've been less confusing
Tell us more about quantum computing
This is the kind of material that _every_ Linear Algebra course needs to have as mandatory viewing along with Gilbert Strang 👏👏 I'd love to see more such videos 👍👍 Thanks for sharing these 🙂
I was aware of the application of graph theory to electrical circuits -- Ernst Guillemin (1953) "Introductory Circuit Theory" and Wikipedia: Topology (electrical circuits) -- but in just one minute, the relations between graphs, loops, and trees are clarified. Beautiful job, sir. Thank you for posting this video.
There is a problem with visualizations from 2:45 and so on. The linear equations which are homogeneous(i.e. have 0 on the right) always correspond to the planes WHICH COME THROUGH THE ORIGIN (because all-zeros vector satisfies such equations), and it is not the case in the video. Similarly, at 6:36 row space and null space must always contain the origin as they are LINEAR SPACES, not AFFINE SPACES
This was very enlightening. Thank you.
This one video teaches me better than my linear algebra professor did for a full fkn year. I don't even really need them when you and 3Blue1Brown doing so well.
Between them, kurzgesagt, and brilliant. I am paying less than $200 a year for a better education for myself and my 2 kids, than we could get in college. My proud dad moment came when my curious 7yr old solved a Coulumb's law based problem on her own. Shes obsessed with physics and has completed several brilliant physics courses
@@thezyreick4289 Amazing
Great explanation of the four fundamental subspaces ! I love your videos!
Great visualizations! I’d like to see a video on matrix applications in probability. Would continue the stuff with the circuit here.
So good :D So accuratte representation of interesting relations not explained in class.
OMG. I loved it. Thank you for this great explanation. Do you have notes for this video? or any text document to refer to these concise fundamental concepts?
My Physics major in the 70s and 80s would have benefitted greatly by tutorials such as this, and KZhead and online resources in general. Great presentation !
Thank you ! Superb Explanation! Could you please make a video on coordinate transformations on vectors, rotation? It would be easy to grasp those concepts. Thanks
That was EPIC! I know how to solve all these problems, but adding the visual part makes it easier to get it and to explain it to my students.
I was having some difficulty understanding nullspaces until this video. This definitely helped for my analysis of engineering systems class!
animation staffs that u used are obvious and really understandable...3d perception helps to figure out linear matrix clearly... god bless you and your family...
The geometric interpretation can be useful in the context of 2D and 3D graphics, but I find it even more confusing when talking about higher dimensions. There is another interpretation that is used in signal processing and that I find is much more intuitive for a large number of dimensions. You can find the full explanation in the first few lectures on Dynamic Linear Systems from Stanford, by prof. Stephen Boyd. This is particularly useful in the context of neural networks. It goes something like this: Suppose you have a linear system with a n-dimensional input and a m-dimensional output. Then this system can be fully described by a m by n matrix *A* . The entry *A_i,j* (i-th row, j-th column) just tells you how much the i-th output is affected by the j-th input. The corresponding equation is *Ax=b* , where *x* is a n-dimensional column vector and *b* is a m-dimensional column vector. But what if you have an equation *AX=B* , where *X* and *B* are also matrices? Well, *X* is just a series of n-dimensional inputs (each represented by a column vector) and *B* is a corresponding series of m-dimensional outputs (also each represented by the corresponding column vector). Here it's also immediately obvious that *X* and *B* have to have the same number of column vectors.
HOLY COW Zach, this is beautifully edited!!!
Excellent Zach great to see you discovered Adjacency Matrices. Yes graph theory becomes very nice to organize. Great visual software too.
Wonderful Video. Thank you so much. What tool did you use to create the 3D visualizations?
on my physics exam I got a test question with DC circuits and we had to calculate currents. I've done these type of excersises before, so I started to "show off" a bit by using gaussian ellimination to solve it, while my classmates struggled because they were "afraid" to dwelve into linear algebra, as they barely passed it last semester. Linear Algebra really is the subject of corelations, and if you understand it properly by the end of your semester, it will open up so many new possibilities to view things for you
I am really and really loving this series more and more! Cheers man for the information as always!
Hey RC, daddy here
@@RockHardWoodDaddy Heyo!
UGH IM SO HYPED RN THANKS BRO
Hello! I am excited about linear algebra so I will prepare a lecture for my students at school. I liked your visualization. I'll use Geogebra myself, but your program is better. I wonder what program you used. Congratulations on all your YT math videos. They are exceptional! Jure Grgurevič
I also loved this cool application when I saw it in an appendix of Paul Renteln's book.
This is a very intuitive and informative video, I highly recommend it. It has helped me a lot, thanks!
This is the best thing ever. Could you do a similar video on riemannian geometry?
Wow, a ton of information here, I’m gonna have to watch this many times before I start to get what was explained. Thanks.
Great video! Was not expecting to see an interesting explanation of KVL.
This just helped me so much. Thank you more than I can even express.
Good job. I love that software you're using for the 3-D graphing. I use it also!
I use the Geogebra program for 3D drawing. I really like the program that Zach star uses, but I don't know what it's called. I asked him, but he didn't answer me. Can you tell me?
Awesome! Thank you for your great videos, enjoying them once again.
I remember the first linear algebra classes I had back when I was studying pure maths and I cried the first month cause I was like wth?? Now I always use software for the students to understand certain abstract things that might be hard for them when it’s their first approach to the themes.
You should become a maths motivational talker
I feel pretty motivated to do some maths.
The four fundamental spaces have always alluded me. This video helped me finally to visualize them. I was wondering, what software you used in the demonstrations?
Glad it helped! And I linked those in the description
@@zachstar Thank you.
Connectivity is very cool. You point to it as graph theory. Linear algebra has some neat applications. I like Y buses in power grids. It is basically what you laid out with your circuit analysis.
One of the best linear algebra fundamentals ever! you are the best man!
Every aspect of this video is amazing!
I love this video! One quick question: at 13:34, Zach says "if you go through with the analysis, you find that b1 + b4 - b5 = 0 and b1 + b2 + b3 + b4 = 0" What analysis is he talking about here?
ThankU for posting and sharing.
This is nice software animated Gilbert Strang’s way of presenting the material from day 1- row picture, column picture, matrix equation and then leading to the 4 fundamental sub spaces - good job! That is the way to go! Thank you! 🙏 Please make more animations for more other examples! 🙏At 6:41 “this always will be true” can be misunderstood by some students. Better to say in general f+r=n will be true, or Nullity + (dim of Col or Row space) =n. Otherwise some students may think 1+2=3 combination will always be true. Thanks! 🙏
Well explained with clear visuals!
Espectacular! You are amazing. Thank yooouuu!
Great video man, you are not the only one liking matrices the least^^
I thought 3Blue1Brown was ultimate but ZachStar has changed the definition of being ultimate.👏👏👌
This is so inspiring. Amazing stuff
wow. skimming over this video for 3 minutes conveyed what I've been trying to wrap my head around for YEARS using linear algebra. Thank you for making this for those of us that don't do well in the land of make believe and imagination
This is really nice! Thanks!
🎯 Key Takeaways for quick navigation: 00:00 🧮 Matrices represent systems of equations and can be thought of as sets of vectors. 01:20 📊 You can visualize solutions to a system of equations by looking at the intersection of planes represented by the equations. 02:28 🧲 The null space of a matrix represents the set of solutions to a system when all equations equal zero, and it can be a line in 3D space. 04:10 🔄 Linearly dependent vectors in a matrix result in a single free variable and infinitely many solutions. 06:31 🌌 The row space is perpendicular to the null space and contains linear combinations of row vectors. 08:41 🌟 The column space is the space spanned by column vectors, and it can be different from the row space. 09:50 🌐 Connected graphs reduce to trees, and loops in a graph lead to dependent rows in the incidence matrix. 12:23 ⚡ Kirchhoff's voltage law emerges from analyzing the column space of the incidence matrix in electrical circuits. 14:55 🧩 Determining if a vector is in the column space involves checking if it obeys Kirchhoff's voltage law. 15:51 📚 Further learning on matrices and linear algebra can be done through courses on platforms like Brilliant.org. Made with HARPA AI
Wow. Just Wow!! I am currently pursuing Master's in Physics, and I have NEVER been taught the fundamental beauty of linear algebra to this extent. Thank you for making this video!
Thanks, and it's so easy & simple!
Great example, and refresher, on linear algebra. Would like to see one on non-linear algebra ; )
This is sooooo beautiful... Thank you sooo muchh.....
this is WONDERFUL. thank you🙏
Just awesome... Thanks for making such videos..🙏 If possible please make some on norm linear spaces too.. From India.
this is a supper hero explain of linear algegra, and very easy to under .thank you so much for your sharing.
Some undergrad math programs may have two or three “linear algebra” components that never really get much beyond matrix multiplication and Gaussian elimination. It’s a real travesty and does a horrible job preparing students to actually use it in practice or use the key concepts as a foundation for functional analysis. Thanks for helping to rectify this 🙂.
This is the same explanation that Gilbert Strang gives to his students! Nice job, keep going!
12:20 - question: what is 0 by making that vector as the voltages? What I think: I think it means that these voltages are a combination that doesn't make the circuit produce or consume energy; or that for some reason that still make the current 0(?) which I don't think makes too much sense.
You can also use simple FEA like approach to solve steady state currents and voltages in the system using matrices
Thank God for you, Khan Academy and 3B1B. You guys are single handedly keeping the math students community alive
Linear was my absolute favorite class, I also love diff eq
nailed it man!!!
Kudos to all of you that follow and understand all this stuff in real-time, never having to pause for 5 minutes and think about what was just said. Kudos...but also, you're freaks.
Thank you so much, you are a legend! You single-handedly helped me understand my vectors unit in math and circuits unit in physics.
*we are loving this series* Thanks for doing it man
this is the best video i have ever seen ...thank u so much man i can finally have a vague visualization ......
Thank you very much for great explanations and illustrations.
Amazing !!! Really thank you Quick question: How do you graph a R^4 (or more) dimension matrix?
You don't. You can project R4 onto R3 and graph that, but there's not a good way to visualize 4 dimensions. In the study of optimization and dynamic programming you can get interesting special cases that allow you to visualize hyperplanes and planes generated by higher order spaces.
In addition to @Ben Bruland's answer I would like to add: That is the beauty of learning linear algebra. In most cases you don't need to visualize them. Once you learn the mathematics, they work for any number of dimensions.
Can you please tell me what software do you use for these presentations? Thanks in advance and keep it up with such great and cool stuff.
Yeah im interested in that too.
In the description :)
THANKS I'm just starting to study them and this helps
This was great! Literally helped me understand two of my classes in a completely different way. Does brilliant actually have interesting stuff like this I tried curiosity stream but it had a bunch of lame history and environmental crap I want more science and engineering? I would love to see more tricks in linear as I have to teach the class to myself online. And I especially need help in circuits.
You could watch Linear Algebra series from 3blue1brown too
This is simply brilliant! Thank you.
it really helps. Thanks very much.
Thanks Zach, this added another way to understand why row operations do not affect the solutions
This kind of clarification of the connection between technique and meaning is refreshing to say the least.
Can you make a video about subspaces and vector spaces too? Linear Algebra is the hardest topic for me so far
It's really amazing!!!!
This video is like Gilbert Strang with 3D visual tools! Brilliant !!
I really recomend Gilbert Strang 's books for this subjects
we're using his book as our text for my class