The Applications of Matrices | What I wish my teachers told me way earlier

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This video goes over just a few applications of matrices that may give you some insight into how they can be used in the real world. Linear algebra was never explained well to me in school and I had very little motivation to learn matrices at the time so hopefully this helps you if you're in a similar situation. Also note there are so many applications there's just no way to fit them into one video but here you'll find some of my favorite instances of when they come up.
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[@chaselongenecker5398]

Modern mathematics education’s biggest crime is not explaining what the hell anything is useful for before you learn it.

[@MathTravels]

Absolutely. It goes with earlier concepts, too.For example, the context is rarely given for fraction operations (like division) and when it is it's inadequate.

[@MrAlubenco]

Too right... math is extremely fun when it's applications are made known...

[@AyushSingh-be2nm]

Get a dive into the Jee mathematics and Its Teachers. You will Feel It. In India many things are like festival. Onen of them is Jee exams.

[@abyasahasan]

yessss

[@jianhuang0124]

Math itself has nothing to do with applications. It's a pure logic game. Applied cases will distract the core.

[@martinmartinmartin2996]

As an electronics engineer that dabbles in math: For 30 years I have puzzled: WHAT IS an eigenvector ? Finally (for the first time, after 40 years ) I have UNDERSTOOD ! Thank you !

[@lonestarr1490]

Fun fact: "eigen" in German means something that is inherent or a personal attribute by something or someone - not as a verb, but as a prefix. For example, you can translate "peculiarity", "individuality", or "foible" to "Eigenart" or "Eigenheit" in German. And an "Eigenschaft" is something like a characteristic property or an attribute.

[@ccuuttww]

OH what 40 years I think u miss the lesson of PDE and ODE it will explain what is eigen space

[@rigorouselectronics8089]

Look at 3Blue1Brown's linear algebra playlist. You'll get a more full understanding of the concept. While you're there you might as well check out the most beautiful equation, which you and I use alot as electronics engineers (particularly in electromagnetics and signal processing)

[@pentachronic]

Totally agree. We never got an explanation what they were either.

[@StephenGillie]

This video really fills in some of the gaps in connecting Pythagoras to Minkowski, through Einstein, to Kaluza-Klein, String Theory, and beyond. Just keep adding dimensions onto the matrices for different fundamental fields.

[@cfgauss71]

Showing how matrices can be used to blur images was awesome. As a civil engineer, I should add that matrices are also used to analyze entire structures (building, bridges, etc), where a force matrix (ex: the loads the individual beams or columns experience) is equal to the product of a stiffness matrix (reflects the material properties) and displacement matrix (how much the members of the structure move). In effect, it's Hooke's Law, only taken to an extreme level using matrices.

[@JohnFekoloid]

You're talking about FEA. But really, it's all coded into the computer software. Most likely, it will be easy for the average user not to even know about any matrices in there and just use the software.

[@GoldenAgeMath]

Would these "matrices" really be tensors? Asking as someone who studied physics but not engineering.

[@rkus07]

@@GoldenAgeMath Theory of Elasticity uses a lot of tensor math.

[@alibaba-kv1ww]

@@GoldenAgeMath Hm I would say not every matrix would be really considered as a tensor of a 2nd order.I mean, mathematically they are but let's just take an example of a 3dimensional state of loading.The Hook's law would be represented with a 4th order tensor(material properties)acting on a 2nd order tensor(strains) and that would give 2nd order tensor (stress). But take the node matrix for example. Is there any physical meaning behind or not? It's just a convenient way to write all nodes in a Matrix form and therefore we won't really say - it's a "node tensor" right?

[@twinplayerz]

I am taking Analysis of Structures right now and this is the last topic that is being covered in this class. In a conversation with other peers who have taken this class in the past, they didn't even cover this topic!! Thanks for sharing your insight!!

[@gracialonignasiver6302]

"There may not be a zombie infestation anytime soon, but this kind of math could be used to analyze how a virus could spread throughout a population, for example." That aged well...

[@meatbleed]

math is only useful with accurate data though

[@jaredgreathouse3672]

No kidding

[@chimero3806]

You could say it aged too well...

[@brendancooney9401]

@@meatbleed LOL, no matter how accurate the data there are plenty of zombies who still dont believe.

[@thanhvinhnguyento7069]

Like fine wine

[@khari_baat]

Dear Zach, I thank you from my heart. This video alone is worth more than years spent in mathematics. I remember multiplying matrices by matrices and answering questions in exams without ever understanding what are they trying to do. The education system sucks big time.

[@randomrandom316]

As a fellow Indian, we the students suck just as much. When a student is really interested in the class and asks the teachers why we are learning a particular topic, what its applications are and so on, most students simply roll their eyes and look at that student as if he/she is a criminal. Most students don't mind copying assignments and are literally willing to cheat on the exam to get a higher score, never stopping to think if they are pursuing education for learning or for the sake of some place in social hierarchy. Education system indeed sucks big time, but frankly, a lot if it is really a reflection of the students themselves, if students held themselves to some principles and standards maybe they can legitimately seek redressal from the system, else the system knows where our priorities really are and they are more than willing to exploit it.

[@impolitedirector3595]

@@randomrandom316 Can't agree more

[@josefrancis7126]

ESPECIALLY IN INDIA

[@DoctrinaMathVideos]

You have to understand how to work with matrix computations and understand vector spaces before you learn the applications and this is the goal of a beginning linear algebra course. After developing a solid foundation in linear algebra then you can take a more advanced linear algebra course and this is where you will learn all about the applications of linear algebra (even more than what's covered here in the video).

[@ungarlinski7965]

Or you're an idiot for not recognizing it's use. Or you missed that part due to being on your phone in class. Or you never read your book. Or you should have just gone to trade school.

[@Eromasin]

I dispised matrices and avoided learning about them like the plague. I was given 0 contexts as to why they were useful. I managed to avoid matrices questions in papers all the way up until my Masters degree (I even managed to get past an EE BEng without understanding them - I was into digital not analogue so it didn't really impact me not knowing them). I'm now doing Robotics. Guess what. ALL THE MATHS IS MATRICES. EVERYTHING. Kinematics, Dynamics and Control is the bread and butter of making robots - and almost every equation using a matrix of some sort. I'm now so far up **** creek trying to catch up on 5 years of matrices education on top of doing a Robotics postgraduate. If you're a teacher, please for the love of god teach your student *WHY THEY NEED TO KNOW WHAT YOU'RE TEACHING THEM*. Teaching a kid all of the cool reasons as to why they need to know something is to me almost more important than the teaching itself.

[@faustin289]

This is even more important in the age of internet where information is out there open for everyone to access freely. If one knows why something is important, then it'll be easy to teach oneself. The role of the teacher is therefore to guide the pupil on what is important and why.

[@timmurphy5541]

I think they often don't really know themselves - I mean they're teaching not doing robotics. They need to get people like you to come and explain the whys.

[@GaussianEntity]

To be fair, the applications are so wide that it's very difficult to even begin to describe what they can do. I didn't understand their geometric interpretation despite reading about them a several dozen times. Even now, there are still things that I'm learning about them.

[@ableone8956]

No Shit!

[@cruxer666]

Totally agree. I saw no point in all this "garbage" only when I started to work and I have seen what this is used for it made sense and is actually interesting and easier to understand, now trying to catch up. Education systems are so broken, they cannot keep up with what is happening and why it is happening.

[@patrickmayer9218]

Notes: *Vectors that are only scaled by matrices are called eigenvectors of that corresponding matrix. *How much the vector is scaled by is called the eigenvalue. *Blurring, distorting, and sharpening images all use very specific matrix math on the pixels to get the desired result. Thanks for the video, Mr. Star!

[@dineshk9803]

I wished, I watched it before entering into high school, now I'm pursuing engineering. I was like "WOW". Thanks Man

[@sabin97]

yeah. as an engineer i never quite grasped what eigenvector and eigenvalue were......never....until today.

[@dineshk9803]

@Deal Negrasse Bison that too watch it bro 😁

[@minh9545]

Well i was not taught about the use of matrices in high school. Only when we enter university, would we be taught this. I believe this is the perfect moment to know about this as uni is going to start in our country in about 1 to 2 weeks

[@vibinjacob4610]

Spring mass system, Eign Values and Eign Vector kzbin.info/www/bejne/rqunYXepdqeAn7c

[@vibinjacob4610]

@@sabin97 Spring mass system, Eign Values and Eign Vector kzbin.info/www/bejne/rqunYXepdqeAn7c

[@felipeflores5403]

Man, you talked about pandemics and riots and I'm here watching this in June 2020 convinced you're an actual prophet

[@brian8653]

I know, he's a prophet or something

[@gerooq]

@@brian8653 This reminds me of the time mathematicians in the old ages were called witches

[@aschmitt89]

THIS!!!

[@aschmitt89]

If mathematicians in old ages were called witches, and today we're calling them prophets... Is our universe really real? Or could it really be that we're all part of a massive calculated simulation?!?! (I've literally only heard this conspiracy theory once before in my life, but DANG Math can be scary accurate at times!!!!)

[@stevencvisuals]

I see how matrices can be used now lol we're stuck in a pandemic

[@Zilberlex]

He did not explicitly say it, but seeing the number of graph triangles in the last part basically allows you to calculate for potential threesomes.

[@anothrto1045]

It also explained how p{2,3,4,5,6} got STIs from p1

[@matthewcuriel991]

Lolol

[@cancel1913]

@@anothrto1045 I thought it was STD's but what do I know

[@anothrto1045]

@@cancel1913 infections also covers things that aren't diseases and in addition some syndromes aren't diseases but result from infections

[@bobloblaw1021]

Bow chicka wa wa

[@justingolden21]

Zach: "There may not be a zombie infestation any time soon" 2020: "Don't test me child"

[@danpal6737]

nah its almost over 2020

[@betterfly7398]

@@danpal6737 2019 was almost over when COVID-19 appeared.

[@angqingxiong]

1:14 Vectors 4:51 Electronics 5:13 Zombie predict 9:17 website ranking 9:35 assault victim 15:09 Computer graphics

[@jonny2575]

*No one:* this video is getting long and uninteresting.. *You:* “As much as I’d love to keep going into depth on different subjects, this video is already quite long.” You sir are a gentleman and a scholar. PLEASE make a part 2.

[@dennispetrov9628]

I for one just recalled how I lost track halfway into the lecture, just like I did this time with this video. Now it would take me hours to digest that second part, that is, if I were in the mood. Is there a math course for bipolars with ADD? I mean, serving it in small chunks is a real help for some.

[@timinator1178]

@@dennispetrov9628 do one video a day from Khan Academy is what I'd suggest. One video a day, solving one problem a day, and eventually you'll grease that math groove enough that you can move to two problems a day, then three, then two videos a day, etc. I don't know, just a suggestion. I know Khan breaks everything into tiny bite sized videos overviewing single concepts.

[@dennispetrov9628]

@@timinator1178 thanks, maybe I will. It's not that I'm plain stupid, it's just that I'm a bit bipolar and a bit ADD , that's what's slowing me down

[@timinator1178]

@@dennispetrov9628 I understand completely. Everyone learns different ways. Some are more advantageous than others, but there's always different a method that works better for someone. Personally I think it's very interesting, all the different ways people learn. One of the problems of this internet age is that so many people go through school thinking they can't learn or they aren't meant to be knowledgeable, when the real problem is that they simply don't fit in a one size fits all approach to learning. Many people need an optimized approach, and finding that approach isn't always easy.

[@theonionpirate1076]

"Any vector that is only scaled by a matrix is called an eigenvector of that matrix. And how much the vector is scaled... is known as the eigenvalue." Wow, those two simple facts would have been nice to have known when I was taking linear algebra.

[@robertcaspari9827]

Right?!?!? Dude i had so many math classes and never really got it this well

[@guitistic2732]

Exactly. I took engineering mathematics in my 2nd year of engineering and had no idea what eigen vectors or eigen values meant. If only they explained it like this, it would have been easier.

[@IzumuK]

Dude that blew my mind. I had no idea what the heck eigenvector and values even was. I was just blindly calculating them and wondering why the heck I was doing that.

[@alanbouwman5627]

If you paid attention to the definition of eigenvalue and eigenvector this would have been really obvious. In fact, your professor most likely mentioned this while you were not paying attention. Fun fact: "Eigen" translates from german rough as "own" (adjective). But it is just as useful to think of the word own in the verb sense, I think. A matrix (linear transformation) "owns" a vector space when any vector in that vector space stays in the same vector space after that transformation is applied. So saying "eigenvector of a matrix" is similar to saying "the vector owned by a matrix."

[@maxsimes]

Yes this absolutely clicked just because of that

[@Bald_Zeus]

I'm studying to become a mechanical engineer and I've really struggled with linear algebra, mostly because I haven't grasped how I need to view it. This video helped me immensely

[@dixonhill1108]

Yeah I can use it, but I never had a successful way of seeing it in my head. Pretty rare in mechanical engineering to not be able to imagine the thing, that's a problem for electrical nerds.

[@panself]

3blue1brown makes much better visual explanations.

[@BigDaddy-je2nq]

When I took linear algebra nothing had a purpose, it was just memorizing a pattern, I didn’t even realize what a matrix was until I went off on my own experiments with information I didn’t even have to pay for

[@The_InfantMalePollockFrancis]

I'm studying mathematics and physics for an entrance exam to a university for an engineering degree, but really I need the scores for pilot school and I WANT to visualize them--at least to a degree that a good pilot could use--and I'm finding it quite difficult

[@The_InfantMalePollockFrancis]

I'm studying mathematics and physics for an entrance exam to a university for an engineering degree, but really I need the scores for pilot school and I WANT to visualize them--at least to a degree that a good pilot could use--and I'm finding it quite difficult

[@turpialito]

Zach, people like you should really have a voice in public education policies. My sincere congrats.

[@tyereksmith8946]

The internet is so wonderful just imagine being a million miles away and find a great teacher like you who isn't boring. Great job nuff respect from kingston jamaica 👍🏾

[@cubingcubez6678]

i m from India

[@cboy-ou2hr]

The internet truly is amazing I envision a day where traditional school settings will finally be obsolete and u truly get a education worth while

[@BangMaster96]

+tyrek smith You can't possibly be a million miles away from anyone on Earth, the Earth's diameter is less than 8k miles.

[@bobloblaw1021]

Imagine all the people, living for todaaoouuy. You may say I'm a dreamer, but I'm not the only one. I hope some day you'll join us, and the world will live as one. PS silly Google doesn't know that song. Hey look ma I made it!

[@meshachconnell5595]

Trinidad

[@Cuzilla47]

After almost 53 years, I finally REALLY understand eigenvectors and eigenvalues! I can also declare: "I wish my teachers told me way earlier."

[@Hr1s7i]

Pft some of them probably had no idea what it's practically used for either. If people do internal field research, they don't get invested in the real world. Math as it's own field doesn't really have a purpose other than to teach people how crude virtual logic models function.

[@louisrobitaille5810]

@@Hr1s7i Trying to couple maths and applications is like asking an expert in a field to only teach the basics to beginners for the rest of his life. It's possible, but will be incredibly dull. - The applications of maths is such a tiny tiny tiny TINY fraction of the field that the one doing the teaching will only teach the bare minimum out of boredom while the one learning won't learn anything because the teacher is better at making them sleep than transfering his knowledge. Idk if you got my analogy. I did my best to explain how I see it 😅.

[@whannabi]

​@@louisrobitaille5810 maybe but that makes it more engaging to people that don't necessarily see the reason behind learning it. And you talked about math but most people searching for applications are probably in applied math fields like engineering or computer science y'know?

[@arikayemusic]

YES. I feel like so many courses teach you the computational aspects of linear algebra, but you don't think about what's really going on with it. Amazing vid

[@esPeSkull12]

Please keep doing these kinds of videos! I remember doing highschool math and coming across your "common math mistakes students make", which helped greatly. I'm doing Linear algebra atm and this makes a lot more sense than trying to keep with the 300% speed trail of thought that my teacher has.

[@darknightninja9773]

Matrices are a crucial part of 3d graphics. In my graphics game engine that i am writing from scratch, all the mathematics related to transforming, rotating, scaling and even projecting the scene onto the screen is handled using matrices. I too was really surprised that matrices can be so useful when i was first exposed to graphics programming. Its sad that they dont tell you stuff like this in school.

[@skilz8098]

Most of those who don't work with 3D Graphics don't understand the significance of the MVP... And not it's not Most Valuable Player either. It is the combination or multiplication of the Model, View and Projection Matrices. And depending on how you defined your 3D coordinate system L-hand or R-hand system will determine the order of linear transformations that need to be applied such as translation, scaling and shearing and rotation about an axis. This also doesn't account for the fact of transforming one set of data in one coordinate system to another. For example; transposing the coordinate values of a Model's vertices, normals etc. from model space into world space and from world space to camera space and from camera space to culling and clip space and finally from Camera or Eye space to Screen Space in which you can not forget the perspective divide otherwise 3D Images mapped to a 2D Screen will looked stretched and skewed. This still doesn't account for triangulation and the vertices winding order for back face culling, this also doesn't account the Z-Depth buffer and the order of rendering of objects that are dependent on two major factors, first the depth and second if it has transparencies or not... Then after all of that comes all of the lighting calculations as there are 3 major types of lights; point or positional light, directional light and spot light as they all share some similar commonalities but each have their own distinct properties. For example a point light could be that of a street lamp as it has a position and the light goes out in all directions. A directional light such as the sun or moon in a sense doesn't have a relative position but comes in from the same direction. The spot light is like a flash light as it has both position and direction but it also has attenuation and cutoff angles. Lights also can contain colors which can be defined as a vector of values. This also doesn't account for all of the different kinds of lighting and shading techniques that one can achieve such as bump map, glow, blur, reflection and refraction, bloom, and so much more. So if you are trying to use modern OpenGL with GLSL, DirectX with HLSL or Vulkan with Spir V and you are trying to build a 3D or even a 2D application while using Shaders to be sent to a modern GPU and you don't understand linear algebra, geometry, vector and matrix calculations along with trigonometry and vector calculus you are going to be at a loss and you might start to pull out your hair trying to get everything to work correctly in your final scene. Now if you are getting into motions and mechanics within 3D you can use basic Euler Angles for simple rotations but if you start to rotate a single object along multiple axes at the same time you will come into problems such as trying to rotate an airplane long both its x and z axis (horizontal) considering y to be vertical you can end up with situations where you will lose a degree of rotation and the freedom of a full dimension as this is called Gimbal Lock. To avoid that we can use Quaternions instead which is very useful and there is an excellent math library that provides all of these operations for you so you don't have to write your own, it's called GLM as it will provide all of the needed vectors, matrices, quaternions and all of the necessary calculations between them including the dot product, cross production, normalization, finding inverses and transpose matrices and much more. But yes; matrices are extremely important in fast and efficient calculations and they are not going away anytime soon. If you have a hard time believing that just watch The Matrix, soon they'll be taking over!

[@darknightninja9773]

skilz8098 Wow, interesting stuff dude! To be honest i dont fully understand the math behind it all, and like you said i use GLM to handle almost everything and treat it sort of like a black box. I understand the significance of mvp ( i dont use the model matrix for my project since i am making a voxel engine and i directly pass the world coordinates into the VS), and the general math behind it but i never really bothered reading its derivation.

[@huverdoose]

Extremely useful for ray tracing as well. ...and visual recognition software. ...and predicting future attacks of Somali pirates. ...and...

[@researchandbuild1751]

Writing a graphics engine from scratch is a waste of your life

[@komodot9249]

Exactly why schools need to teach better at why you are learning something.

[@sdeerwester]

I'm one of the authors of latent semantic indexing/analysis, which is a technique that some folks in natural language processing, machine learning and search engines have found useful. It's based on some pretty funky linear algebra. I just wanted to say that, even with the major role that matrices in general and linear algebra in particular have played in a very important chapter in my life, I still learned things from your intro. Very nicely done.

[@polyhistorphilomath]

LSA? You’re the man

[@aBradApple]

I curiously took an online course for linear algebra thinking it should be a walk in the park and LSA blew my mind. Though unintentional, I found it improved my programming and understanding of matrices. Thanks a lot for your contribution to society.

[@douglasstrother6584]

My first exposure to Linear Algebra was a Sophomore course from the Math Department & it was a dry as dust, definition, theorem, proof course. (Granted, I was a 19-year-old at the time!) "Basic Concepts of Linear Algebra" by Isaak & Manougian was the text, which I still have. My second exposure was a year later in a very applied Mathematical Physics course using "Mathematical Methods in the Physical Sciences" by Mary Boas. Approaching Linear Algebra from an applied perspective made it much more interesting and increased my appreciation of the Mathematician's perspective. Linear Algebra is the starting point of learning to visualize higher-dimensional spaces.

[@andrewgjennings]

Suddenly, matrices aren't so scary. Thank you for putting this video together. As a computer programmer, we deal with this all the time. Because we don't call them matrices - we might call them something different, like tables, arrays, grids, etc - I got the idea that they are completely different disciplines, but you just proved they aren't. Spectacular video and explanation. I wish I'd known this decades ago. Well done!!

[@pspmaster2071]

I took a class called linear algebra with applications....THERE WERE NO APPLICATIONS! Edit: Thanks for all the likes!! Great to know others think the same.

[@iannieves5346]

Try Quantum Computing, a very potentially lucrative application.

[@huey1153]

Ian Nieves he’s not saying LA doesn’t have applications he’s saying they didn’t use any

[@projectjt3149]

Ian Nieves Or parallel programming with GPUs

[@lordx4641]

Typical university education

[@pspmaster2071]

@@lordx4641 I know right. Tell me about it. College is a place to put stuff on a transcript and get credentials. Learning happens outside of class by yourself. Teachers are good to fine tune.

[@mhh5002]

Your and 3b1b ‘s videos are what keeps me motivated for learning math and data science. Thanks and keep up the good work 👍🏻

[@harjotsinghbaidwan2204]

Well said !!

[@haleshs66]

3b1B

[@Blox117]

yeah theres a lot of channels that cover solving math problems but not as many conceptually explaining why and how it works.

[@bithisarkar1412]

What about mathologer?

[@maryamali5087]

thank you so so so much, i am a senior in highschoool and i am taking advanced maths, i lmost forgot how passionate i am about maths because how boring our curriculums are, you helped me remember my love for maths again

[@nourjouini803]

I am studying to become a software engineer and this video has given me the motivation to ace my linear algebra course , I can't stress enough how this video has made me fall in love with what I'm studying and it's applications . This truly has opened my eyes to how amazing and useful the understandding of mathematical concepts is to our lives . Thank you so much for this breathtaking content.

[@georgecaplin9075]

The times in this video when he said “I haven’t got time to go into it now” or “ this video’s gonna be too long” or whatever, I just wish he’d kept going. I was really enjoying it.

[@anymaths]

watch my helpful maths videos.

[@SivaramSarma]

Imma be honest ,u just told me the best explanation for eigen vector and eigen value ,my teacher told me 'just read what's in the book ,u don't need any explanation'😭😑.

[@lydianlights]

Same for me! I know a lot about pieces of linear algebra for computer graphics and related things, but still never really understood exactly why eigenvectors were important. Now I have a better grasp on what they actually are!

[@user-lv1wn5wq7n]

I thought that this happened only in Hindia

[@andrewsalazar98]

Should've told your professor, "Excuse me? This class is called 'Linear Algebra with APPLICATIONS.' If you're not going to teach me some applications, then you're just an overrated tutor."

[@ashokjadhav9904]

Ditto here! 😊👍 Most of the maths teacher aren't interested in the subject. Many of them, don't understand it, in the first place. So..... But this guy is great! Not only does he understand, but he has a knack to explain it smoothly.

[@dahawk8574]

Exactly what Ashok said. Teachers dismiss questions because they don't understand the stuff themselves. Like ask anyone why a negative number times a negative number is a positive number. Almost no one can explain it. And this is one of the most basic things we are "taught" in math. The reason why so many students end up hating math is because they have this pile of unanswered questions which eventually becomes so heavy that they end up calling _bullshit._ And they quit. They've checked out. They've had enough. Math is "taught" in a way that is quite similar to religion, in that there are these things that are not logically and rationally explained. The learner reaches a breaking point, and then walks away. The difference is that in math, explanations exist. And the earnest seeker will find the answers, just like MajorPrep found his answers. Well, maybe that isn't exactly a difference. It is said, "seek and you will find."

[@eleazararce5716]

When I got to time 4:20 I had to pause and recognize that this video has been more insightful than any other resources I've used to understand these concepts. I'm currently registered to take this class after many year of going through it. This time it will be more enjoyable! Please keep making great video like this!!!

[@RunOs3]

This is by far the best explanation I have ever had in my entire career as a math student.

[@Jocularious]

I'm a third year math major. And finally I know what a eigenvalue and eigenvector mean. Like doing the math, no worries. Asking to explain what it is... now that was another problem. Love the channel, keep the hard work up mate

[@Blox117]

"its that thing to do the other thing which gets you the correct answer..."

[@vittoriopiaser9233]

Jordan Turner you should definitely watch 3Blue1Brown playlist “Essence of Linear Algebra”, you’ll love it 👍🏻

[@gflow8357]

You really need to step up your game.

[@anothrto1045]

Ask a stranger what pi is, why it is, and why is it important

[@oliverclennan3449]

G Flow 👎👎👎👎👎

[@BharCode09]

Regarding you recommending 3B1B: Every time a youtuber recommends another youtube channel's video, purely for the sake of it's quality content and not bcz it's sponsored, the humanity is restored!

[@cassied9327]

Thank you for making this. I was having a hard time 'engaging' with matrices. I looked up the applications because I thought it would make the concept more interesting and exciting to learn. It helps a lot to know the applications/why we might want to do this.

[@james_pilgrim]

This is such a great video - thank you! I'm inspired to dig into linear algebra again, now decades after my college years.

[@ir2001]

Thank you for releasing Season 2 of 3Blue1Brown's essence of linear algebra. His videos included the visualisations of theoretical concepts and yours showcase the wonderful applications of matrices!

[@randomdude9135]

He should make a series on Group theory.

[@randomdude9135]

Like the one by Socratica.

[@randomdude9135]

Or Ring theory.

[@mayankraj2294]

.

[@mayankraj2294]

.

[@mra1685]

I never thought i would be this happy about understanding something i have learned without actually understanding it. Thank you for the taking the time to make such great videos. i think i found my new favorite channel :) .)

[@rayromanov]

I struggled so much with matrices in school, and then this guy makes sense of it in a single 25 minute video. Incredible.

[@eugeniures7593]

I also watched many of 3b1b videos, but your explanations regarding linear algebra are the best and the most intuitive I have ever seen. Amazing, excellent job!

[@primajump]

This is one of the best presentations that help to appreciate the value of matrices. The simple explanation of eigen vectors and eigen values was just brilliant. The speed was just prefect that I did not need to go back and forth. Thanks for posting this.

[@KeithRowley418]

best introduction to matrices I have ever seen. This sould have given me so much more initial insight an understanding when I was learning engineering math!

[@johnquinn6]

Zach, I am coming to the end of year 2 of my undergraduate degree in mathematics. In year 1 we did linear algebra and covered eigenvectors and eigenvalues, and in year two we are using said linear algebra in computational mathematics. I have paused this video less than 3 minutes in so I can leave this comment. In less than 3 minutes, you have explained more about the role of matrices, and more importantly, what eigenvectors and eigenvalues actually are. It is absolutely astonishing that I am paying so much money for an undergraduate course when there is better tuition to be had on youtube. I appreciate this video and I have subscribed to you channel.

[@LorenaGonzalez-fr1fm]

The first 5 minutes helped me to definitely understand what I’ve been doing for years without knowing the reason why. Big thanks!

[@earavichandran]

All my expression, reaction etc after watching this video is AWESOME. Unbelievable explanation. Hats off to you. I am now your big fan.

[@zachstar]

Edit: Made a part 2 to this video! kzbin.info/www/bejne/n2mppp6Bm7J-eas I'm honestly blown away by all the positive comments this video has gotten, so glad you guys enjoyed! I've gotten a lot of people asking for a part 2 so I'll definitely put that on my list of videos to get done. As I mentioned here, the video I did on the PageRank algorithm (kzbin.info/www/bejne/p6monIxuhLiWj7s ) is good follow up if you enjoyed this as it's all about matrices and goes into another application of the eigenvector. It was meant for this video but taken out due to how long it was. I'll work on making a script dedicated to more applications though! In the meantime let me know if there are any other topics you want me to discuss. Laplace transform and z transform videos coming soon!

[@HighestRank]

MajorPrep nix the colloquialisms.

[@borntodoit8744]

I'm about to watch this video purely because i did an Engineering degree which required us to learn matrix algebra. I loved it back then (30 years ago when i was a youngster). It is still all useful and relevant today ...see TensorFlow by Google (re: AI / Machine Learning / Neural Networks training). Maybe you can do a video on TensorFlow too? We referenced 2 Books at degree level (surprisingly well written) for anyone hoping to learn. Google them:- 'Engineering Mathematics' (see ISBN10 # 0333448871) 'Further Engineering Mathematics' by K A Stroud' (see ISBN10 #0333657411).... well worth having as a bookshelf resource. Useful when youre a child learning calculus, as an adult and when your a child again (pensioner) who doesnt want to do Suduko ;-)

[@bk-sl8ee]

U have no idea what u just did there!!! Sir. Our prof. Taught us to find eighen vectors and values but never told what fuck it is!!!

[@henryreese7236]

Hey, while using the zombie example you seemed to imply any vector will be rotated to a eigenvector after successive matrix operations. Is there any result which implies that? I searched for it but couldn't find any.

[@rs45888]

Oh my God.. this is so well explained.. this is the best video I checked on that topic.. i could never really understand this eigen thing until i saw this video.. thank you so much!!!!!

[@LightningStrike691]

As soon as you mentioned a changing system, I immediately knew you were going to talk about Markov Chains. I loved that section of my Linear Alg class.

[@jakem3043]

Your example on graph theory and matrix algebra was so enlightening and made me realize my professor was terrible. I was studying uniquely pan-cyclic graphs for my senior research project and had no clue what I was doing. This has helped me. Thanks!

[@RecursiveTriforce]

22:50 To those who want to know how the "divide by 6" was derived: As he said the Trace(A³) is just how many paths are between a person and themselves. But this little triangle can be walked through... 1. clock- and counterclockwise (2 Options) 2. starting from every vertex (3 Options) So you will count every triangle 6 times.

[@jonnamechange6854]

What's the point of this? Is it for setting up a ménage a trois, or some such group sex?

[@zachstar]

@@jonnamechange6854 Yeah in the dating app case it tells you how many genuine 'love triangles' or potential threesomes there are where everyone is happy.

[@jonnamechange6854]

Well done for making mathematics sexy. When I started learning matrices we didn't have the internet or advanced computers, so all these applications had not yet been created. I always asked the teacher what application there could be but he told me not to ask. Everything on the subject's syllabus had to be known and committed to memory without even the slightest explanation as to why.

[@goosecouple]

Doing things = Operation

[@nel_tu_]

The formula for squares is wrong. idk how you arrived at it.

[@tuskofgothos2637]

I absolutely love this video, and I am only halfway through! I have a fear of maths, but videos like yours give mathematical concepts much needed context and take the edge off that fear.

[@jwriccardi]

Wow, what a great piece of content! Very well thought out and superbly produced. I wish this had been around when I was in high school

[@000998poi]

You're a saint for explaining things so transparently and for giving real world case scenarios.

[@fnors2]

Good job making things clear and interesting without going into too much details. It was enjoyable to watch even as a Math person. Matrices are so useful, it's amazing. They show up almost everywhere in Math and Computer Science. You definitely learn to love them when working in those areas of science.

[@prabhatchanchal]

"As much as I’d love to keep going into depth on different subjects, this video is already quite long.” Sir ,It doesn't matter how much long the video. I can watch your video entire day long. please keep it.....

[@nishudahiya927]

Dear Zach, Thanks for providing us with such a valuable thing as when you get the feel of the topic you are much more interested in doing it

[@C.K.Productions]

I often feel very insecure about my math. My school district wasn’t the best, and by the time I got to high school everybody was trying to introduce common core but my teachers wouldn’t abide by the curriculum. Half the time they wouldn’t even show up. I remember I understood absolutely nothing on the math portion of the ACT. I’m almost done with my undergrad degree but math still haunts me. I still feel very inferior. Thank you for your videos

[@zeeehnbugattiisyeah8294]

I've been following this channel for a while now and I just wanted to give a huge thank you for raising my interest in math and its apllications even more and more! I'm starting university next week and you just motivated me so much for my linear algebra class. Greetings from Germany ;)

[@BangMaster96]

OMG.... I just took a course on Linear Algebra and Differential Equations, the whole time, i never understood what the hell are Eigenvectors and Eingenvalues, i just did the calculations using the given steps and formula, but never knew what it meant. Finally, all the dots are connected, it makes so much sense now. Thank You MajorPrep, this was so good and interesting to learn.

[@otrrsuns3t]

This was assigned to be watched in my University class. Writing a test tomorrow, might as well procrastinate productively ;)

[@nykhalil21]

I was vaguely familiar with the concepts but your explanation made it real for me. Good job.

[@alexlemarchant4376]

About a week ago I asked my teacher this exact question and she just brushed over it thank you

[@rameji-chan737]

Eigenvalue and Eigenvector both have the german word "eigen" in them, meaning "own". So "Eigenvector" means "A vector owned by the matrix". Makes it easier for me to remember what these two things are doing.

[@JonathanMandrake]

Yeah, as a german, these two words made much more sense when learning about them

[@WatDoino]

Man I hated those terms in my class. I understood absolutely nothing about any eigen crap. I also had a russian math teacher whose voice made me so sleepy.

[@fernandomoraescrescencio6428]

in Portuguese their called "autovetores" auto -own vetores -vectors

[@statisticallylaura]

Gosh, even as a German with that understanding, it took me a while to grasp their full meaning. Cannot imagine how abstract it must be to not know that from the start - never occurred to me that not every teacher brings this up immediately.

[@dylanparker130]

Loved this video, especially as I'm currently working on an image processing project! Thanks Zach!

[@charliegarrett5993]

This video is soooo good. Thank you for making it. Just about to start a computer science msc and this type of content really gets me excited for the maths!

[@barmimandal7466]

This is how we get interested to study more about the topic... When some videos like this open ur eyes to see the significance of the topic in reality. Thanks a lot!!🤗

[@Y45HV1N]

as a non mathematician who has been struggling to understand matrices, this has helped a lot ! thanks!!

[@chrisengland5523]

Superb. I've used matrices on a few occasions, but never really studied them beyond the basics. This video explains a lot that I was vaguely aware of, but never really got to grips with.

[@nganpham135]

this is the most amazing video of explaining linear algebra that I have ever watched. Thanks so much

[@JohnAlanWoods]

Super vid. After 4 years of engineering I didn’t understand Eigenvectors. 20 seconds here I do. 👍🏽

[@aman2426]

This blew my mind. I have a recommendation that you should have given a short summary of the video at the end. That would have made it more effective.

[@crimsonalchemist856]

Also, Zach matrix factorization has some really good applications in recommender systems. Thank you for your video. A very intuitive way to look at the square and cube of a matrix.

[@rthelionheart]

Linear Algebra is by far one of my favorite classes taken as an undergraduate student. Very interesting class.

[@oh-yt9ug]

I was having flashbacks to my linear algebra class watching this video

[@randomdude9135]

This was the fastest 12mins 47.5 secs in my life so far. Interesting video. This channel is pure gold.

[@kentlab3850]

I think you meant 12min37.5sec

[@randomdude9135]

@@kentlab3850 No. I had to rewind some parts so they took some extra time.

[@chintakgurjar8426]

I have taught these mathematical topics to students in college. You gave me new insight. Thank you 🙏

[@hassanyare9963]

i am planning to take a new course on image processing an this helped me a lot thank you so much for all these quality videos

[@CodicologiaUCM]

Thank you. I am taking a course on Linear Algebra and I really couldn't make any sense of what I am learning there. This video has been more helpful than many hours of study there, because now at least some things start to click.

[@richardkulling1432]

Wow this made me pumped to learn more Linear Algebra, much appreciated!

[@harshsharma5768]

Linear algebra came in my 2nd semester electrical engineering and i used to calculate eigenvalues and eigenvectors of matrix blindly without knowing why am i doing it. Now i came to know what they really mean and why do we get infinite number of eigenvectors for a particular eigenvalue. Please make more videos on these kinda stuff!😃

[@souravpaul2179]

Msc paas kar liya kab ka mujhe toh aaj bhi samajh nahi aaya...shaabash

[@andycopeland7051]

I followed a rabbit hole into this video, didn't think it would be very interesting, and was pleasantly surprised! This was interesting! I had no idea there were so many applications for matrices. Cool vid man keep it up

[@abhijitmajee6754]

Thank you, Sir. You are great. You completed my logic gap related to matrices. Now it makes much more sense.

[@douglasgorden3843]

I love the zombie example. I'm going to use that in my chemistry classes to introduce the topic of equilibrium.

[@itsrandeep]

If we had more teachers like him... The world would be different!!

[@preetammeena593]

We now have and just need to search the internet, where as it was really hard to find good teachers prior to the internet.

[@Meuduso1]

Oh, pretty cool. I just started Uni as a mechanical engineer and this is the first time I've had to do anything with matrices. I gotta say that I actually quite like them so far, though we haven't done much yet of course but I have been wondering where you even use them and now I trip over this video. Thanks Zach

[@srikuhere]

wow, this is how education should fell like . learnt matrices in school , did well in the exam however stayed matrices ignorant , now i am educated in matrices . your video will help me in my data science education . thanks man

[@thatguyben7754]

I knew a good amount of this, but that one about the dating site was really cool and interesting. It’s one of those things if I know where they’re used, just not how

[@brunojani7968]

I share this opinion. I knew about adjecency matrices and A^k showed me the nr of paths of length k. But the application in terms of the dating app was a real treat.

[@arpanchatterjee2596]

Glad someone made this stuff, the title too included in the delight.

[@TheOzpad]

No, Zach I want to learn MORE about how matrices are used so please do a longer video! The network and image processing examples BLEW MY MIND

[@sgringo]

That was a great explanation. I learned about matrices in a university finite math course. I learned the material, but like you said, I was left with absolutely no context for how/when/why to use them. I acquired that context a few semesters later, during a computer graphics course assignment to create a "spaceship" (really just a simple triangle with three vertices) that flew in a large circle on the screen. The challenge was to ensure that, as the ship moved around the circle, it reoriented itself correctly, to always be pointing in the direction of travel. This was accomplished by multiplying x,y for the three vertices through a hardcoded 2x2 matrix to get the next set. (The matrix was more complex than the example you provided; it utilized trigonometric functions). That assignment was the only context I ever had for matrix operations... until now! 👍

[@SM-qk7jv]

I love these kinds of videos. You're awesome, MajorPrep. Keep up the great work. 🙂👍

[@Vyantri]

If I was a lecturer, I would play this video for my students during their first linear algebra class.

[@sharmisai]

This is one of the, if not THE best explanation of eigenvalue and eigenvector I have come across.

[@omniscienceisdead8837]

your video in only 4 minutes has put so many things into perspective, Bravo!

[@kmax3339]

I wish this channel existed during my student years. It really would have made a difference.

[@becomepostal]

I’m not sure it’s enough.

[@retiredmeme2751]

that was the most easiest-to-understand definition of eigenvactors/values that no other resource could provide

[@hrivera4201]

This one is equal or better: kzbin.info/aero/PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab

[@colfaxschuyler3675]

I wish this has been around in high school. It would have helped me tremendously, and made learning much easier, seeing the simplified explanation and the practical uses.

[@calaious]

Amazing video, please make a series out of this video. I really like the linear algebra approach to neural networks.

[@roygalaasen]

When I saw that 20% of humans turns to zombies, and 10% of zombies turn to humans, I thought to myself that it couldn’t be so simple as to be twice as many zombies as humans. Alas... (This is a great repetition video on linear algebra!)

[@parrotraiser6541]

The one thing that is missing from all the "Introduction to Matrices" that I've tried to follow is a simple. real world example to use as a mnemonic and check for basic operations. Converting a realistic example to matrix form would ground it in memory, and knowing the answer would serve as check when calculating.

[@lakshmits4675]

I generally don't comment on videos. But this explanation is so awesome, i have to write back to appreciate this effort. None of the univ courses have taught it so intuitively. Thanks a lot for the effort 🙏🙏

[@M3t4lik]

Thank you for this great video! It helps to visualise the concepts, components and operations of matrices in a way as to show clarity of how they work and as to how it is such an amazing mathematical tool for just about anything one can think of. Bravo.