Linear Algebra - Matrix Transformations

Matrix multiplication and linear algebra explained with 3D animations.

[@EugeneKhutoryansky]

To see subtitles in other languages: Click on the gear symbol under the video, then click on "subtitles." Then select the language (You may need to scroll up and down to see all the languages available). --To change subtitle appearance: Scroll to the top of the language selection window and click "options." In the options window you can, for example, choose a different font color and background color, and set the "background opacity" to 100% to help make the subtitles more readable. --To turn the subtitles "on" or "off" altogether: Click the "CC" button under the video. --If you believe that the translation in the subtitles can be improved, please send me an email.

[@Tiago211287]

If I had these videos available since my childhood, I would never ever thought I am stupid with math. These explanations were exactly what I missed.

[@Feynstein100]

+Tiago Bruno Castro Seems like your English isn't that good either.

[@Tiago211287]

+Feynstein100 absolutely

[@pendalink]

+Feynstein100 that can happen when Portuguese is your native tongue

[@Feynstein100]

+pendalink Where did you come from? And don't give me that nonsense. English isn't my native tongue either but my English is reasonably good.

[@pendalink]

***** You having good English doesn't mean you can assume anyone will have good English. If you're going to come out of nowhere to call a guy out on his English, then you can expect someone else coming in to call you out on your unjustified assumptions. Welcome to public commenting

[@RangaNanayakkara]

Man with people like you we can start by teaching calculus to primary students. Great Work !

[@haristydes1166]

I dream it too

[@seandafny]

The care has to be there

[@ssn8]

🙋 same here .

[@Anonymous-ui1mo]

Me too

[@EugeneKhutoryansky]

I just created a Patreon account and added the link to my KZbin home page. Also, in case, you have not already seen them, I uploaded several other videos recently. As always, for each video that you like, you can help more people find it in their KZbin search engine by clicking the like button, and writing a comment. Lots more videos are coming very soon. Thanks.

[@afederdk]

Yeah, please put it in video descriptions or I will forget to click.

[@geoffreywekesa6206]

Asante sana, Your graphics + explanations are Supa!

[@anshulsharma5756]

didn't understand the stretch and flip part. it was stretched only in z direction, while in x and x direction it was just flipped. explain please

[@lumek4513]

what is there to understand? A transformation that stretches in z and flips y and z in certain way

[@vinsavi]

Anshul sometimes in this video even mult by 1 is called stretch. it is not . Only when 1 0 0 becomes 0 -1 0 then x flips to -y . But still this is a video in the right direction.

[@napomokoetle]

You must be the best teacher I've ever encountered! You must be God sent... Thank you!

[@EugeneKhutoryansky]

+Napo Mokoetle, thanks for that really great compliment.

[@EugeneKhutoryansky]

You can help translate this video by adding subtitles in other languages. To add a translation, click on the following link: kzbin.info_video?ref=share&v=IrggOvOSZr4 You will then be able to add translations for all the subtitles. You will also be able to provide a translation for the title of the video. Please remember to hit the submit button for both the title and for the subtitles, as they are submitted separately. Details about adding translations is available at support.google.com/youtube/answer/6054623?hl=en Thanks.

[@malm7arb]

where was this channel when i was in school 😭 . i used to solve these equations without even understanding them , i just learn the rules and how to multiply them and went to the exams, but the visua representation makes it so easy to understand . I was taught to do it like this but i didn't know why should i do it like this, but now i know why

[@EugeneKhutoryansky]

+AL_ muharb, I am glad to hear that my visuals make it easy to understand. Although these videos were not available when you were in school, hopefully this will make life easier for the next generation of students. Thanks.

[@fmartin59]

+Eugene Khutoryansky I had the same problem in my Algebra 2 class. My teacher just taught procedures; he didn't even bother explaining what we were doing. I'm glad you're here.

[@ull893]

You are not alone. There are millions of students and even teachers and professors and engineers who had no idea why they were doing it the way they were doing ???!!! But you are the first one to honestly admit that fact. The only difference between you and them is that you are honest and frank about it. Good luck with your studies !!

[@TheMIRO1962]

The same case was with me.I was in fact good in solving this equations but had no idea what was it about. :)

[@shipwreck9146]

Late comment: I'm a physics major, and I think a lot of people don't realize that literally all math can be visualized in some way. I'm in my third time taking linear algebra right now. First time I dropped it. Second time I realized why I was doing so bad; linear algebra is taught completely wrong, but by the time I realized that, it was too late and I failed it. This time around, I have a 98 in the class, everything makes sense now because I've had the time to teach it all to myself in a way that makes more sense. Anyone in the class can tell you a generic procedure, but almost none of the students can tell you about why they're doing what they're doing. It's just abstract math to them. This is why I think there needs to be 2 linear algebra classes. One specific for visual learners, the other specific for non-visual learners.

[@starwarsjk99]

Your channel is 100 times better than Khan Academy! You actually explain Matrix transformations and one way they can be derived. Channels like Khan Academy spoon feed you information like multiply this number by that number and this number by that number. They don't tell you why you should do that. Your video on Linear Algebra demonstrates WHY we multiply matrices by vectors in a certain way. Before I saw this I thought Matrices were something weird we just made up that did not behave like normal mathematics. After watching this I realise that matrices are truly a part of mathematics and they are made to solve real mathematical problems, not made up as some weird form of arithmetic. If all of youtube was deleted and I could only choose to save one channel, yours would be the one I keep. Good JOB!

[@EugeneKhutoryansky]

+jk991234, thanks for that really great compliment. I am glad that you like my videos that much.

[@mdmobashshir432]

+jk991234 Khan academy made my life. It was the stepping stone, I don't have to be a stupid anymore to my classmates. These Eugene videos are awesome but they shed light on the same topic from different perspective.

[@vinsavi]

True but Khan academy was the beginning of the democratization of education.

[@ravirajh6658]

khan academy does a good job on conceptually explaining things. It had never occurred to me as spoon-feeding

[@michaelrosen4916]

jk991234 Khan Academy actually has many videos on proofs and is good for people that actually want to learn something unlike other channels like vsauce.

[@ArpanD]

What I love about this channel is its unique and diverse content. Like 3Blue1Brown focuses only on math, but in Professor Khutoryansky's channel you will get physics, math, philosophy, engineering etc.., all visualized in an epic manner. This channel is a revolution, it really is!

[@ambientscience2951]

true

[@slavakotelnikov2440]

Eugene, what a fabulous explanation. I have been struggling (with no luck) for about 2 months to understand how the transformation matrix is built, but then I saw your video and it all became crystal clear to me. Please continue with matrix algebra, you definitely have got the right talent to do teaching!

[@EugeneKhutoryansky]

Thanks. Glad to hear that my video was helpful.

[@mypoostank775]

Your videos are so helpful! Why doesn't my teacher just show us your videos!! A confusing week's worth of class was simplified in this 20 minute video. The creator of this is a genius and should be recognized more! Thanks for these awesome videos!

[@EugeneKhutoryansky]

+mypoostank, thanks for the compliment. I am glad that you like my videos.

[@Alexandre152]

My favorite teacher!!! You made me enjoy the Universe in a very unique way. Thank you!

[@SinanAkkoyun]

For everybody struggling with why rotations are not linear but still are a linear transformation: You do not rotate a vector on a fundamental level. With a rotation matrix, you describe a non-linear function (sin/cos) OUTSIDE the transformation. The actual _linear_ transformation is linear. You compute the amount of linear cominations required for something that acts like a rotation in advance, the act itself _is still linear_ What I am saying is, that when you apply a rotation matrix (e.g. 30° around x axis) to a vector, it snaps the vector to the 30° position. IT DOES NOT ROTATE inbetween when only applying one 30° rotation matrix once! A vector can make any sorts of non-linear movements, but these are just non-linear because you compute the matrix for each step outside with non-linear inputs (This is just a complementary explanation, the video is right in every way and is awesome!)

[@benjaminasmad9806]

Great!! During my college days, I had a hard time of understanding the concepts of matrices and vector space, now, you make it simpler and understandable! Thank you so much!

[@EugeneKhutoryansky]

Thanks.

[@albertabd3206]

Я влюблён в вашу великолепную графику. Продолжайте!

[@tarikbadi1837]

This is the best video I have ever seen on youtube.

[@EugeneKhutoryansky]

Thanks for that really great compliment about my video.

[@bradleycollings8176]

Such a fantastic resource as always! Thank you and may more people find your channel.

[@EugeneKhutoryansky]

Thanks.

[@kaikleinbard7544]

i love how the video takes it's time to show you the ideas! that helps me grasp it. Thanks!

[@dhavaldeshmukh3653]

Its the greatest channel that has everything. Physics , electronics ,maths

[@EugeneKhutoryansky]

Thanks for the compliment.

[@orozcoapaza1660]

Amazing channel, I wanted to remember Linear Algebra but didn't find my books and ended up here :) , thanks for creating this kind of amazing content...

[@winterclimber7520]

Amazing. Truly amazing. Thank you for making this.

[@EugeneKhutoryansky]

Thanks. Glad you liked my video.

[@metcas]

To be quite honest, I was initially taken aback when I stumbled upon your channel and its content. It was a little strange at first (the animation, primarily,) but I've come to really, really enjoy your videos. Thank you very much for the extra clarification on many subjects. It really helps fill in the blanks for me. It is in such straight forward manner as well. Bravo.

[@EugeneKhutoryansky]

+Metcaslix, I am glad to hear that you enjoy my videos and that they help clarify the subjects. Thanks.

[@__rikaisuru]

I love these videos. A 3D visual representation is so much more intuitive.

[@EugeneKhutoryansky]

Thanks.

[@kuhataparunks]

This is done with astounding elegance and beauty. This channel is one of the few creations in existence which “perfect” accurately describes without any exaggeration or hyperbole.

[@EugeneKhutoryansky]

Thanks for the compliments.

[@VaughanRussell]

Please do a video on string theory. I love your videos!

[@Trackman2007]

Fantastic explanation! Thank you! Finally i now understand the most fundamental thing - what stands behind the matrix concept.

[@EugeneKhutoryansky]

+Trackman2007, thanks for the compliment, and I am glad that you found my video useful.

[@joshevans4322]

These videos are great! Very well done and the explanations are clear and easy to understand. Thanks. I hope you continue to make many more.

[@EugeneKhutoryansky]

+Josh Evans, thanks for the compliment. Many more videos are on their way.

[@Mysoi123]

Eugene, you're my best math teacher since I got expelled from school! Thank you for all the good content you have made!

[@fortoday04]

Can you make a video about eigen values and vectors please? I need an in depth understanding. Maybe, matrix rank, vector spaces, etc. Linear algebra is really interesting!

[@EugeneKhutoryansky]

Yes, that is on my list of topics for future videos. Thanks.

[@mostafahaggag9876]

Eugene i am obsessed with your videos, since that video about theory of relativity and i'm just ridiculously following every video you upload. can't wait to view your upcoming stuff. Thank you, you really helped me and grew my mind really thank you.

[@EugeneKhutoryansky]

+jack hardy, I am glad to hear that you like my videos that much, and that you have found them helpful. Lots more videos are on their way.

[@raedsheyyab915]

fantastic , you can use similar ideas to representing the operators of rotation and transition in quantum mechanics

[@EugeneKhutoryansky]

+raed sheyyab Thanks. Glad you liked the video. I plan on going into more detail on Quantum Mechanics in future videos.

[@JimmyDeLock]

Thanks for your help again Eugene!

[@ericosmostech4903]

Everytime I watch these videos, my jaw drops with Wonder & Enthusiasm... Incredibly amazing this channel is! ♥

[@EugeneKhutoryansky]

Thanks. I am glad you like my videos.

[@ericosmostech4903]

@@EugeneKhutoryansky I wish If I could see who are you guys making this incredibly valuable videos! & extensive research on every topics! I wish if I could participate however but I will do pray that this journey never ends with you!

[@yonghengwang3603]

This is beautiful! I watched many videos. This is the one I am missing! Thank you very much for your effort to explain it!

[@EugeneKhutoryansky]

Thanks for the compliment. I am glad you liked my video.

[@mak741]

An enormous thank you for all these lucid illustrations of abstract concepts. Could you make a video on Laplace and Z transforms too?

[@eliaswrwf5826]

I never would've thought linear algebra would be that easy, too bad i'm going to accountancy after bachillery but I guess your videos are so good and clear that they make everyone want to go to engineering.

[@adhavankadhiravan4132]

OMG!!! For such a long time I didn't understand what matrix transformations are. Now I do. Thank you so much Eugene.

[@EugeneKhutoryansky]

Glad my video was helpful. Thanks.

[@ayushshah4649]

this is what i missed during my engineering....

[@mataiasagm]

great video, and Videos, loved the Animations

[@JordanEdmundsEECS]

As is typical of your videos, this one was excellent. Thank you.

[@EugeneKhutoryansky]

+Jordan Louis Edmunds, glad you liked it. Thanks.

[@harikrishnanchandramohan4209]

Really amazing explanation. Never believed I would understand the crux of these concepts. Please do videos on Computational Fluid Dynamics.

[@EugeneKhutoryansky]

Thanks for the compliment. I'll add Computational Fluid Dynamics to my list of topics for future videos.

[@quantaali543]

Amazing explanation..... even better than 3blue1brown....these 2 channels are the best channels for understanding the hidden philosophy behind any concept......

[@EugeneKhutoryansky]

Thanks for the compliment.

[@hisxmark]

This is much better than a lecturer with chalk and blackboard. Had my math teachers been able to take this approach, I might not have dropped out of high school.

[@jamirahamaahmed5197]

BEAUTIFUL!

[@ElanaSl]

wow! thank you! its so clear! and I like the study music in the background!

[@pallavi_chandaka9480]

Words can't describe how grateful I'm ... Your videos had taken me to see mathematics in a different perspective... Thank youuu sooo muchhh Eugene,🌸

[@EugeneKhutoryansky]

Thanks. I am glad you like my videos.

[@anandsinha9692]

This is correct way of teaching. Best teacher in the world.

[@EugeneKhutoryansky]

Thanks for the compliment.

[@NaeemKhan-px8mc]

Simply amazing as it should be,this channel makes maths so beautiful ❤

[@EugeneKhutoryansky]

Thanks for the compliment.

[@azapent]

one of the best and most useful channel i found in youtube thank you!

[@EugeneKhutoryansky]

Thanks for the compliment.

[@Cosmalano]

Awesome video. Do you plan to make a video about tensors in the future?

[@EugeneKhutoryansky]

+electrocat1, thanks for the compliment. Yes, tensors are on my list of videos that I would eventually like to make. Thanks.

[@Cosmalano]

Eugene Khutoryansky great, I will be waiting!

[@guitarheroprince123]

+Eugene Khutoryansky yay I want them for gr!

[@rickandelon9374]

Please make them Eugene i will recommend your channel to all my class we are really having trouble with tensors i have already suggested them this channel but they say that they want tensors please upload it within july we are having test i would be really grateful to you although I don't have money to support you on patreon i know my friends are obviously going to support you. Thx!!!!!!!

[@ishworshrestha3559]

Ok

[@vkila8788]

Best ever matrix video....understood really well. Thank you so much and keep making your amazing videos

[@EugeneKhutoryansky]

+V Taylor, thanks. More videos are on their way.

[@davishilone1922]

You are a God sent blessing to students! I thank you from the bottom of my heart.

[@EugeneKhutoryansky]

Glad you like my videos.

[@markostojiljkovic7100]

Everything is easier with visual representations!!! Great video.

[@EugeneKhutoryansky]

+Marko Stojiljkovicm, thanks. I am glad you liked my video.

[@mansursgmn]

Thank you for help me understanding this things for real. I think this kind of videos are the future of education and you are really good at it. Your videos will help me a lot in the exam this year in Turkey.

[@EugeneKhutoryansky]

+Mansur Segmen, thanks for that really great compliment, and I hope you do well on your exam. Thanks.

[@twlson49]

wow... what a badass video... excellent eye opener

[@EugeneKhutoryansky]

Glad you liked my video. Thanks.

[@zeeshanrahman481]

man you need get biggest award in education the way you explained is exceptional never in my life i had seen such unique and crystal clear way of teaching

[@EugeneKhutoryansky]

Thanks for the compliment.

[@1DR31N]

How beautiful videos, this is an art masterpiece. When I was a student I had to imagine all of this mathematical representations in my mind. You made it real. Adrian

[@EugeneKhutoryansky]

Thanks for the compliment about my videos.

[@kyoung1686]

Awesome. Thank you!

[@Haroldolino]

It's really a worthwhile video! Helped a lot! Thanks for sharing 🙏

[@EugeneKhutoryansky]

Thanks. I am glad my video was helpful.

[@astropike]

More videos about linear algebra, please. It's so hard to understand it without good examples

[@curtpiazza1688]

Excellent presentation! Thanx!

[@EugeneKhutoryansky]

Thanks.

[@bruno_sjc_]

you rock! You have a big fan down here! Keep rocking!

[@EugeneKhutoryansky]

Thanks. I am glad you like my videos.

[@sweethome3938]

Very Helpful! Thank you! :)

[@a.p.ketchum8033]

Thanks so much for this!

[@ling6701]

It was enlightning for me, thanks.

[@karllaurenzg.diamante1701]

This video is a godsend.

[@EugeneKhutoryansky]

Thanks for the compliment. I am glad my video is helpful.

[@moroccanrapnightcore716]

its just soooo beautiful

[@EugeneKhutoryansky]

Thanks for the compliment.

[@Masardirasa]

Nice video . Keep going on this road we love u

[@lorenzoemanueletomasello215]

Excellent video!

[@EugeneKhutoryansky]

+Lorenzo Tomasello, Thanks. I am glad you liked it.

[@DlcEnergy]

4:56 to 5:24 this was the eureka moment for me. hands on my face in joy at how simple it is to understand matrices now. and yes, i watched the whole video. 17:33 the final example just to show a regular vector being transformed with the vector translations of the matrix. now all that built in opengl glRotatef(degrees, x, y, z); glScalef(x,y,z); makes sense. they use a 4x4 matrix so they can add a final column with an x,y,z to translate the output vector. so i'd guess you'd only need a 4x3 matrix. but they'd prefer using perfect square.

[@EugeneKhutoryansky]

Glad you liked my video. Thanks.

[@H0peAnL0ve]

You have the Best lessons on the web

[@EugeneKhutoryansky]

Thanks for the compliment.

[@dipankarbanerjee1130]

Actually what I was thriving for.... Thanks ma'am / sir ...

[@EugeneKhutoryansky]

Thanks.

[@bharathreddygudibandi492]

simply............. it is superb explanation!!!!!!!!

[@EugeneKhutoryansky]

+bharathreddy gudibandi, thanks.

[@khn.majid1212]

I love the videos. And expect more topics of maths to be cover in such visualization. It helps me to see what maths is. It benefited me a lot in my teaching and studies.A famous quote perfectly fits here (And what is the use of a book," thought Alice, "without pictures or conversation?). I mean maths books lack pictures and conversation.

[@rgl8109]

Eugen your videos are amazing!!

[@EugeneKhutoryansky]

Thanks for the compliment. I am glad that you like my videos.

[@hansenw9366]

I really like this video,thank you.

[@EugeneKhutoryansky]

Thanks.

[@suleshangodoyin]

With many thanks for introduction of Matrix, as a branch of Mathematics! It's me, Sule Shangodoyin.

[@fariha3925]

Sule Shangodoyin

[@Yoooomanwhatsup]

6 years later. Still a masterpiece

[@EugeneKhutoryansky]

Thanks for the compliment.

[@JamesManjackalMSFS]

Maybe you would mention a property of linear transformations which is alineated points are conservated after a linear transformation. I know that talking about "points" is to deal with afin spaces but is very easy to associate a vectorial space to an afin space. You could show a video with an object changing...you don't know the equations of the transformation, but you can say it is a linear transformation if you see that the alineation of points are preserved! this is why these transformations are call "linear"... just because they preserve lines! Congratulations for your work!

[@mindovermath2042]

I found you through the suggested videos. Your videos are vey informative! I just made a KZbin Channel due to your inspiration. Keep up the Videos! Just Subbed!

[@kwgm8578]

Hello Eugene! I love your animations, and I have a suggestion to improve the clarity of the addition and multiplication by unit vectors section. When you dissolve the column(s) of multiplicands into their scalar values, the new student will be puzzled about as to the source of each value, especially when you use two zero values in the initial column. This doesn't become clear until you have an example with unique values around minute 16:XX or so. What might help is to place the scalar at its source column, and have your fine narrator, Carol(?), describe how to distribute each addition or multiplication operation within the matrix, or its column vector representation. I hope that I've been clear! I know that I could do so, quickly with a pencil and paper while the two of us sitting together. By the way, you two make a good team! I'm grateful for your ability to show me each process, visually, which can be very useful to all of us who can use multiple communications methods to understand a complex scientific concept. My visualization skills were poor to begin with, and often I relied on the mathematics to prove a concept. When I was made to teach in graduate school, I was forced to ask my graphic artist friends to help, and often hired the work done. Later in my own start-up consulting, business my partner and I realized early that a good graphics department was essential for success. Anyway, I realize that your work is essential to so many students just beginning to learn their craft in Engineering, the Sciences, and pure Mathematical fields, and for that, I thank you. 🙏🏼

[@nash-aliyyi6948]

It's Excellent to visualize

[@jatinsachdeva2342]

This video is just awesome .....Must watch it guys...

[@Enigma4you405]

Excellent video for visual learners and a refresher on matrices! Please continue making all sorts of math related videos. Also can you please add in your description any music you use? I find it very soothing.

[@EugeneKhutoryansky]

Thanks for the compliment. All the music in this video is from the free KZbin audio library, and the names of the songs are the following. Wigs Clouds Horses_to_Water Mast C_Major_Prelude

[@peterrenhult3822]

Exelent Voice.. Exelent Work .. Exelent exlained ... !

[@kamalkamal0123]

Thanks alot.. this is amazing..

[@diegomartinez2414]

Love your videos 😄

[@EugeneKhutoryansky]

+Diego Martinez, Thanks. I am glad to hear that you love my videos.

[@noem1977]

you are legend sir. please keep going.

[@EugeneKhutoryansky]

+noem1977, thanks for the complicment. Lots more videos are on their way.

[@siqbal988]

Educational videos is a great work, plz more

[@EugeneKhutoryansky]

Thanks. More videos are on their way.

[@dan1608]

thanks for make the videos ! are amazing!

[@EugeneKhutoryansky]

+Daniel Cortes Diaz, I am glad that you like them. Thanks.

[@yarooborkowski5999]

Great. Maybe You can try to show other spaces and other reference frames? By the way.... What kind of program do You use? Best regards

[@zaneaussie]

Wow very bizarre. Not sure my brain can digest all that fully but great video and love the background music! I sure missed some math classes lol

[@ozzymandius666]

I have recently stumbled upon your work, and I am impressed. I have subscribed, and I click "like" on each one I watch. If I my be so bold as to make a request, is a couple of videos say starting with curvilinear coordinate systems and then going on to tensors possible in the future? Thanks again.

[@EugeneKhutoryansky]

+Michael Bishop, thanks for the compliment on my videos, and I am glad to have you as a subscriber. Curvilinear coordinate systems and tensors are on my list of topics for future videos. Thanks.

[@sousahugo1000]

Great job.

[@EugeneKhutoryansky]

+Hugo Sousa, thanks.

[@LonglongGuitar]

OMG, your videos are so informative, amazing!!!!!!!!!!

[@EugeneKhutoryansky]

Thanks.

[@DANIELALEJANDROROMERONAVARRO]

I love this videos.

[@EugeneKhutoryansky]

Thanks!

[@NeuralEngin33r]

Please do a video on eigenvectors and eigenvalues. Great set of videos--thank you.

[@EugeneKhutoryansky]

+NeuralEngin33r, eigenvectors and eigenvalues are on my list of topics for future videos. Thanks.

[@ankurkumar9660]

Whenever I'm not understand something , I come before you❤️❤️❤️❤️....for such a nice explanation euginee 🥰🥰🥰

[@EugeneKhutoryansky]

Thanks. I am glad you like my explanations.

[@ricardocesargomes7274]

Espetacular!

[@EugeneKhutoryansky]

Thank you.

[@kirpal7709]

Excellent work

[@EugeneKhutoryansky]

Thanks for the compliment.