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.
***** 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
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
+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.
+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.
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 !!
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.
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!
+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.
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.
I just created a Patreon account and added the link to my CZcams 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 CZcams search engine by clicking the like button, and writing a comment. Lots more videos are coming very soon. Thanks.
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.
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!
You can help translate this video by adding subtitles in other languages. To add a translation, click on the following link: czcams.com/users/timedtext_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.
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!
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!
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!
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...
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!)
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!
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.
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 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!
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.
Amazing explanation..... even better than 3blue1brown....these 2 channels are the best channels for understanding the hidden philosophy behind any concept......
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.
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.
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!!!!!!!
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.
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
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
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.
I found you through the suggested videos. Your videos are vey informative! I just made a CZcams Channel due to your inspiration. Keep up the Videos! Just Subbed!
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.
Thanks for the compliment. All the music in this video is from the free CZcams audio library, and the names of the songs are the following. Wigs Clouds Horses_to_Water Mast C_Major_Prelude
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!
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.
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.
+Tiago Bruno Castro Seems like your English isn't that good either.
+Feynstein100 absolutely
+Feynstein100 that can happen when Portuguese is your native tongue
+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.
***** 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
Man with people like you we can start by teaching calculus to primary students. Great Work !
I dream it too
The care has to be there
🙋 same here .
Me too
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
+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.
+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.
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 !!
The same case was with me.I was in fact good in solving this equations but had no idea what was it about. :)
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.
You must be the best teacher I've ever encountered! You must be God sent... Thank you!
+Napo Mokoetle, thanks for that really great compliment.
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!
+jk991234, thanks for that really great compliment. I am glad that you like my videos that much.
+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.
True but Khan academy was the beginning of the democratization of education.
khan academy does a good job on conceptually explaining things. It had never occurred to me as spoon-feeding
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.
I just created a Patreon account and added the link to my CZcams 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 CZcams search engine by clicking the like button, and writing a comment. Lots more videos are coming very soon. Thanks.
Yeah, please put it in video descriptions or I will forget to click.
Asante sana, Your graphics + explanations are Supa!
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
what is there to understand? A transformation that stretches in z and flips y and z in certain way
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.
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!
true
You can help translate this video by adding subtitles in other languages. To add a translation, click on the following link:
czcams.com/users/timedtext_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.
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!
Thanks. Glad to hear that my video was helpful.
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!
+mypoostank, thanks for the compliment. I am glad that you like my videos.
My favorite teacher!!! You made me enjoy the Universe in a very unique way. Thank you!
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!
Yes, that is on my list of topics for future videos. Thanks.
Я влюблён в вашу великолепную графику. Продолжайте!
Such a fantastic resource as always! Thank you and may more people find your channel.
Thanks.
Fantastic explanation! Thank you! Finally i now understand the most fundamental thing - what stands behind the matrix concept.
+Trackman2007, thanks for the compliment, and I am glad that you found my video useful.
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...
i love how the video takes it's time to show you the ideas! that helps me grasp it. Thanks!
This is the best video I have ever seen on youtube.
Thanks for that really great compliment about my video.
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!)
Amazing. Truly amazing. Thank you for making this.
Thanks. Glad you liked my video.
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!
Thanks.
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.
+Metcaslix, I am glad to hear that you enjoy my videos and that they help clarify the subjects. Thanks.
Eugene, you're my best math teacher since I got expelled from school!
Thank you for all the good content you have made!
Thanks for your help again Eugene!
fantastic , you can use similar ideas to representing the operators of rotation and transition in quantum mechanics
+raed sheyyab Thanks. Glad you liked the video. I plan on going into more detail on Quantum Mechanics in future videos.
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.
+Josh Evans, thanks for the compliment. Many more videos are on their way.
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.
+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.
I love these videos. A 3D visual representation is so much more intuitive.
Thanks.
Everytime I watch these videos, my jaw drops with Wonder & Enthusiasm... Incredibly amazing this channel is! ♥
Thanks. I am glad you like my videos.
@@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!
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.
Thanks for the compliments.
Please do a video on string theory. I love your videos!
This is beautiful! I watched many videos. This is the one I am missing! Thank you very much for your effort to explain it!
Thanks for the compliment. I am glad you liked my video.
Amazing explanation..... even better than 3blue1brown....these 2 channels are the best channels for understanding the hidden philosophy behind any concept......
Thanks for the compliment.
Really amazing explanation. Never believed I would understand the crux of these concepts. Please do videos on Computational Fluid Dynamics.
Thanks for the compliment. I'll add Computational Fluid Dynamics to my list of topics for future videos.
An enormous thank you for all these lucid illustrations of abstract concepts. Could you make a video on Laplace and Z transforms too?
It's really a worthwhile video! Helped a lot! Thanks for sharing 🙏
Thanks. I am glad my video was helpful.
Best ever matrix video....understood really well. Thank you so much and keep making your amazing videos
+V Taylor, thanks. More videos are on their way.
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,🌸
Thanks. I am glad you like my videos.
BEAUTIFUL!
Everything is easier with visual representations!!! Great video.
+Marko Stojiljkovicm, thanks. I am glad you liked my video.
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.
OMG!!! For such a long time I didn't understand what matrix transformations are. Now I do. Thank you so much Eugene.
Glad my video was helpful. Thanks.
great video, and Videos, loved the Animations
Its the greatest channel that has everything. Physics , electronics ,maths
Thanks for the compliment.
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.
As is typical of your videos, this one was excellent. Thank you.
+Jordan Louis Edmunds, glad you liked it. Thanks.
Simply amazing as it should be,this channel makes maths so beautiful ❤
Thanks for the compliment.
With many thanks for introduction of Matrix, as a branch of Mathematics!
It's me,
Sule Shangodoyin.
Sule Shangodoyin
this is what i missed during my engineering....
Excellent video!
+Lorenzo Tomasello, Thanks. I am glad you liked it.
Awesome video. Do you plan to make a video about tensors in the future?
+electrocat1, thanks for the compliment. Yes, tensors are on my list of videos that I would eventually like to make. Thanks.
Eugene Khutoryansky great, I will be waiting!
+Eugene Khutoryansky yay I want them for gr!
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!!!!!!!
Ok
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.
+Mansur Segmen, thanks for that really great compliment, and I hope you do well on your exam. Thanks.
Awesome. Thank you!
one of the best and most useful channel i found in youtube thank you!
Thanks for the compliment.
wow! thank you! its so clear! and I like the study music in the background!
You are a God sent blessing to students! I thank you from the bottom of my heart.
Glad you like my videos.
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
Thanks for the compliment.
More videos about linear algebra, please. It's so hard to understand it without good examples
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
Thanks for the compliment about my videos.
Thanks so much for this!
Great job.
+Hugo Sousa, thanks.
Nice video . Keep going on this road we love u
you rock!
You have a big fan down here!
Keep rocking!
Thanks. I am glad you like my videos.
you are legend sir. please keep going.
+noem1977, thanks for the complicment. Lots more videos are on their way.
Excellent presentation! Thanx!
Thanks.
It was enlightning for me, thanks.
Very Helpful! Thank you! :)
wow... what a badass video... excellent eye opener
Glad you liked my video. Thanks.
Exelent Voice.. Exelent Work .. Exelent exlained ... !
This is correct way of teaching. Best teacher in the world.
Thanks for the compliment.
Superb.. I am a teacher and often face doubts on this from students.. Thank you Mr. Khutoryansky for making the less accessible concepts so clear :)
Thanks.
Educational videos is a great work, plz more
Thanks. More videos are on their way.
It's Excellent to visualize
Eugen your videos are amazing!!
Thanks for the compliment. I am glad that you like my videos.
OMG, your videos are so informative, amazing!!!!!!!!!!
Thanks.
Thanks alot.. this is amazing..
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.
Glad you liked my video. Thanks.
thanks for make the videos ! are amazing!
+Daniel Cortes Diaz, I am glad that you like them. Thanks.
This video is just awesome .....Must watch it guys...
Please do a video on eigenvectors and eigenvalues. Great set of videos--thank you.
+NeuralEngin33r, eigenvectors and eigenvalues are on my list of topics for future videos. Thanks.
I found you through the suggested videos. Your videos are vey informative! I just made a CZcams Channel due to your inspiration. Keep up the Videos! Just Subbed!
Excellent work
Thanks for the compliment.
You have the Best lessons on the web
Thanks for the compliment.
Love your videos 😄
+Diego Martinez, Thanks. I am glad to hear that you love my videos.
simply............. it is superb explanation!!!!!!!!
+bharathreddy gudibandi, thanks.
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.
Thanks for the compliment. All the music in this video is from the free CZcams audio library, and the names of the songs are the following.
Wigs
Clouds
Horses_to_Water
Mast
C_Major_Prelude
its just soooo beautiful
Thanks for the compliment.
THANKS, IT'S REALLY GOOD
+楊漢軒, glad you liked it.
i have no words for thanking you , best experimental Explanation thnksu :) :)
Thanks for the compliment.
Physics Videos by Eugene Khutoryansky Welcome
best of the best
Super video!
[by
I really like this video,thank you.
Thanks.
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!
Actually what I was thriving for.... Thanks ma'am / sir ...
Thanks.
very very thankful to you for your reply
Great introduction to matrix transformaions
+Bart Kwezelstaart, thanks. I am glad you liked it.
amazing video
+R Vass, thanks. Glad you liked it.
This video is a godsend.
Thanks for the compliment. I am glad my video is helpful.
your channel has very useful videos .....thanks a lot
Thanks.