This channel is criminally underrated, watching the video i was baffled by how well explained and how WELL EDITED THIS IS, I cannot fathom the amount of work that went into this especially the visualisations ! I thought this channel had millions of subscribers and i was wondering how i missed it, only 9K ?!!! I'm sorry this is outrageous. PS: I think 3Blue1Brown would definitely be proud.
You just brought the whole linear algebra of quintals of books of one full academic year less than an hour of enjoyable stuff to the senses of even common people. Kudos. Thank you very much.
In these 2 parts I could say I've learnt more about matrices than (sadly) during the high school and even university courses on linear algebra. It gave me so much needed intuition behind all those theoretical concepts. I could only wish our educational system was so engaging and demonstrative like these videos! Thank you very much! This channel deserves much more attention! ❤
I deeply appreciate your videos on matrix visualization, bro. I always "struggled" with matrices and linear algebra. Now you gave powerful tools for interpretation and intuition behind every single concept behind them. Great content. Impressive animations and sound effects. A piece of art.
Dude unfortunately we are only told to calculate them without any meaning. Now that I know the meaning it all makes sense. Thank you pouring so much information into one single video
It's a great piece to summarize a hard topic. He has also made it very simple with the computer graphics. I've learnt linear algebra from Prof G. Strang and this just gives it life to explain those hard topics.
for those confused why is sqrt(2)/2 a unit vector, cuz at least I thought it should be equal to 1, here's the proper explanation: A unit vector is a vector with a magnitude of 1. we find magnitude using |v| =√(x^2 + y^2) or also known as the distance formula between points OR ALSO KNOWN AS THE PYTHAGORUS THEOREM-ish ("-ish" cuz Pythagoras theorem is for right-angle triangles but it's fine it works here as well cuz when finding distance from two points, we can just imagine the distance to be a hypotenuse of an imaginary right-angle triangle and then apply the formula using √(x^2 + y^2) where y is the 'rise' and x is the 'run') . normally, |v| =√ (1^2 + 0^2) = 1 OR |v| =√ (0^2 + 1^2) = 1, hence they're a unit vector, but it's not always limited to that. The expression √2/2 can represent the x or y component of a unit vector in two dimensions. For instance, normally if you have a unit vector pointing in the positive x-direction, its x-component would be 1 and its y-component would be 0. However, if you have a unit vector pointing at a 45-degree angle from the positive x-axis aka √2/2, both its x and y components would be √2/2, and the magnitude of the vector formed by these components would indeed be 1. This is because when you square and add the components, you get 1, then √1 is still 1, hence satisfying the condition of being a unit vector.
Projection matrices are idempotent matrices. What you considered are orthogonal projection matrices i.e., symmetric idempotent matrices. Orthogonality requires an inner product but general projectors exist in every vector space not just inner product spaces.
Seriously, you're going from basic visualisation to spectral decomposition I'm the second vid?! Brave man. I don't mind you skipping Gaussian elimination but i do think you should have shown what a linear combination is. Also independent and dependent bases. Was a lovely video though and, like you, I prefer the geometric interpretation.
It's not true that all orthogonal matrices represent rotations. Some orthogonal matrices are reflections. Specifically, orthogonal matrices with determinant = 1 are rotations, and orthogonal matrices with determinant = -1 are reflections.
@@mb59621 I wouldn't say so. A reflection inverts the orientation of the space, hence the negative determinant, whereas a rotation does not. Reflections are also their own inverse. Applying a reflection twice lands you back where you started, whereas it doesn't with most rotations. Since orthogonal matrices are defined as those with an inverse equal to its transpose, this means that the transpose of a reflection matrix is equal to itself (since it is its own inverse). Therefore, reflection matrices are symmetric across their main diagonal (taking the transpose doesn't change the matrix), whereas most rotation matrices are not.
@@APaleDot good points to ponder while still learning the subject. At first I thought reflection is just a rotation about the normal , but ty for your reply , i found an inaccuracy in my intuition that a single rotation matrix built on this principle cannot reflect anything more than 1 point ! Which is what makes reflection matrices a bit unique.
These videos, and other visualization videos of a similar nature, are very helpful in understanding much of underlying concepts. One thing, however... is it possible to perform some spell checking within the visualizations themselves? Sometimes the misspellings diminish the impact of the video. Is spell checking within the visualizations difficult, or is it difficult to correct the visualizations after a misspelling is detected later in the production process? Anyway, good work.
I asked ChatGpt and it could not compete with this guy's videos. This is Gold. Whoever did these videos should share their name. A great honour beholds them. 🎉❤
great video but here is one error/correction: at 3:16 it is stated that orthogonal matrices produce pure rotations with "no reflection" - this is in fact false - orthogonal matrices can indeed produce reflections. The defining property of an orthogonal matrix is U^T U = UU^T = I; check this property for any reflection matrix and you will find that it is satisfied. Orthogonal matrices can produce both rotations and reflections. Note that in 2D, a product of two reflection matrices is also a rotation matrix.
in case anyone is wondering, the subset of orthogonal matrices that produce pure rotations (not reflections) are those that have determinant(U) = +1 (and with -1 a reflection is additionally involved in general). for orthogonal matrices the determinant is always +/- 1 so this covers all cases.
@noitnettaattention @rhaegartargaryen0947 basically what he is saying is that for a vector ( column of the matrix) to be a unit vector then it needs to have a module of 1. The module of a vector is just like doing a Pythagoras theorem with x and y to find the hypothenuse. this said, generally people don't go that far and just use the following formula: module = sqrt(x^2 + y^2). the module of the vector is therefore sqrt(1/2+1/2) = sqrt(1) = 1 thus making it a unit vector. this vector is very commonly seen on many applications since it has a simple 45º angle with the axis but any vector you can imagine that goes from the origin to a radius 1 circumference really qualifies as a unit vector (in R^2, 2 dimensions). What he wants to explain with this is that there can be matrixes that have two unit vectors, but to have them orthogonal is yet another important and distinct characteristic that he shows some seconds after. The good thing about the orthogonal unit vectors is that they are called "eigenvectors" and together with "eigenvalues" they will set a new set of axis for this transformation, as if the whole original axis is being rotated, as to have two perpendicular axis in a 2D space you can basically say that they are just some rotation or inversion of the original two axis. Hope I helped! Cheers
at 2:07 , orthogonal matrices need not have a unit column vectors, only orthonormal matrix, anyway which is a subset of orthogonal matrix, has unit column vectors. Am i right? If not, please enlighten me. Thank you for the awesome video.
Your videos are brilliant, they should be 1K times more popular! But why do you need jazz in the background, it just distracts? ). Thanks, helped me a lot!
For orthogonal matrices, I don’t believe that implies they have unit vectors for columns. Isn’t that reserved for orthonormal matrices? Just want to correct my understanding it it is wrong, great video!
A unit vector is any vector with a length of 1. We can find the length of a vector using the Pythagorean theorem: (length)² = x² + y² = (sqrt(2) / 2)² + (-sqrt(2) / 2)² = (2 / 4) + (2 / 4) = 1 The square root of 1 is just 1, so we conclude that the length of the vector is 1, a unit vector.
![ทุกครั้งที่หลับตา (Lucid Dream) - AYLA's [ Official MV ]](http://i.ytimg.com/vi/4i1OBAQyv58/mqdefault.jpg)
This channel is criminally underrated, watching the video i was baffled by how well explained and how WELL EDITED THIS IS, I cannot fathom the amount of work that went into this especially the visualisations ! I thought this channel had millions of subscribers and i was wondering how i missed it, only 9K ?!!! I'm sorry this is outrageous.
PS: I think 3Blue1Brown would definitely be proud.
This is magic. You are an incredibly talented math communicator. A billion thanks for your content.
The world needs you bro
You just brought the whole linear algebra of quintals of books of one full academic year less than an hour of enjoyable stuff to the senses of even common people. Kudos. Thank you very much.
you made 4 videos on topics i didnt fully understand and just dipped lmao. legend
In these 2 parts I could say I've learnt more about matrices than (sadly) during the high school and even university courses on linear algebra. It gave me so much needed intuition behind all those theoretical concepts. I could only wish our educational system was so engaging and demonstrative like these videos! Thank you very much!
This channel deserves much more attention! ❤
I deeply appreciate your videos on matrix visualization, bro. I always "struggled" with matrices and linear algebra. Now you gave powerful tools for interpretation and intuition behind every single concept behind them.
Great content. Impressive animations and sound effects. A piece of art.
thanks for making me fall in love with math all over again
Awesome, simply awesome. Only 51 comments for such a fabulous job is just not fair.
Please do not stop doing videos, are just invaluable.
Looking forward to Chapter 2 !!
Dude unfortunately we are only told to calculate them without any meaning. Now that I know the meaning it all makes sense. Thank you pouring so much information into one single video
This video was beautiful and emotional. Thank you
The music is unexpectedly cool for a Math video
This is really great. You will get a better understanding at 3:39 with a unit circle.
I like the character development of the potato!
Man, these videos are gold!
It's a great piece to summarize a hard topic. He has also made it very simple with the computer graphics. I've learnt linear algebra from Prof G. Strang and this just gives it life to explain those hard topics.
Shared it with all my friends as a token of gratitude.
Keep doing these videos man, We just love them!
Please upload more videos these are extremely helpful!
A lovely channel. Personally I wish there were no music, but the clear examples are golden.
for those confused why is sqrt(2)/2 a unit vector, cuz at least I thought it should be equal to 1, here's the proper explanation:
A unit vector is a vector with a magnitude of 1. we find magnitude using |v| =√(x^2 + y^2) or also known as the distance formula between points OR ALSO KNOWN AS THE PYTHAGORUS THEOREM-ish ("-ish" cuz Pythagoras theorem is for right-angle triangles but it's fine it works here as well cuz when finding distance from two points, we can just imagine the distance to be a hypotenuse of an imaginary right-angle triangle and then apply the formula using √(x^2 + y^2) where y is the 'rise' and x is the 'run') .
normally, |v| =√ (1^2 + 0^2) = 1 OR |v| =√ (0^2 + 1^2) = 1, hence they're a unit vector, but it's not always limited to that.
The expression √2/2 can represent the x or y component of a unit vector in two dimensions. For instance, normally if you have a unit vector pointing in the positive x-direction, its x-component would be 1 and its y-component would be 0. However, if you have a unit vector pointing at a 45-degree angle from the positive x-axis aka √2/2, both its x and y components would be √2/2, and the magnitude of the vector formed by these components would indeed be 1. This is because when you square and add the components, you get 1, then √1 is still 1, hence satisfying the condition of being a unit vector.
magical video
insane - many thanks for this!!!
Thank you for this fantastic video!
These videos are invaluable! Thanks a lot. Please create more of such videos.
nujabes in the background with anime characters to explain. 10/10 banger
Thank you so much love from india. You sorted out Lot of things
Visualizing matrix helped me to lock in e knowledge and make sense of it. Thank you 🫡
Incredible work!!
perfect.
Phenomenal work! I'm very thankful to you for such a great content
Nujabes music in the back and Watanabe characters to describe. I like this channel!
awesome videos and I love the graphics, nice to see some of my favorite anime characters while studying linear algebra ;)
Projection matrices are idempotent matrices. What you considered are orthogonal projection matrices i.e., symmetric idempotent matrices. Orthogonality requires an inner product but general projectors exist in every vector space not just inner product spaces.
excellent, why is this channel not producing more videos like these.
Nujabes = Chefs kiss.
Love the detail of the least squares normal equation when talking about the data matrix (10:46)😂😹
This is so good.
very clear. like it very much. Thanks.
"Unfortunately no one can be told what the Matrix is" - Morpheus
“- you have to see it for yourself.”
Amaizing content!!!!!!!!!!
sublime
Seriously, you're going from basic visualisation to spectral decomposition I'm the second vid?! Brave man.
I don't mind you skipping Gaussian elimination but i do think you should have shown what a linear combination is. Also independent and dependent bases.
Was a lovely video though and, like you, I prefer the geometric interpretation.
these are wonderful videos!
Long live the king
Using cowboy bebop to teach linear algebra is something i didn’t know i needed
It's not true that all orthogonal matrices represent rotations. Some orthogonal matrices are reflections. Specifically, orthogonal matrices with determinant = 1 are rotations, and orthogonal matrices with determinant = -1 are reflections.
A reflection is a rotation too , right ?
@@mb59621
I wouldn't say so. A reflection inverts the orientation of the space, hence the negative determinant, whereas a rotation does not.
Reflections are also their own inverse. Applying a reflection twice lands you back where you started, whereas it doesn't with most rotations.
Since orthogonal matrices are defined as those with an inverse equal to its transpose, this means that the transpose of a reflection matrix is equal to itself (since it is its own inverse). Therefore, reflection matrices are symmetric across their main diagonal (taking the transpose doesn't change the matrix), whereas most rotation matrices are not.
@@APaleDot good points to ponder while still learning the subject. At first I thought reflection is just a rotation about the normal , but ty for your reply , i found an inaccuracy in my intuition that a single rotation matrix built on this principle cannot reflect anything more than 1 point ! Which is what makes reflection matrices a bit unique.
@@mb59621 a reflection is a discrete transformation whereas rotation is the sum of infinitesimal transformations.
These videos, and other visualization videos of a similar nature, are very helpful in understanding much of underlying concepts. One thing, however... is it possible to perform some spell checking within the visualizations themselves? Sometimes the misspellings diminish the impact of the video. Is spell checking within the visualizations difficult, or is it difficult to correct the visualizations after a misspelling is detected later in the production process? Anyway, good work.
brilliant videos, thanks a lot
0:42 you can always work out the equations and it will be quite apparent what is going on!
Thank you for sharing your video. Wish you health and wealth.
Thank you for your information. Thanks for telling us about Manim. I want to make videos to teach 7 years old kids math with animation.
Is that a reference to 二向箔 :)
7:01 Death's end reference?
I wish you could also do the same for Statistics and calculus topics. What can ML/AI/DS students ask for?
I asked ChatGpt and it could not compete with this guy's videos. This is Gold. Whoever did these videos should share their name. A great honour beholds them. 🎉❤
Great great great
Orthonormal
great video but here is one error/correction:
at 3:16 it is stated that orthogonal matrices produce pure rotations with "no reflection" - this is in fact false - orthogonal matrices can indeed produce reflections. The defining property of an orthogonal matrix is U^T U = UU^T = I; check this property for any reflection matrix and you will find that it is satisfied. Orthogonal matrices can produce both rotations and reflections. Note that in 2D, a product of two reflection matrices is also a rotation matrix.
in case anyone is wondering, the subset of orthogonal matrices that produce pure rotations (not reflections) are those that have determinant(U) = +1 (and with -1 a reflection is additionally involved in general). for orthogonal matrices the determinant is always +/- 1 so this covers all cases.
vsauce music was a nice touch!
how you made a [ -sqrt(2)/2, sqrt(2)/2 ] into unit vector remains complete mystery to me....
i was stuck there too. can somebody explain this to me
@noitnettaattention @rhaegartargaryen0947 basically what he is saying is that for a vector ( column of the matrix) to be a unit vector then it needs to have a module of 1. The module of a vector is just like doing a Pythagoras theorem with x and y to find the hypothenuse. this said, generally people don't go that far and just use the following formula: module = sqrt(x^2 + y^2). the module of the vector is therefore sqrt(1/2+1/2) = sqrt(1) = 1 thus making it a unit vector.
this vector is very commonly seen on many applications since it has a simple 45º angle with the axis but any vector you can imagine that goes from the origin to a radius 1 circumference really qualifies as a unit vector (in R^2, 2 dimensions).
What he wants to explain with this is that there can be matrixes that have two unit vectors, but to have them orthogonal is yet another important and distinct characteristic that he shows some seconds after.
The good thing about the orthogonal unit vectors is that they are called "eigenvectors" and together with "eigenvalues" they will set a new set of axis for this transformation, as if the whole original axis is being rotated, as to have two perpendicular axis in a 2D space you can basically say that they are just some rotation or inversion of the original two axis.
Hope I helped!
Cheers
What do you use to visualise the transformations?
10:36 - From this on matrix is wrong. You seem to place numbers into the matrix quite randomly.
7:12 is the big reveal of dark forest
Nujabes ❤
Sir, What is Geometric interpretation of Trace of a Matrix. Kindly make a video.
Bravooooooooo
I know this transformation, 3 bodies, reduce dimension 😁
at 2:07 , orthogonal matrices need not have a unit column vectors, only orthonormal matrix, anyway which is a subset of orthogonal matrix, has unit column vectors. Am i right? If not, please enlighten me. Thank you for the awesome video.
Agreed
Your videos are brilliant, they should be 1K times more popular!
But why do you need jazz in the background, it just distracts? ).
Thanks, helped me a lot!
your content is amazing! but i feel you can probably rename your videos so they aren't discriminated by the algorithms!!!!
Woow...
For orthogonal matrices, I don’t believe that implies they have unit vectors for columns. Isn’t that reserved for orthonormal matrices?
Just want to correct my understanding it it is wrong, great video!
yes I had the same thought. Too lazy to check now...
three body reference??/
Where is the captions 🥺
wait can someone tell what's special about projection matrices?
bro is the asian 3b1b
nujabes + maths
background music too loud
3:13
brilliant!!
Where are you!!!
How is root 2/2 , - root2/2 a unit vector
A unit vector is any vector with a length of 1. We can find the length of a vector using the Pythagorean theorem:
(length)² = x² + y²
= (sqrt(2) / 2)² + (-sqrt(2) / 2)²
= (2 / 4) + (2 / 4)
= 1
The square root of 1 is just 1, so we conclude that the length of the vector is 1, a unit vector.
二向箔哈哈哈