Change of basis | Chapter 13, Essence of linear algebra 

Wow, I didn't notice 😂😂😂

I was looking for this comment :)

@@paprikar same lol

He's a backbencher 😅

Tensors are dependent on the choice of basis though :)

she could ask the same of us. i hat and j hat being perpendicular only really makes sense once we have a dot product, so without that, any (linearly independent) choice of basis vectors should work equally well

From her perspective, we're the ones complicating things.

for her, we are the complicated ones. it's like different units of measure, except in more than one dimension.

While the linear transformation used in the video was indeed a lot more complicated from her perspective, there will be other transformations which are far easier from her perspective then from ours! (This will be a part of next video I believe)

Ask women.

Glad I could help! It's always super rewarding to read a comment like this.

:)

So there's an isomorphism between vector combination problema (cool plural for problem I made up) and linear combination problema. That said with your coordinate grid (basis vectors) being an axiomatic method of viewing space or information in mind, is there an isomorphism between matrices and tensors? Probably a better question would be, what are tensors?

Incredibile come rimanga la memoria di quel grande artista del quale porti il nome, anche dopo la sua morte!

Sempre libero.

Hva studerer du, og hvor?

Vet ikke helt hvorfor det gjør han 'humble', videoene har veldig bra kvalitet og er en god måte å få bedre forståelse på, og som du sa er det ikke akkurat som det er faktisk del av pensum. så ville ikke helt sagt det var humble, men var i hvert fall kult gjort.

👏👏😇

I thought "I wonder what application would make Jennifer use that system instead of the "standard" one?" and BAM I found your comment! Haha thanks for sharing!

It's almost like universities are just institutions that suck up our money for degrees when people don't necessarily need them to learn

BAM im glad i stumbled upon this comment

BAM

.... Chapter 200 Renormalization in QED

The Physicist Cuber But this is a mathematics channel, not a magician channel.

WTF?!

The Physicist Cuber We're talking about renormalization in quantum electrodynamics, correct? I.e., the process of making infinities magically go away, correct?

lol xD

@Swagger Kekker lol yes

I think people often mistake watching these videos for a full understanding that you would get from an LA course, people just get different things out of it, for example someone who just watches this series might be able to conceptually describe all these concepts, but not be able to actually do the problems associated with them like someone who's taken a linear algebra course, but that second person may have a worse conceptual grasp. they're just different, this isn't a replacement of a uni course, it's more a useful supplement to it.

It surely gives us an easy understanding of the basics, but it is not enough for our own comprehension. Managing to get the point oneself, and then watching this video will give us a much advanced and thorough view of linear algebra: the intrinsic relation between calculation and visualization(intuition). We tend to use our intuition for having the point of mathematical actions, which means that studying this topic isn't enough by only a bunch of videos. It is necessary to make the bridge between the two of ourself.

God I do a course of LA on Coursera using these videos as a teaching material. I could NEVER be able to do this if I wasn’t watching these, since the teachers do a much worse job of explaining than 3b1b does. I would encourage everyone to do this to get the best of both worlds AND support 3b1b financially (as I do)

​@@MKWiiLuke4TW​Very well put, I think it's accurate.

I like your perception basis vector

Bhai hamare studd hai

My algebra teacher turned off the lights and read from slides. Course notes saved my ass, wish I'd seen this at the time

Hell yes!

Could you be any more dramatic?

I am LITERALLY on the floor having a seizure. My essential bodily functions are shutting down rapidly. The beauty of mathematics is just too overwhelming. Please send help.

Ah c'mon, maths can be beautiful but don't exaggerate. Tearing up because of it seems a little too dramatic.

its called Stendhal syndrome when when you see something so beautiful you enter a psychosomatic state. its a good thing usually

Thanks

great

Wow, that's a great image to remember how the transformation works. Thanks a lot! However, I would recommend to write it the other way around (8:35) since matrix multiplication is in general not commutative; when you write it as follows you can extrapolate from it to 2D-space one to one. her/our language = her/our coordinates 5 * 1 = 5 her basis 1D-vector in our language * 1D-vector in her language = same 1D-vector in our language her basis 2D-vectors in our language * 2D-vector in her language = same 2D-vector in our language (as a matrix rerpresenting transformation) 1/5 * 5 = 1 inverse of her basis 1D-vector in our language * 1D-vector in our language = same 1D-vector in her language inverse of her basis 2D-vectors in our language * 1D-vector in our language = same 2D-vector in her language (as a matrix rerpresenting transformation)

Thanks! I *almost* had it down intuitively at the end of the video, but was still slightly confused. This made it sink in properly :) Well worded!

You can use dimensional analysis to make it even more obvious! 1 Jennifer vector * (3 our vector / 1 Jennifer vector) = 3 our vector. 3 our vector * (1 Jennifer vector / 3 our vector) = 1 Jennifer vector. I like how you simplified it to 1D :)

Nah, the standard basis is special. 0 and 1 are part of the definition of vector spaces.

@@MrCmon113 You frickin racist!

Vectorist not racist

Sorry, but I find transvectors creepy.

Look at vectors while the basis vectors i and j are fixed most of the other vectors are fluid And have both i and j charachter They can also transform themselves if they want to So much to learn for humans from these vectors

well he has a bitcoin adress so you can send him some coins ?under about in his youtube channel

he also has a patreon: www.patreon.com/3blue1brown

Best way nowadays is donating BATs if you use the Brave browser.

This is what which is called as a similarity transform - I guess!

We used this to convert local coordinates to global coordinates in matrix analysis of structures.

The inverse of Ask Me Anything?

@@totheknee There are two ways of finding that inverse: The "Don't ask Me Anything" phrase, which could look like the person just doesn't want to answer whatever you want to ask him. The "There exists a topic which I don't want you to ask me about" phrase, which sound unnatural, but is equivalent (logically speaking) to the previous phrase (even tho It doesn't have the same connotation), which seems like the person just doesn't want to talk about "something", but anything else would be fine. So, if A^-1MA = Ask Me Anything, then the inverse of the entire thing (A^-1MA)^-1 = (A^-1)(M^-1)A means one of two things: I don't want to talk about something, or I don't want to talk about anything. I'm sorry, I'm bored :3.

Same here. Our book barely described the intuition behind all this. I had to memorize it all until I found this vid.

isn't milk expensive protein? beans and rice seem cheaper...

The video I'm planning to do after this series will be related to topology. As to a whole "Essence of topology" series, maybe later.

@3Blue1Brown, thanks you so much. Even when I've already knew this things about linear algebra from university it is good to construct intuition of visual represented connections of all this concepts. It is even more nice to know of your plans about topology videos.

Oh my god, that's great. Looking forward to it.

+3Blue1Brown I am looking forward to it!! :)

nst as storix or not

yeah

Yess very very important point.

Nah. It doesn't matter. Tell me why he's showing that picture of Jennifer's grid after using inverse change of basis matrix for the rotated matrix? No matter which grid you use. The position of that vector doesn't change. Isn't it?

@@Robocat754 the position is relative you can not tell the position unless you have a grid so if the gird change, the position must change you think it did not change because the screen did not get distorted and it is fixed to our world

I think this part is quite hard to grasp. Specially because the matrix has two meanings in there, and it coincides to be the same matrix. 1. The matrix describes her basis in the standard basis notation (and coincidentally our basis is the same as the standard one) - this is the "her basis" matrix; 2. The matrix also transforms a vector from her basis into our basis, but this is only because our basis is on the standard one - this is the "her basis to our basis" matrix; I think that if our basis wasn't the standard [1,0;0,1] then the matrix for (2), the basis "translation" from her to ours matrix, wouldn't be the same as the matrix for (1), her basis matrix. For example, if our basis were [2,0;0,1], ie. our î axis is stretched by 2 times, the the transformation from her basis into ours would need to take our stretch into account, and thus the matrix representing that transformation would not be the same as her basis matrix.

This sounds indeed similar! Conjugating (thats the name for "multiplying something by something else from the right and the inverse of that fromj the left") appears in more areas than just with matrices

Random story but a great lecturer of mine put quite a bit of emphasis on the "perspective" perspective of conjugation, but when I talked to my tutor about it, she had never thought of it that way... and she was an algebraist. Surprising!

In Stefan Pochmann's methods (both original Pochmann and M2/R2, both typically used for blindsolving), the "setup moves" are a change of basis, essentially. If S is a setup move, and A is an algorithm that switches the desired pieces once set up, then S^-1•A•S denotes the switch of the desired pieces in situ.

You might find this interesting: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-syyK6hTWT7U.html

@@BlueGiant69202 what is the blue giant your username is namesake to? Is it Ymir?

Mabelcorn N. Most courses don't have the time / resources. Also this is meant as an addendum to students of linear algebra, not as standalone linear algebra course. How do you visualize that R is an infinitely dimensional Q-vectorspace?

There are other fields besides mathematics, such as electrical engineering which I studies, that are using linear algebra and who do not need to visualize such things. But yes, this will not teach how to perform calculations, but would certainly help understanding many applications where linear algerbra steps in.

@@msergejev yeah... I am learning Lin All for mechanical engineering (also in a top 10 university), but the prof only shows the calculations and does not provide us the understanding of what we are really doing. This series is a blessing for me...

@@Casskario Its a blessing for all of us, tho for me it came about 10 years too late :)

@@msergejev hmmm so how would you visualize linear algebra's Analogies in EE? like Eigenfunction for example?

确实讲的挺好的

I'm glad you brought this up. It was in the original script, but I cut it out to stay, as you said, in the essence. I remember being truly surprised at the fact that the dot product is, in some sense, a choice that we make, and least in the context of an abstract vector space.

I think that deserves a footnote video.

How does she?

Out of my head I could imagine two obvious options: One practically inherits the dot product: You would take the dot product of two of Jennifers vectors by first transforming them back to "normal" vectors and rhen taking the normal dot product. The other option would be to just take tge dot product of Jennifers vectors like normal. There's an interesting difference because with the second option, jennifers basis vectors will always be orthogonal with length 1, while they normally wont with the first

Really well-said! Both interpretations are interesting. Casting aside our "mathematical empathy" suppose we say that our dot product is "right" and hers is "wrong." Then in order for Jennifer to take the "right" dot product, she would have to do what you just said: convert her vectors from her coordinate system to ours before computing. On the other hand, if Jennifer insists that her vectors are, as you say, orthogonal with length 1, then we're at an impasse: she will simply get different answers than we. By the way, you could also ask the question, if we were to "empathize" with Jennifer and agree that her dot product is "right," then you could ask, how would we go about doing that? How could we modify our dot product formula so that we get the same answers she would get? You'll find that there's an interesting difference between the first question and the second: one requires us ultimately to take an inverse matrix. Do you see which one? It's kind of like, if I'm talking to a European (I'm an American) and I want to describe the weather, I'll convert the Fahrenheit to Celsius for them, because neither system is "wrong" per se, but they'll understand it better if I do that.

This is the part I feel like is absolutely awesome... coordinates become vectors and vectors become coordinates , and they span everything.

Haha didn't notice that, that's very cool haha

I totally agree with you. The video part after 9:50 explains how Jennifer would see our 90-degree transformation. But if we want to see Jennifer-90-degree transformation in our perspective, first we should rotate the general (x,y) vector by 90-degree, ie, by multiplying by that [0,1,1,0] matrix and then converting it to our grid by A-inverse

Sida Liu No, if you look at the angles it's still 90 degrees, unless her measurement of angles is messed up now

Deepto Chatterjee - maybe her angles are messed up! If you look into an arbitrary vector space (over the real or complex numbers), sometimes you can develop a notion of an _inner product._ An inner product is a generalization of the notion of the dot product. It is an operation that takes two vectors in your vector space as an input and outputs a real number, subject to certain properties. A vector space together with a specified inner product is called an inner product space. In an inner product space, we _define_ the length of a vector to be the square root of the inner product of that vector with itself; symbolically, ||v|| = √(in(v,v)). And we _define_ the angle between two vectors to be the arccosine of the inner product of those vectors divided by the product of the lengths of those vectors; symbolically, θ = arccos(in(v,w)/(||v||∙||w||)). Knowing this, if Jennifer decides to use the "dot product" with respect to her basis vectors (which would be an inner product on R^2), then her basis vectors would appear to her to be orthogonal and of length 1! And our basis vectors would appear to her _not_ to be orthogonal or of length 1. Of course, Jennifer could also choose a different inner product from the "dot product" with respect to her basis vectors, in which case her basis vectors may not be orthogonal or of length 1 using that inner product.

Nope! Here, everyone sees space the same way. A 90 degree rotation to us is a 90 degree rotation to Jennifer. What's different is how she DESCRIBES space. So, she needs a different matrix to describe a 90 degree rotation.

@@rynin8019 6 months late, but from the video, what looks like a 90 degree angle in our space looks like a greater than 90 degree angle in space from her perspective. So while the rotation is equivalent in both coordinate systems, I am not sure we all "see space the same way."

Great question! Almost all of the time, the angle between a pair of vectors that are described with some basis B will change whenever that same pair of vectors is described with a new basis B' His next video in this series, the one on eigenvectors and eigen values, explains this concept well!

The "Eigen" is used in English. This is because Hilbert coined the prefix "Eigen-" for characteristic (intrinsic, coordinate independent) properties. (faql.de/etymologie.html#eigen ) I guess he was the one who really started to figure out "Eigen-"properties on a higher, more abstract level than just linear algebra.

Probably not. There haven't ever been any German mathematicians of note... ever. 😀

iSquared Trolling, aren't you? What about Riemann, Gauß, Euler, Hilbert or Cantor?

+Elchi King (Maddemaddigger) Relax, I was making a harmless joke. Half of mathematics comes from Germany haha.

iSquared while mathmethicans isn't what Germany is most famous for (other than physicists, but these all had to be good in mathematics too btw), there certainly are Germans that are famous for their mathematic archivements. Ever heard of Gauß (or like you would probably write it Gauss)?

Linear transformation works in one basis system (coordinate system). A change of basis is a transformation from one coordinate system to another. They are similar in that you can use a linear transformation to define your new basis vectors. They are both the same thing, it depends on the context

Hence the intro quote

Put another way, a change of basis is an _application_ of a linear transformation to change coordinate systems. Sometimes a problem is easier to reason about in a different coordinate system, and a specific linear transformation lets you do that.

My thoughts exactly.

thanks man i was confused at this point but your insight cleared my doubt

I think the point he makes is more general. It may be easier for you to invert your matrix by hand, but what if you need to invert a 1000x1000 matrix whose values change depending on a previous set of steps? Do you then have the system print out that matrix and wait for your input to provide it the new inverted matrix? In reality, you should work to form an equivalent statement that does not require you to invert a matrix wherever possible (find symmetric matrices maybe) and then bring down the number of computations you'd have to do. However, It is almost never worth inverting these matrices by hand.

I had the same doubt, I thought shouldn't Jennifer multiply the same rotation matrix if she wants the vector in her coordination system to be rotated by 90 degrees.

This took me a while to figure out too, I think it's one part that could've been better explained. It finally hit me that the two vectors in some arbitrary basis might not be 90 degrees from each other, so the orthonormal basis transformations will be encoding 'flips' of basis vectors that are NOT 90 degree rotations. To get what a 90 degree flip is you need to find out your basis vectors are in terms of her basis and then flip them according to the result.

Yeah me too

Exactly! hahaha I didn't see that :P

We both see the exact same thing. The only thing that changes between us and Jennifer is the number used to describe it. But Jennifer knows that her base is not orthonormal. That's why for her just as for us, a 90-degree rotation doesn't bring b1 to where b2 was.

MemeLord why are crying?

too many dead memes

Welcome in affine geometry

that's not a linear transformation.