Cramer's rule, explained geometrically | Chapter 12, Essence of linear algebra
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- Опубликовано: 5 сен 2024
- This rule seems random to many students, but it has a beautiful reason for being true.
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Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld
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6:52 immediately got so excited when you went to 3 dimensions, because I knew I was going to get to hear you say "parallelepiped"
"Parallelepiped" -- a solid body of which each face is a parallelogram.
Next time please add your teacher’s Russian accent.
@@no-body-nobody hahaha!! parariririid
@@no-body-nobody hahaha!! parariririid
Sometimes it sucks being German. I have to call these "Spat"
Perhaps some of you are wondering why, 2.5 years later, I've come to insert a video into this series. Does it mean the start of an extension to the series? Er...no. Or rather, not yet.
I'd been sitting on this video for a while, thinking I'd wait to put it out until there was a larger batch of new linear algebra content. But other plans have risen above that in the project list, so it seemed a bit silly to keep it unpublished for too much longer.
In a few weeks, I'll start putting out some content for a miniseries on differential equations, so stay tuned for that! And after that...well, actually, I have a bad habit or breaking promises, so I'll keep the forecasting to a minimum here :)
Fun little challenge puzzle: Use Cramer's rule to write down/explain the formula for the inverse of a 2x2 matrix. What about 3x3? 4x4?
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Edit (correction): In the video, I describe matrices which preserve dot products as "orthonormal". Actually, the standard terminology is to call them "orthogonal". The word "orthonormal" typically describes a set of vectors which are all unit length and orthogonal. But, if you think about it, dot-product-preserving matrices *should* be called orthonormal, since not only do they keep orthogonal vectors orthogonal (which, confusingly, several *non*-orthogonal matrices due as well, such as simple scaling), they also mush preserve lengths. For example, how confusing is it that we can say the columns of an orthogonal matrix are orthonormal, but a matrix whose columns are orthogonal may not be orthogonal. GAH! Maybe my casual mistake here can help nudge the tides of terminology towards something more reasonable, though of course that wasn't the intent.
3Blue1Brown Hi 👋
I greatly look forward to a differential equation mini-series
Pls pls pls make a video on measure theory and integration....
thanks we will look forward to it
It’s already 2.5 years??!
God, I love how I understand everything but after 5 minutes after watching the video I forget everything.
That is because you do not "Pause and Ponder" 😎
You gotta pause and ponder, take notes and try to explain yourself what you think you've just understood on each step of the way. If you don't check that what you've just been told really makes sense to you, by explaining it to yourself, you will never know if you truly understood it.
Bhai concentration Sab kuch nhi hota tujhe practice karna padega
Oo
use it or lose it :D
This channel is truly amazing- so original and so much work put into it.
Keep up your amazing work!
Thanks!
i totally agree - it's not only my favorite math channel but has actually become my favorite channel overall :-)
Ok
@@3blue1brown thank you so much! Without u my maths will only be memorizing stuff. I like how ur videos explain things so clearly!!❤🎉
taking linear algebra this semester with an extremely difficult professor. your whole series has helped me in ways you will never know. thank u so much.
This channel is a continual reminder for why I love math.
true
i love how 12min can become 5 hours
I had to pause and ponder every 5 seconds 😂
If not 2 days depending on ur schedule lol
In Poland we used to have spoken math exams when we needed to explain everything that we're doing and why. And the method from this video is called "metoda wyznacznikowa" (determinant method).
When one student was asked why it's called that, he answered "it's because Viznachnikov invented it".
Cramer's rule (written as the product of A and its adjusted equalling the determinant of A times the identity matrix) is not just important for the reasons given in the beginning of the video but also for other reasons.
For instance, if your matrix is made of integers and the determinant is +/-1, then you know that its inverse is also made of integers. This is useful when dealing with matrices whose entries belong to a general ring.
Such an intuitive explanation of what we learn only in the abstract mode in our schools. Thanks for existing 3b1b.
Can you also do Hilbert Space and its application in Quantum Mechanics?
Be honest, is Sayan Mondal your alt acct? Cause you both asked for the same thing! lol
If you really want to learn about Hilbert Spaces or any other higher level applied mathematics, you check out Faculty of Khan, he does some great work on RUclips.
Technically he already has, as a Hilbert space is just a complex (or real) inner product space that's also a complete metric space. Euclidean space (ie what 3blue1brown is using in all of his videos, namely vectors being arrows on a grid in Cartesian coordinates) IS a Hilbert space with the inner product being the dot product. All that changes in quantum mechanics is that your basis vectors are now the solutions of the Schrodinger equation and inner product is defined by the basis functions being square-integrable. As such, every video 3blue1brown has on vectors, change of basis, dot products, etc applies to quantum mechanics as well with at worst a few small modifications (mainly do to the inner product being different from the standard dot product, that fact that you're almost always dealing with an infinite dimensional vector space, and because quantum mechanics demands an additional criteria in that all the solutions to the Schrodinger equation must be normalized).
now he did. see hilbert curves
Just when I was going to sleep
Sleep can wait
Lets be honest you weren’t going to sleep anyways
True
It's morning here
I don't need sleep, I need answers.
Faeze Heydaryan ... followed by a few follow-up answers, an extended nap, and a pint of coffee, right?
I studied Cramer's rule since my high school days including determinants and matrices but never took it seriously thinking that its just a fancy way of writing numbers and performing operations and now I realize how important it is to the world of mathematics.
love this channel.
You're getting so close to Geometric Algebra! (Oriented volumes: just get the wedge product involved and you're basically there.)
Take it all the way! We're ready! We need it! :)
Was just thinking that. It'd be grand if he could do a series of videos on Geometric Algebra; feels like it's a subject he would really get a lot of mileage out of.
Man I always wondered in my math class how this was possible. We never had any visual intuition, neither our teachers wanted to show us. That's how freakish bad educational system is here. Thank you man. Grant, I wanna thank you in person! 🙏
I would cut them some slack. Imagine creating content, with this level of clarity, every week, to probably a couple hundred students. It's not actually easy. And there's no incentive for them to do so.
I think it's not a case of "bad educational system" but a discussion of the best way to teach the introduction to Linear Algebra. I'm not a mathematician, so don't quote me on what I'll say. Geometric representations are great to teach some basic concepts of Linear Algebra, but are very limited. You can only represent up to 3-dimensional operations, leaving behind some very important abstractions like n-finite and infinite dimensions and continuous-space operations. Also, I think teachers worry about conditioning their students to always expect a geometric representation or intuition for every concept in Algebra, and that could lead to frustration when these students arrive at advanced topics of Linear Algebra (or even Abstract Algebra). I love this 3b1b series but you should always follow a Linear Algebra book for a deeper understanding of those abstractions.
@@biscoitom excuses. if shitty teacher. just say shitty teacher.
You just made sense of a lecture I struggled through 35 years ago. Thank you, it now makes sense.
I was struggling with this video at first. I don’t know why but I found this idea a little bit hard to grasp but after watching this video for four times I finally understood what you were trying to state. And it was utterly beautiful.
same for me. Somehow i missed the obvious that y = Area / det(A) simply means "old area" = "new area" / "scalar"
@@reZss thank you man I finally understood it lol
ig it's the first part of the video that makes things confusing where the x and y values turn out to be the dot product with the basis vectors, that throws you off into a weird pattern of thinking. but it works in that case, because the space hasnt changed dot product happens to be the projection of the output vector on the unit vector and gives the x, and y values of the input vector. but this is not the case for when the space has been changed because then the dot product would not be the same as projection, nor would it give x and y values of the input vector
1:46 - 1:57 - That simple visualisation alone would be a perfect way to teach kids the meaning of those equations in school. Instead, children are often told not to visualize algebra, which is a missed opportunity for many of them, if you'd ask me.
the entire series points in that beautiful geometric interpretation and makes matrices beautiful,i used to think they where the most boring thing when i was a student,now i think matrices mixed with a little bit of calculus are the most beautiful achievement of human thought.maybe the hidden geometry is the key to make the most complex ideas understandable.like they were always there but we didn't notice before.
Perhaps those who say so don't know what visualization is, hence unawaring of the beauty of it.
a SEINFELD REFERENCE in a 3b1b video MY LIFE IS COMPLETE
Kramer: “The important thing is that you learned something.”
Jerry: “No, I didn’t.”
This was just brilliant! Couldn't ever think cramer's rule could even have such a relation with geometry!
I am so dumb, I need to listen to the video several times to get the whole idea but I love it
3b1b, thank you so much for your work!
Just as every student with some topic ;)
My teacher teaches me how it works and you teach me why it works.
HUGE THANKS TO YOU SIR. please keep up the good work
Thank you, love from India
You are an artist !!!
Kids in grad school everywhere will learn so much faster because of how visually you can communicate ideas.
Everytime I watch your videos all my sadness and depression goes away. I'm very happy and amazed at the amount of clever ness went into these concepts. I wish i learnt all this in my high school. "Ur videos make me wanna live to see this beauty".
I wish I had teachers like you in school and at the university. You present everything in such a fascinating way with the visualizations. Maybe I wouldn't have lost interest in computer science program, if I knew how this all relates to geometry and space. Keep up the work man, your videos are gold!
That might happen 🥲
What makes me really REALLY happy, is that other than most math channels, you have black background.
Almost every other math channel has bright white background and i am just trying not to go blind.
I got my phone on lowest brightness settings and i still have to flinch my eyes to be able to watch them without physical pain.
This video is sooo good!! We just briefly rushed over Cramer’s rule in one day in my precalc class, with no actual understanding at all. This makes it so much more clear and satisfying! Keep up the amazing content :)))
why u upload at 12:35
now I gotta deprive myself of sleep.
the essence of calculus and linear algebra series(serieses?) are truly amazing ...extensions of them would be so cool and appreciated ;D
Can you please, make a video on Hilbert Space and its application in Quantum Mechanics?
I didn't know quantum mechs were a thing already O:
please*
its*
you forgot the question mark
@@JorgetePanete added...
@@Reydriel You weren't aware that quantum mechanics is a thing already? It's a field that's been in development since the 1920's, with the first hints being discovered in the 1900's
lol I was just telling someone about a 3blue1brown video and here comes another one!
And it explains Cramer's Rule!
The ability you have to convey almost anything in a clear and intuitive way really shows how smart you are. You and Richard Feynman made me love mathematics
This explanation is gold. Much more illuminating than the straightforward but obscure proof using the properties of determinant.
Wow, I never thought I would be this early for a video. Sure it's gonna be great. The whole Linear Algebra series is fantastic!
Santino Motta
I assumed this video was uploaded months/years ago until I read this.
You can generalize Gaussian elimination from solving systems of equations to finding inverse matrices, just by doing it on multiple columns at once. Applying the same with Cramer's rule gives A^(-1)=adj(A)/det(A). Neat!
I was just thinking about this topic yesterday, and how it works. Thanks for the in depth guide.
When You figure it all out, it feels like suddenly someone just taught you the magic of nature, Thank you Sir, You are doing a wonderful Job.
I watched the whole series again because this video came out, and it just so happens I’m also concurrently taking a rigorous linear algebra course. It’s thrilling to me how in depth this series goes (and how little of that depth I picked up when I watched this 2 years ago) and seeing these topics I understand in a very different perspective. I’m very excited for the differential equation series to come, since I’m taking that in the fall!
Brilliant. I wondered why Kramer's rule worked since highschool and I finally got (and understood) the answer
this is really good explanation I mean i was never taught why cramers rule works and its really comforting to understand these things
Oh dang dude, you were completely right! Did a few Cramer Rule exercises and the concept of the dual vector clicks! Appreciate it!
Rainy Sunday morning, coffee, chocolate and this video. There is nothing else I could ask for! This is perfect!
Your videos of explaning these concepts in the simplest and an intuitive manner will have such huge positive ripple effect in this world.... Thank you for your selfless service to the humanity. 👏🏼👏🏼👍
Amazing video! So the key idea is that the determinant of any matrix M basically represents the area of the shape whose edges are the column vectors of M.
And we learned from previous lesson that det(A) is the area scaling factor of any shape in the original vector space.
Combining these two principles we have det(T(i), T(v)) = det(A) * det(i, v), where det(T(i), T(v)) represents the area of parallelogram whose defining edges are T(i) and T(v).
and since det(i, v) = 1*y - 0*x = y, we get det(T(i), T(v)) = det(A) * y, and consequently y = det(T(i), T(v))/ det(A).
Quite amazing how the formulation of this rule is so easily understood under visual interpretation. Keep the videos coming please!
Enlightening. Just purely enlightening! I think the key to understanding here, as pointed out in the video, is that under linear transformation all areas (or volumes in 3d case) change in the same way, so that the RATIO of change is the same. Cramer's rule is really all about this change. Rearranging the equations to reflect this ratio of change really helped me digest this one.
I've never taken any linear algebra class before, but this brilliant series makes me really want to learn much more about the subject. To enlight, not to daunt, students, is the only golden standard of teaching. Can't imagine how much happier and more satisfactory students could have been if they were taught this way in school. Oh man, this even makes me want to become a teacher like him.
Keep up the enlightning process, please!
Hi Sir!! could you please make video on convolution and correlation?
That would be cool, some stuff related to signal processing
I'm not sure if you are into Mathematical Logic but I 'd really love to see a video from you on Gödel's Incompleteness theorems. Your channel is amazing, thank you and keep up the good work!
2nd best moment of the day: "Ah, a new 3b1b video!" Finger on the mouse button goes _click_
Best moment of the day: About halfway through the video, "Ah, I see where you're going with this!" Brain goes _click_
Could you please do a series on Abstract Algebra? (Groups, Rings, Fields etc.). Thanks!
I always questioned myself about why doing that proccess i can correctly calculate the variables values, It is not intuitive, its so beautiful to finally understand It, i just feel like some kind of gift has been given yo me, thank you!!!!!!!
considering i got this right, it is amazing that cramers rule also works in 1D, where it comes down to just a linear equation (a*x=b), where x=b/a. x=det(b)/det(a)=b/a
Was taught this in high school out of context, felt pretty detached from reality, glad to see the sense behind it
I figured this out myself at 0:53 and was so happy. Thank you so muchhhhhhh 3b1b
Your channel is the only place where we can see and feel Mathematics rather just scribbling equations! Just loved it! Good job!
We study all these things in highschool but we're never told about their use in this field, for this reason I find these videos mindblowing.
Laplace transforms and ODE series!!! (I loved the Fourier transform too) please...
Christopher Marley
Read the pinned comment. Hes doing differential eqns next.
I have probably covered a month of coursework with this channel within a day (counting exercises). I have donated to a couple of videos but honestly, I cannot pay you enough for your service
3b1b's contents have always been really articulate. The topics in the past uploads have been very complex :(
For 3 x 3, we have
z=det (i, j, mystery)
y=det (i, mystery, k)
x=det (mystery, j, k)
Then after the transformation, we have that
x det A = (output, j, k)
ydetA=(i, output, k)
zdetA=(i, j, output)
And the rules follow for x, y, z
Linear algebra really is generally taught poorly. When I took my first college-level linear algebra course, one of the questions on the final was "compute the determinant of this 5x5 matrix by hand"
That's just useless.
I am so glad this series exists. I have learned more about linear algebra from watching this series than a half dozen engineering courses that heavily use linear algebra.
this is the first time i have understood why cramer's rule works. i have looked for explanations for ages and nothing got through. thank you so much for this. this is so freaking clever
3b1b video! About Cramer's rule! Explained geometrically! On my birthday! 🎉
Please do a video on Laplace Transform!
Just when I was about to say Cramer's rule was impossible to understand geometrically, 3b1b has come in to save the day!
You truly are a blessing to mankind
ax+by=c where a and b are vectors
wedge each side with b(find the area of the parallelogram formed between the vector and b)
(a^b)x+(b^b)y=c^b
a vector wedged with itself is 0
x=c^b/a^b
repeat with wedging a on the left to get
y=a^c/a^b
sweet. Just started Lin Algebra1 at uni and your videos are a big help
Did you pass the exam?
Guess what? I probably will never forget Cramers Rule again. Thanks a lot. Amazing lectures.
Brilliant exposé as usual. I struggled around 9:27 with the reasoning leading to the numerator Area to be understood as a newly constructed determinant. It took me too long to grok that any parallelogram shaped area corresponds to a stretching of the i- plus j- hat square by an amount defined by the determinant of a square matrix whose column vectors define the parallelogram
. So just as y is unknown so also is Area unknown. But y is equal to Area/det A. Area is the determinant of a new matrix constructed as the known transformed i-hat column vector (first column of A) with the known transformed {x,y} which is the RHS of equation ie. the known coordinates of where unknown {x,y} ends up. Very obvious: after my struggles. These videos are priceless because they offer beauty also and even to those with my very modest math skills.
Thank you very much!!! Your comment was the final piece of the puzzle for understanding the big picture) I struggled around the determinant as a scale factor too)
this is because if you think of two points in space as vectors then the determinant of the matrix containing those vectors gives us the area of the parallelogram formed by those vectors but if you think of the matrix containing those vectors as a transformation then the determinant tells you about how that transformation can scale an area when it the transformation is applied to the space
My student days are far behind me. Having memorized and used the tools of linear algebra throughout my life, I am delighted to be taught the geometrical intuition behind all those machinations, specifically those tools to solve systems of linear equations in many variables, with square coefficient matrices.
But I would like to see this taken a step further.
I have sometimes had the need to estimate solutions to a large number of nonlinear equations, with a relatively small number of independent variables, typically involving coefficients based on experimental data. These types of systems can be treated with least-squares analysis, and successive approximations. The formalism generates a square matrix, where the solutions are corrections to previous estimates.of the independent variables.
While the analytical approach to generating these matrices is pretty intuitive, it would be great to see a geometrical interpretation. I have ni doubt that the creator of 3Blue 1Brown could add a lot of insight.
The best explanation for it! I have never seen such kind of explanation but the old "a matrix is a function of a determinant"...
Usually when I'm watching lectures on RUclips I would turn on 1.5x and watch as fast as possible. However when watching 3b1b's video, I never skip a single second.
@ 3:00 -ish when you show graphically the det(A)=0 solutions was profound.
Seeing the many solutions coalescing onto a single point just nails home the eigen value / eigen vector relationship, IMO.
Ok - this merits a second viewing when it's not bedtime, AND where I have time to do the 3d exercise at the end! Thanks so much for making this Grant, it's awesome.
I will be delighted if you start a serie about differential geometry and curvature
A 3b1b and a Mathologer video on the same day! What have I done to deserve this?
Oh my god, working through the ways I could find to "find out what happens to the volume of this ppp" challenge at the end made my brain *work* holy shit. Got there in the end. Found 3 ways of doing it in the end. Anyone else try this with the numbers given for the example on screen?
EDIT: So another thing I found with some poking around is that you don't even need to do Cramer's rule to work out the initial vector.
If you take the transformation matrix (M) and replace on of its columns with the output vector ( Mv) (it doesn't matter which).
Then dot the Inverse of M with that resulting matrix ( M^-1 . Mv).
The output matrix will be the identity matrix, but with the column you chose earlier now containing with full input vector!
No need to go through each term! Super neat. I was shock when I noticed it.
Was like.. Nahhhhh, that's too easy I must have made a mistake.
But I reasoned it through geometrically after and from what I can tell it should just work every time [there is a single solution].
Just saw this new video when I was reviewing for my linear algebra final!!
Usually you can think of simultaneous equations as 2 lines and finding the point of intersection, but you can also think of it as 2 points and finding the line that connects them. I did some calculations for this a while ago and ended up with a determinant on the denominator and it’s nice to see why that happens
More linear algebra for the win! Thanks for the awesome video Grant! You should do a video on differential forms and the generalized Stokes' theorem. That would just be fantastic! I just finished reading Vector Calculus by Hubbard and Hubbard and would love to see some geometric intuition into the crowning theorem of that text. Thanks again, you're the best!
It amazes me how humans can think of these sort of nuances and actually discover something while doing it
Thank you so much. I knew the Cramer's Rule. I can find x,y,z.. But now, Ican imagine the (x-y-z) with your intuitive explanations... Ten times...Thank you.
One of my life wishes is to have one video like this explaining laplace's theorem on determinants
Currently wearing a 3b1b shirt :)
wow I'm subscribed to your channel, what a surprise
I am in high school and my teacher just taught me Cramer's rule via cross multiplication method .
And , I was like , I have seen this stuff but don't recall it . Here it is , a way through determinants .
after meditating over this for a while with pen and paper, straightforward and really cool explanation
8:39
the determinant before transformation is 1. for unit vectors
so the change of signed area is y*determinant of transformation
1 goes to 1*determinant value
y goes to y*determinant value
A very nice geometric understanding of Cramer's rule, that I didn't see at all until now. It was just algebra for me. Thanks.
1:34 But Gaussian elimination is also pretty geometrically! You change the basis of the target space to the standard basis so that finding the solution is easy, but at the same time since you're doing row operations you don't change the row and null spaces so you're left with the same solution to the re-posed problem. I think that's rather neat.
I quickly forgot about Cramer's rule when I was taught it, but now I'll never forget it. Thanks 3b1b!
I love this series so far... It's clear, interesting and encouraging! Sometimes I even pause the video and try to figure out by myself beforehand, which I never do during class. All thanks to the enlightenment of this video. You really make me change my way of thinking maths. Frankly, this is the first time ever in my life I think maths is actually interesting. Thank you.
As lovely as always :)
Could you perhaps cover tensors in the future?
Smoothly progressed on this series until this chapter -- it just felt a bit hard to follow. Maybe revisiting it tomorrow is a good idea. Amazing job, a real eye-opener. Thank you!
this is channel is the best thing happened to maths
What a beautiful proof! I was taught some "abstract" derivation of Cramer's rule which I instantly forgot, as it only used abstract linear algebra (LA). But isn't LA all about the abstract background behind visual geometry?
0:00 Intro
0:50 Why learn it?
1:28 The setup
2:37 Types of answers
3:14 A mistake to learn from
5:26 The take-away
I love your channel, it certainly makes me enjoy learning and visualize everything. An small quotation. Crammer's Rule is actually awesome when you dont have a numerical matrix but one that uses variables, such as the ones we use to define regressions
I'm very much looking forward to an "Essence of Differential Equations" series of videos if that's what you're planning on.
to digest 12 mins content i need 12 hrs of rigorous stduy .
This is god level explanation
Fantastic!! I was struggling to understand that why the "Ay" will be the area of parallelogram when the Matrix is applied. Thanks again! I am trying to learn Machine Learning and hence going through Linear Algebra rigorously.
This channel truly is a bless. I remember I watch this series when they were posted, just before entering engineering school, and it really gave me interest in math, and in particular the intuition you give is great. Thank you
I'm so sad that there is no korean subtitle but still this is best lecture to understand linear algebra