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Cramer's rule, explained geometrically | Chapter 12, Essence of linear algebra
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- เผยแพร่เมื่อ 15 มี.ค. 2019
- 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.
@@RoselineJerryA hahaha!! parariririid
@@RoselineJerryA hahaha!! parariririid
Sometimes it sucks being German. I have to call these "Spat"
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
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 But other plans have risen above that in the project list, so it seemed a bit silly to keep it unpublished for too much longer.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??!
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
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.
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?
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".
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 TH-cam.
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
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
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.
This was just brilliant! Couldn't ever think cramer's rule could even have such a relation with geometry!
You are an artist !!!
Kids in grad school everywhere will learn so much faster because of how visually you can communicate ideas.
You just made sense of a lecture I struggled through 35 years ago. Thank you, it now makes sense.
Your channel is the only place where we can see and feel Mathematics rather just scribbling equations! Just loved it! Good job!
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 :)))
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.
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
I was just thinking about this topic yesterday, and how it works. Thanks for the in depth guide.
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!
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.
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
This explanation is gold. Much more illuminating than the straightforward but obscure proof using the properties of determinant.
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.”
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 🥲
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.
the essence of calculus and linear algebra series(serieses?) are truly amazing ...extensions of them would be so cool and appreciated ;D
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. 👏🏼👏🏼👍
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 ;)
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.
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
why u upload at 12:35
now I gotta deprive myself of sleep.
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
Rainy Sunday morning, coffee, chocolate and this video. There is nothing else I could ask for! This is perfect!
Oh dang dude, you were completely right! Did a few Cramer Rule exercises and the concept of the dual vector clicks! Appreciate it!
Just started learning this in my linear algebra course, perfect timing thanks!
I watched your playlist before the beginning of my Linear Algebra course this semester and it gave me a great geometric intuition, but at this point in the course the intuition has started to get swallowed by all the weird math stuff. Excited to watch (when I get the chance) this video to hopefully reignite that wonderful intuition!
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!
lol I was just telling someone about a 3blue1brown video and here comes another one!
And it explains Cramer's Rule!
You're a genuine genius!Take love.
I enjoyed this one along with all the previous videos.
Wow!!! Thank You so much, really missed you these last few weeks!!
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!
OMG I'm shocked and really excited with this series, thank you for doing this
I learned matrices last year and never understood Cramer's rule.
thanks for the vid :)
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".
The best explanation for it! I have never seen such kind of explanation but the old "a matrix is a function of a determinant"...
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.
Your channel & content represents an essential milestone in the evolution of maths education
This channel is amazing! I love everything on it
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
after meditating over this for a while with pen and paper, straightforward and really cool explanation
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
Just saw this new video when I was reviewing for my linear algebra final!!
This video series is truly core of linear algebra I think.I really thank 3blue1brown team to make me higher level!
For the first time I understood the WHY behind Cramer's rule. Thank you so much!
I quickly forgot about Cramer's rule when I was taught it, but now I'll never forget it. Thanks 3b1b!
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.
It's an amazing geomatric understanding of Cramer's rule!!! Thank u so much
sweet. Just started Lin Algebra1 at uni and your videos are a big help
Did you pass the exam?
Amazing job!!! It feels great to get to finally understand the logic behind the equation. Many thanks :-)
great channel, helped me understanding the concepts of linear algebra.. thank youu sooo much
Brilliant. I wondered why Kramer's rule worked since highschool and I finally got (and understood) the answer
Amazing content, thank you sir, you have a special place in my life to reveal the bright side of mathematics
3b1b's contents have always been really articulate. The topics in the past uploads have been very complex :(
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!
Was taught this in high school out of context, felt pretty detached from reality, glad to see the sense behind it
Eye opener, thank you so much for your hard work!
the linear algebra is so beautfiul! thank you for showing us that!!
I've never seen a better explanation awesome !!!
This channel is such a gem
amazing video with amazing animations. also that pi in beginning was so cute!!!
Loved it, this is where Concepts are born..!
Wonderful explanation. I'm taking linear algebra this semester and we saw the Cramer's rule a few weeks ago, but we weren't explained where it came from. This video is like Christmas to me
My deespest gratitude to this gorgeous videos, you and : smartereveryday; brian douglas; zach star and jim pytel are awesome youtubers
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!
Oh my gosh! What a wonderful way to intuitively understand Kramer’s rule!
These are really good. It turns on the intuition and imagination.
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
It should be a crime for a series this thorough and amazing to leave out SVD
I'm very much looking forward to an "Essence of Differential Equations" series of videos if that's what you're planning on.
I will be delighted if you start a serie about differential geometry and curvature
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.
I finally understand Cramer’s rule. Thank you so much!!
You'll definitely be the first channel I'll support on Patreon once I'll get my degree and a job, ... in a month :D
Just when I was about to say Cramer's rule was impossible to understand geometrically, 3b1b has come in to save the day!
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.
What a nice way to start Sunday morning, with a 3Blue1Brown video
As lovely as always :)
Could you perhaps cover tensors in the future?
I wish, I had you as my math teacher in my high school! Please keep your video coming...
I have no idea what you're talking about, but your videos are just the fucking best. Keep it up, my man.
One of my life wishes is to have one video like this explaining laplace's theorem on determinants
Great video as always. If you have plan to further extend the linear algebra series, can you do a video about singular value decomposition (SVD)? It is something that I struggled to understand for a long time.
Usually when I'm watching lectures on TH-cam I would turn on 1.5x and watch as fast as possible. However when watching 3b1b's video, I never skip a single second.
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!!!!!!!
Guess what? I probably will never forget Cramers Rule again. Thanks a lot. Amazing lectures.
A 3b1b and a Mathologer video on the same day! What have I done to deserve this?
wow. just wow . completely blew my mind. thank you, grant.
this is channel is the best thing happened to maths