Gaussian Elimination
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- Опубліковано 13 чер 2024
- Algorithm Archive chapter: www.algorithm-archive.org/con...
I thought this was a cool visualization to show you guys.
Examples of Gaussian Elimination:
- math.dartmouth.edu/archive/m2...
- • Algebra 54 - Gaussian ...
- • Gaussian Elimination w...
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Unfortunately, the deadline has passed. I'll let you guys know about future opportunities, though!
All music was from Josh Woodward: www.joshwoodward.com/ - Наука та технологія
At first I thought this was 3blue1brown video, but then I've noticed it's only 5 minutes :)
hahaha that's a great way to tell isn't it
Yeah. I guess we theme our thumbnails similarly. Maybe I should update my style a bit.
I feel like that was more of a compliment than a complaint ;)
So 5 mins vids are not 3blue1brown? Well 3blue1brown ever made vid"s" 5 mins less so yeah... (and yep it is not a 3blue1brown)
Great visualization! Thank you.
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Thanks a bunch! Let me know if there's ever anything I can help you with.
@@LeiosLabs Can you help me with the math of comparing the coefficient [comparing the equation] that we do it, in -- |integration by parts|.
I'm on my first semester of engineering and we just learned gaussian elimination. This visual representation has opened my eyes to what I'm really doing when applying the algorithm. thanks a, lot great content!
I'm glad it was helpful!
Wow I wish I saw that visualization while I took linear algebra as an undergrad. It makes a lot of sense geometrically.
2:00 “sorry for the messy chalkboard” ha! I wish my chalkboard was that organized when I TA.
I'm glad the visualization was helpful! Also: yeah. I did a number of takes on the chalkboard this time.
OMG im a math major and it's final week and im losing interests in math (cuz of the stress), this got me interested in it again! thank you!
Haha, I am really happy to hear that!
Nice work, LeiosOS. Especially the visual explanation :-)
I'm glad it was useful!
beautiful stuff ! thanks for the visual representation which is never taught in schools
I'm glad you liked it! As for the visual representation... I think a lot of people just don't try to visualize these things because it's not clear how to do so. This one took me a while to get right, but really helped me in the end.
I love this format! Great video.
Awesome animation
Amazing video! Thank you, for this and all your videos!
I just want to provide a quick explanation about something that confused me when I was learning this. It's mostly for me if I watch this in the future, but maybe it helps someone else!!
Starting at 3:10
"Each row in our matrix is itself an equation for a plane"
That is, the (x,y,z) solution to each row of our equation gives us all (x,y,z) vectors whose endpoints lie on a plane.
To elaborate, all vectors whose dot product with [2,3,4] is equal to 6 have endpoints on the blue plane. All the vectors whose dot product with [1,2,3] is 4 have endpoints on the red plane. And all vectors whose dot product with [3,-4,0] is equal to 10 have endpoints on the green plane.
"Their points of intersection is the solution we found before"
The intersection of the three planes is the endpoint of a vector whose dot product with the first row of our matrix is 6, whose dot product with the second row of the matrix is 4, and whose dot product with the third row of the matrix is 10. It satisfies all three requirements of our matrix equation, so it's the solution to the equation.
"No matter how we change the planes with Gaussian Elimination, the solution remains the same."
Let r1 denote the first row vector of our matrix, r2 the second row, and r2 the third row.
Let x denote the solution.
r1 dot x = 6
r2 dot x = 4
r3 dot x = 10
Now, what's a dot product? What's the geometric interpretation of r1 dot x?
r1 dot x the component of r1 that lies on x scaled by the magnitude of x.
r2 dot x is the component of r2 that lies on x scaled by the magnitude of x.
So, (r1+r2) dot x is the component of (r1+r2) that lies on x scaled by the magnitude of x.
From that interpretation, it becomes pretty geometrically obvious that if r1 dot x = 6, and r2 dot x = 4, then (r1 + r2) dot x = 10
Therefore, the vector x that satisfies (r1+r2) dot x = 6+4, or any other linear combination of the rows, must be the same vector that satisfied the original system of equations.
"The planes wobble about until one of them is parallel to 2 of the 3 axes"
Going back to the dot product picture, we're linearly combining the row vectors of our matrix until one of them lies entirely on one of the axes - in this case, where we're ending up with an upper triangular matrix, on the z axis.
Once a vector (r3) lies entirely on the z axis, its easy to solve for the z compnent of x that satisfies the new dot product equation.
Thanks! Gaussian elimination always seemed to me like one of those math tricks that work but that no one takes the time to show why. This visualization made everything clear!!
Amazing explanation man.. I searched everything on UA-cam for a clear explanation and finally found it in this video which I in fact skipped a couple of times…!!!
I've never seen that visualization before, and it was quite nifty.
This is very helpful to see elimination visually. Very cool and elimination makes more sense now
Love your visualization. It would be really nice to see when I first learned to solve the matrix.
You are doing an amazing job explaining hard topics intuitively. You need a hit video. I know it is hard but a video solely based on animations with an interesting topic, that might be presented to the general audience can give your hit. Good luck and keep up the good work!
I am glad you like the content and definitely agree that if I want to succeed on youtube, I need to attack topics that have a broader appeal; however, I don't know if I really want success on youtube right now. I'm kinda happy with this as a side-job for now and instead working on topics I find interesting. We'll see if that changes in a few months or years.
@@LeiosLabs You are really being helpful to others and doing great, especially for someone doing it as a side-job. Hope everything goes great for a young brilliant guy like you and you find what suits best for you. Keep rocking.
Your videos are fantastic, James!
Wish this guy still made content. Feel likes he's the kinda guy who would steadily improve his content, if he was able to be consistent. Might become something special
I still make content... it just takes a while
Great video :D
That's a very pretty visualization :D
Thanks! Nice Channel you got there!
@@LeiosLabs Thanks! Channels like yours are a huge inspiration for us :)
WoW! Thanks for this video... I'm Electrical Engineer and now I understand the Gaussian Elimination for you! The 3D animation was amazing and very illustrative :D
These are some of the best math videos on youtube.
Thanks...Now I understood what's the concept behind solving equations through matrices
Awesome video! :D
Thanks!
That was cool.. specially the graphics part.. hey boy !!, U r making maths fun for me.. keep growing
It's very nice to see different visualizations for the same thing. You showed gaussian elimination by looking at the matrix row-by-row, where each row yields a plane equation. But you can also look at it column-by-column, where each column is the vector of the basis of that matrix. For those interested, keep reading :)
In effect, the matrix is a transformation function that can be applied to a point, and we know only the answer of applying it to a point. We want to know what point the transformation was applied to. In other words, we want to do the inverse transformation to the given point.
Applying the matrix to the usual (i, j, k) orthonormal basis yields a warped basis: i is mapped to the first column of the matrix, j to the second and k to the third. Gassian elimination operations can be seen as shearing and scaling transformations applied to both the warped basis and the point. After applying all the operations, the warped basis is back to the original (i, j, k) basis, and the point has been transformed to the answer we are looking for.
Whew, this is hard to describe in text... Anyway, I got inspired for this explanation by 3blue1brown's Essence of Linear Algebra, especially this video: ua-cam.com/video/uQhTuRlWMxw/v-deo.html
thanks this visualization help me a lot
Cool.. No mess.. Very helpful visually. 😊
Thanks for the amazing explanation !!
This was SO helpful. Thank you
I was curious to see how the elementary operations geometrically change the planes in each equations. And how put together the algorithm might have a geometric intuition
I already took linear algebra but I had never seen this before, thank you!
I love the video, I learned quite a lot from this.
PS: You made a Typo in the algorithm archive
"This creates a matrix that *sometiems* resembles an upper-triangular matrix;"
Thanks for the catch. I'll go ahead and fix it on a local branch.
Dude its super cool....Thank you
This is such an excellent visualization of the effect of gaussian elimination, much better explained than my professor!
Great visualization :)
I like the literal plug you drew
Beautiful 3D graphic! Thanks
It might look nicer if visualized using disks.
That's interesting. Why?
@@LeiosLabs That way you wouldn't have the edge of a square poking out, and rotation along plane normal doesn't/shouldn't matter.
@@redline6802 You might be right. It's something to think about next time I do the visualization. Thanks!
Great one
Thanks for this video
This channel is cool B']
hey this is interesting,,, definitely need much more from u regarding mathematics ..related to engineering u will becoming a teacher ..love from an engg
👌
I heard Gauss himself was able to solve this in third grade, which is absolutely crazy
seasong Considering the guy is gauss i am not surprised.
@@nejlaakyuz4025 Haha, you guys are great!
Nice topic.
This is what I’m talking about, thank you
wowowow so the matrix turns into an identity matrix... which totally makes sense duh
Yup. I was kinda blown away by this too when I saw the visualization for the first time.
Awesome!
awesome, thanks a lot
Very Nice, plz make some thing whit cuantic fizic
I thought the visualization was _very_ cool. 👍
Thanks!
It would be interesting to do a video showing the three to types of row operations don't affect the point of intersection.
I was thinking about following up a lot of my videos with a more in-depth visual. This would be a good one to start with!
Does Gassian Elimination or Gaus Jordan Algorithm have an effect on the Eigenvectors/values ?
Hello brother can u will able to give the visualization of inverse matrix,
Why do we inverse it I am not able to visualize please can u will able to do video on it
Please come back
Well, at least I learned what is Gaussian elimination.
I remember, that we heard about it last year in high school, and they told we'll learn it in university. Then we learned how to solve linear equation systems by adding/subtracting multiples of the equations from each other... Basically, we used the same thing, but named differently...
that is cool
A\b
I’m curious as to what it would look like if there wasn’t a solution.
At first, there would be no point where the three planes touch each other. After doing Gaussian elimination you'll probably end up with at least two parallel planes.
Well, if you do Gaussian Elimination in that case you should get something like 1=0 in one equation so it would mean probably that one of the planes disappeared and there's no longer any point which belongs to all 3 planes.
On the other hand when you have infinite amount of solutions one or more of the planes should be streched infinitely on the whole 3d space
Spaghettificated I thought if there are infinite solutions then all the planes will be the same plane after Gaussian elimination.
Javulicraft wouldn’t you still have two eigenvalues for a 3x3 matrix.
@@tsgoten Yes, there still would be 3 eigenvalues (if you count 0 as one of them), but what is your point with that?
ok..... even i didnt get all of it properly i am satisfied asuming that i have got answer of question that always run in my mind..........
Q.What would my most of the science and math theoritical problem look like in visual form????
How is each row a plane?
Which software did you use for the visualization?
Just blender
@@LeiosLabs Thanks.
Where were you when I was busy getting a crap grade in linear algebra?
Thought this was a 12Tone video with your voice.
That's a huge complement.
Gaussian blur
Hold up Marius stole my last name
Oh my word. Please tell me you guys are related somehow.
Cool...!!! Why you have so less likes and less subscribers... !!! share share share...
All mathematics is really geometric. Geometric algebra is the true maths.
Algebraic geometry is awesome, too! Advanced results like Gelfand duality and the Serre-Swan theorem really show the beauty of the connection between algebra and geometry.
@@ZardoDhieldor You guys are awesome!
Too late! The finals have already passed! You had to make this video two months ago!
Ps: just kidding.
Honestly, I wanted the video out 2 months ago too... Life happens.
why can't teachers teach like this ?
It's hard.
way too fast