That is sooo coooll!!!! Linear algebra isn't taught so intuitively in India, it's just largely algebraic without really going onto linear transformations and what they actually mean. It's taught more computationally, through adjoints, cramer's rule etc + a litle bit of geometry on what the planes do, but not what the operations themselves encode. Seeing such a fresh perspective on it is so amazing!
This video is like catnip for mathcats and I love it. Thanks for making this video. Our imagination is the only limit there is. Right now, I'm studying chemistry but I really miss those days when I used to watch your set theory videos for hours and those helped me a lot but I still don't understand anything about ordered pairs or n-tuples. I mean, I know its to represent the order but my heart doesn't accept it. I want a truly satisfying reason for it. Usually wikipedia does satisfy but this it didn't. So, can you please make a really long and detailed video about ordered pair. Thanks if you have read my comment. You(and few other amazing folks) are the reason I love math unconditionally!
I am currently taking an advanced Linear Algebra course at the University of New Brunswick in Fredericton as part of my declared minor. Your explanation is an essential concept to understand the tenets of linear transformations, that eventually is a requirement for understanding advance topics such as the non-commutative geometry.
Surprisingly, I got the intuition and picture for Gaussian Elimination early in my Linear Algebra studies, it’s nice to see a computer visualisation of it though!
If you construct the coordinate system, you know that y is a distribution over x, so a subtraction of the arc length when they exist in the same domain should yield a point symmetry along each possible state as long as it is possible. And actually you can probably just use the coefficients to determine it but it has been a while since I studied the construction of planes or coordinates.
Thank you so much, I don’t like my linear algebra course, they don’t show me this beauty; so well explained, I’m gonna start doing harder in the subject, thanks.
Okay now this video give me some idea to use 3d printing in my linear algebra class. Will apply a series of row operations on an augmented matrix and for each corresponding matrix will print a 3d visualization. And see if students if just by looking at the 3d print figure out the sequences of the visualization.
Nice demonstration. You show it, but don't really say it, and I think it could do with saying: a shear operation, applied to a line, is a rotation of that line. The two operations are indistinguishable, when applied to a line that goes to infinity in both directions (except some rotations of 90 degrees cannot be accomplished by a shear tied to particular axes). That'd tie the two parts of the video together nicely.
Thank you Trefor for this insight. I have been struggling for years to get a visual interpretation. Forgive me for hoping, that there is even more refining available?
"Thank you so much! I want to ask you how we can study math effectively. I mean, no one really explains the simple things, like how to understand definitions or theorems properly. How do you learn math? What steps do you take? Can you provide a roadmap to clarify the path (what should I start with)? Thanks in advance!"
I like what I call "the rabbit hole" method. If your teacher is telling you something you don't understand keep asking "but why" going deeper and deeper into the rabbit hole until you get something you really understand deeply and them climb back out. Basically don't allow yourself to just be mimicing what I do in a video or what a teacher does, make sure YOU really understand it.
@@DrTrefor Thanks to many internet resources, especially the ones you're providing, that rabbit hole method is a lot more doable. Thank you for everything you do.
Great video Dr. Question: we decomposrd the matrix and then took the inverse of these decompositions. Is it always true that there exists an inverse of each submatrix of the decomposition?
That'd be fun! I guess since gaussian elimination gets generalized to computing a grobner basis the geometric intuition would probably be animated in the linear case first, not quite sure what good pictures there are in the general non-linear case off the top of my head.
I've been trying to teach myself groebner basis/buchberger algorithm recently. I think the number of solutions will affect how complex the groebner basis is. For this example ruclips.net/video/De8kigoMiNU/видео.html {x^2+y^2-5=0, x^5+y^2-33=0, x^3-5xy+2=0} reduces down to a single solution point, so has groebner basis {y-1, x-2} which are a horizontal and vertical line again like the linear case shown here. If your system had 2 solutions both with the same y value eg. (1,2) and (2,2), the. you would have a groebner basis with a quadratic eg. {(x-2)(x-1), y-2}. If you were to intersect a sphere and a plane, I guess you would get a circle, something like {x^2+y^2-9}. My next problem is finding all the roots for larger polynomials, for example degree 6+. Perhaps I need to study Galois theory and/or just resort to numerical approximations.
I wanna ask question. When u said "I'm gonna take 1011 Matrix first and then1101 second and then do transformation". Actually i didn't understand using which Matrix u've transform first. Can u just help me with telling first u've taken green Matrix or blue Matrix for linear transformation??
Everything I visualize is from game and raster graphics editors such ad Photoshop… so geometrically speaking, I think of polygons and transformations. Yeah, it actually stems from LA, but by the time the end-user gets it…
That is sooo coooll!!!! Linear algebra isn't taught so intuitively in India, it's just largely algebraic without really going onto linear transformations and what they actually mean. It's taught more computationally, through adjoints, cramer's rule etc + a litle bit of geometry on what the planes do, but not what the operations themselves encode. Seeing such a fresh perspective on it is so amazing!
Glad you enjoyed! To be honest it’s often taught that way here in Canada as well
I swear. Been solving linear equations for over 15 years and I never once realized that the operations were about rotation.
It's taught like that in the US as well
Honestly
This video is like catnip for mathcats and I love it. Thanks for making this video. Our imagination is the only limit there is. Right now, I'm studying chemistry but I really miss those days when I used to watch your set theory videos for hours and those helped me a lot but I still don't understand anything about ordered pairs or n-tuples. I mean, I know its to represent the order but my heart doesn't accept it. I want a truly satisfying reason for it. Usually wikipedia does satisfy but this it didn't. So, can you please make a really long and detailed video about ordered pair. Thanks if you have read my comment. You(and few other amazing folks) are the reason I love math unconditionally!
I am currently taking an advanced Linear Algebra course at the University of New Brunswick in Fredericton as part of my declared minor. Your explanation is an essential concept to understand the tenets of linear transformations, that eventually is a requirement for understanding advance topics such as the non-commutative geometry.
Thank you for these neat geometric (and algebraic) insights into Linear Algebra which, somehow, I had never seen before!
Surprisingly, I got the intuition and picture for Gaussian Elimination early in my Linear Algebra studies, it’s nice to see a computer visualisation of it though!
This insight is a game changer for me. Thank you sir!
I would really have used this when I learn linear algebra 15 year ago.
I love linear algebra, it is such a strong tool to combine with computers.
First time ever I am able to visualize equations...thx!
Watching from Kerala India 🇮🇳 Bincy Elizabeth, Keep it up ...
If you construct the coordinate system, you know that y is a distribution over x, so a subtraction of the arc length when they exist in the same domain should yield a point symmetry along each possible state as long as it is possible. And actually you can probably just use the coefficients to determine it but it has been a while since I studied the construction of planes or coordinates.
Back when I studied math, I got really good at truth tables. I was always trying to solve every possible state which was hard lol.
Thank you so much, I don’t like my linear algebra course, they don’t show me this beauty; so well explained, I’m gonna start doing harder in the subject, thanks.
Fascinating and beautiful. Lovely. Thanks!
Nifty, Desmos finally has a 3D grapher. Pretty big fan, it's like a year old already & I had no idea.
Ya loving it for sure
Game changer for sure!
Well that was freakin' cool
hi Marc / music,py
@@johnchessant3012 o hi 🙂
Okay now this video give me some idea to use 3d printing in my linear algebra class. Will apply a series of row operations on an augmented matrix and for each corresponding matrix will print a 3d visualization. And see if students if just by looking at the 3d print figure out the sequences of the visualization.
Nice demonstration. You show it, but don't really say it, and I think it could do with saying: a shear operation, applied to a line, is a rotation of that line. The two operations are indistinguishable, when applied to a line that goes to infinity in both directions (except some rotations of 90 degrees cannot be accomplished by a shear tied to particular axes). That'd tie the two parts of the video together nicely.
Ya definitely worth saying, it really is two sides of the same coin here
Amazing visualization! Can you do another one about eigenvalues finding process too? (λI-A)
Already done in my linear algebra playlist!!
Im interested to know that can desmos can use to make videos i mean there must be issue of copyright?
I’m interested in how you got Desmos to do these things.
desmos3d is for the 3d graphs. For all it is is sliders. For example x+2y(1-s)=4-3s and then it creates a slider for the s.
Great! Thank you !
Thank you Trefor for this insight. I have been struggling for years to get a visual interpretation. Forgive me for hoping, that there is even more refining available?
Can you visualize the process of Groebner bases. My mathematician told me that it is just gaus elimination but more general
"Thank you so much!
I want to ask you how we can study math effectively. I mean, no one really explains the simple things, like how to understand definitions or theorems properly.
How do you learn math? What steps do you take? Can you provide a roadmap to clarify the path (what should I start with)?
Thanks in advance!"
I like what I call "the rabbit hole" method. If your teacher is telling you something you don't understand keep asking "but why" going deeper and deeper into the rabbit hole until you get something you really understand deeply and them climb back out. Basically don't allow yourself to just be mimicing what I do in a video or what a teacher does, make sure YOU really understand it.
@@DrTrefor Thanks to many internet resources, especially the ones you're providing, that rabbit hole method is a lot more doable. Thank you for everything you do.
Damn! This is so good!
Great video Dr.
Question: we decomposrd the matrix and then took the inverse of these decompositions. Is it always true that there exists an inverse of each submatrix of the decomposition?
Yes, every row operation can be "undone".
Ok, future video: same thing but on the ideals and Gröbner basis!
That'd be fun! I guess since gaussian elimination gets generalized to computing a grobner basis the geometric intuition would probably be animated in the linear case first, not quite sure what good pictures there are in the general non-linear case off the top of my head.
I've been trying to teach myself groebner basis/buchberger algorithm recently. I think the number of solutions will affect how complex the groebner basis is. For this example ruclips.net/video/De8kigoMiNU/видео.html {x^2+y^2-5=0, x^5+y^2-33=0, x^3-5xy+2=0} reduces down to a single solution point, so has groebner basis {y-1, x-2} which are a horizontal and vertical line again like the linear case shown here. If your system had 2 solutions both with the same y value eg. (1,2) and (2,2), the. you would have a groebner basis with a quadratic eg. {(x-2)(x-1), y-2}.
If you were to intersect a sphere and a plane, I guess you would get a circle, something like {x^2+y^2-9}.
My next problem is finding all the roots for larger polynomials, for example degree 6+. Perhaps I need to study Galois theory and/or just resort to numerical approximations.
Please what software are using to visualize this
🙏
Desmos3d!
"Zed" values?
I wanna ask question.
When u said "I'm gonna take 1011 Matrix first and then1101 second and then do transformation". Actually i didn't understand using which Matrix u've transform first.
Can u just help me with telling first u've taken green Matrix or blue Matrix for linear transformation??
May I please know what software or website do you use for visualizing linear transformation? Thank in advance!
Nice video! So, solving some boring equations is just about… bending the universe? 🤯😁🤙
Pretty much lol
Everything I visualize is from game and raster graphics editors such ad Photoshop… so geometrically speaking, I think of polygons and transformations. Yeah, it actually stems from LA, but by the time the end-user gets it…
Cool, now make a course on numerical methods from the burden and faires book, with visuals and codes in python to supplement it.