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Current time:0:00Total duration:9:22

Video transcript

hello everyone so in these next few videos I'm going to be talking about something called the Jacobian and more specifically it's the Jacobian matrix or sometimes the Associated determinants and here I just want to talk about some of the background knowledge that I'm assuming because to understand the Jacobian you do have to have a little bit of a background in linear algebra and in particular I want to make sure that everyone here understands how to think about matrices as transformations of space and when I say transformations here let me just get kind of a matrix on here I'll call it 2 1 and negative 3 1 you'll see why I'm coloring it like this in just a moment and when I say how to think about this is a transformation of space I mean you can multiply a matrix by some kind of two-dimensional vector some kind of two-dimensional XY and this is going to give us a new two-dimensional vector this is going to bring us to let's see in this case it'll be alright kind of 2 1 negative 3 1 where what it gives us is 2 X plus negative 3 times y and then 1 X plus 1 times y right this is a new two dimensional vector somewhere else in space and even if you know how to compute it there's still room for a deeper geometric understanding of what it actually means to take a vector X Y to the vector 2 X plus negative 3 y + 1 X plus 1 Y and there's also still a deeper understanding in what we mean when we call this a linear transformation a linear transformation so what I'm going to do is just show you what this particular transformation looks like on the left here where every single point on this blue grid I'm going to tell the computer hey if that point was X Y I want you to take it to 2 X plus negative 3 y 1 X plus 1 Y and here's what it looks like so let me just kind of play it out here all of the points in space move and you end up in some final state here and there are a couple important things to note first of all all of the grid lines remain parallel and evenly spaced and they're still lines they didn't get curved in some and that's very very special that is the geometric way that you can think about this term this idea of a linear transformation I kind of like to think about it that lines stay lines and in particular the gridlines here the ones that started off as you know kind of a vertical and horizontal they still remain parallel and they still remain evenly spaced and the other thing to notice here is I have these these two vectors highlighted the green vector and the red vector and these are the ones that start it off if we kind of back things up these are the ones that started off as the basis vectors right and we kind of make a little bit more room here the green vector is 1 0 1 in the x-direction 0 in the Y direction and then that red vertical vector here is 0 1 and 0 1 and if we notice where they land under this transformation when the matrix is multiplied by every single vector in space the place where the green vector lens the one that started off as one zero has coordinates two one and that corresponds very directly with the fact that the first column of our matrix is 2 1 and then similarly over here the second vector the one that started off at 0 1 ends up at the coordinates negative 3 1 and that's what corresponds with the fact that the next column is negative 3 1 and it's actually relatively simple to see why that's going to be true here I'll go ahead and multiply this matrix that we had that was C now it's kind of easy to remember what the matrix is right I can just kind of read it off here as 2 1 negative 3 1 but just to see why it's actually taking the basis vectors to the columns like this when you do the multiplication by 1 0 notice how it's going to take us to so it's 2 times 1 that'll be 2 and then negative 3 times 0 so that'll just be 0 and over here it's 1 times 1 so that's 1 and then 1 times 0 so again we're adding 0 so the only terms that actually mattered because of this zero down here was everything in that first column and similarly if we take that same matrix to one negative 3 1 and we multiply it by 0 1 over here by the second basis vector what you're going to get is 2 times 0 so 0 plus that element in that second column and then 1 times 0 so another 0 plus 1 times 1 plus that one so again it's kind of like that 0 knocks out all of the terms in other columns and then like I said geometrically the meaning of a linear transformation is that grid lines remain parallel and evenly spaced and when you start to think about it a little bit if you can know where this green vector lands and where this red vector lands that's going to lock into place where the entire grid has to go and let me show you what I mean and how this corresponds with maybe a different definition that you've heard for what linear transformation means if we have some kind of function L and it's going to take in a vector and spit out a vector it's said to be linear if it satisfies the property that when you take a constant times a vector what it produces is that same constant times whatever would have happened if you apply that transformation to the vector not scaled right so here you're applying the transformation to a scale vector and evidently that's the same as scaling the transformation of the vector and similarly second property of linearity is that if you add two vectors it doesn't really matter if you add them before or after the transformation if you take the sum of the vectors then apply the transformation that's the same as first applying the transformation to each one separately and then adding up the results and one of the most important consequences of this this formal definition of linearity is that it means if you take your function and apply it to some vector X Y what can split up that vector as x times the first basis vector X times 1 0 plus y let's see Y times that second basis vector 0 1 and because of these two properties of linearity if I can split it up like this it doesn't matter if I do the scaling and adding before the transformation or if I do that scaling and adding after the transformation and say that it's x times whatever the transformed version of 1 0 is and I'll show you geometrically what this means in just a moment but I kind of want to get all the algebra on the screen plus y times the transformed version of 0 1 0 1 so to be concrete let's actually put in a value for x + y here and try to think about that specific vector geometrically so maybe I'll put in something like the vector 2 1 so if we look over on the grid we're going to be focusing on the point that's over here at 2 1 and this particular point and I'm going to play the transformation and I want you to follow this point to see where it lands and it's going to end up over here okay so in terms of the old grid right the original one that we started with it's now at the point 1 3 this is where we've ended up but importantly I want you to notice how it's still 2 times that green vector plus 1 times that red vector so it's satisfying that property that it's still x times whatever the transformed version of that first basis vector is plus y times the transformed version of that second basis vector so that's all just a little overview and the upshot the main thing I want you to remember from all of this is when you have some kind of matrix you can think of it as a transformation of space that keeps gridlines parallel and evenly spaced and that's a very special kind of transformation that is a very restrictive property to have on a function from 2d points to other 2d points and the convenient way to encode that is that the landing spot for that first basis vector the one that started off one unit to the right is represented with the first column of the matrix and the landing spot for the second basis vector the one that was pointing one unit up is encoded with that second column if this feels totally unfamiliar you want to learn more about this it's something that I've made other videos on in the past but in terms of understanding the Jacobian matrix where we're going with this and kind of getting a geometric feel for it that short overview that I gave should be enough to get us going so with that I will see you next video