 In this video I want to talk about what a shearing map or sometimes call a transvection, what this does to planar geometry. Now this is honestly going to be probably the weirdest of all of the geometric transformations we talked about in this lecture here and that's just because this one might feel less intuitive but its connection to the triangle of matrices is quite solid and it's worth mentioning. So what does one mean by a shearing map or by a transvection? Well the formal definition is that this is going to be a linear map that displaces each point in a fixed direction by an amount proportional to its sine distance from a line that is parallel to that direction. Like I said that sounds kind of weird right? So what we're going to do is the following. We're going to come over to Desmos to try to illustrate what one means by a transvection and so the transvection is going to be determined by some scalar value m. Right now I've set m equal to zero and so you don't see the line right here. Right now you see that the y-axis is highlighted in blue. I'm going to adjust that in just a second. So as m is allowed to get bigger we're going to see that this line is bending right here. The idea is if we have like this rigid rod of some kind as m gets bigger it moves along and in this point it looks like it's spinning around the origin which is correct. The point of a transvection would be something like the following. If we take a point let's color it right here so this is the point zero comma two. The transvection says the following. If you are at zero comma two you're going to move over here to the point which is now going to be where we along the x-axis this is going to be three comma two. This is going to move over to that point right there. On the other hand if you started off with the point let's say zero comma one you're going to move over here to be the point one one and a half right. Let me fix that. 1.5 comma one and then if you're at this point up here let's do like say four you're going to get zero comma four. This would move all the way over here to this point on the line which we would end up with six comma four right there and so this is how transvection works is that you move if we're doing a horizontal shear right now so this causes points in the plane to move to the right or to the left like for example if you're below the x-axis here you're at the point zero comma negative two. This actually causes you to move to the left to this point right here and so we would then get negative three comma negative two as the point it moves over to. That's why I meant by sign distance. You're going to move a distance either positive or negative proportional to where you are with respect to the x-axis. Here was the x-axis. Those things that are above the x-axis are going to move to the right like we see with these three examples right here. Things that are below the x-axis are going to move to the left. That's the one we went by sign distance and the things that are farther away from the x-axis will move farther to the right or to the left. So like this point right here is four units above the x-axis. It moved four times as far to the right as opposed to this point right here which was one unit above the x-axis and so everything shears. It's like we're bending space here. Things above the x-axis been to the right things below the x-axis been to the left. So let's let's clean this thing up a little bit and so then I want to show you what this thing does to the unit square. Again let me get this stuff off the screen. There we go. So if I were to turn on the standard unit square you see this blue little square right there nothing has happened to it whatsoever. I'm going to put back my rod to the y-axis there. And let's turn on this thing right here. So you now see a square although the blue line kind of is coming up the left side of the square as I allow this thing to bend to the right you can see what's happening to our unit square. The original square was right there right in blue and so as we bend this thing as we increase this factor of m so if we shear by a factor of one you can see what happens is that this square has been bent. It starts leaning over forming this parallelogram which you can see there on the screen right here. And so imagine it's kind of like it's kind of like the following idea. If you think of your square as like a stack of cards like you have a deck of cards in front of you if you kind of push it and it starts to lean the things on the bottom that is those points that are on the x-axis they don't move at all. Those things that are above the x-axis will move proportional to how far away from the x-axis they are. That makes a difference. We're going to zoom in a little bit here. And so as we continue to increase the factor for which we're shearing this parallelogram will get more distorted getting stretched in that direction and then things over on the bottom below the x-axis would go the other way around as well. If we were to shear by a negative amount this causes it to go the other way and so a shearing map is a map that's going to turn a it's going to turn a rectangle or in this case a square it'll turn us this square or rectangle into a parallelogram. And so that's what a shearing map does geometrically. So how does one capture that algebraically? What type of matrices cause these transvections? Well turns out these are going to be triangular matrices so the matrices I want you to consider are going to be some of the following. Take the upper triangular matrix 1M01 so this is a this is a so-called unit upper triangular matrix upper triangular matrix like so. And the other type we're going to talk about over here this would be a lower unit triangular matrix. It's unit triangular because you have ones along the diagonal upper triangular because everything below the diagonal is zero. Like I said the other one's going to be lower triangular. Now these type of matrices are in fact elementary matrices. These are these are elementary matrices of replacement type. This first one right here that's circled on the screen these are the types of replacement matrices you would use in the backwards phase of Gauss-Jordan elimination. These are going to be the upper triangular matrices. On the other hand these ones over here these lower triangular matrices these are going to correspond to the forward phase of Gauss-Jordan elimination. And so what the difference between a backwards phase and a lower phase that is the difference between an upper triangular matrix and a lower triangular matrix in this situation is that these matrices will will shear the plane but one will do it horizontally and the other one will do it geometrically. And so for example consider if we take this upper triangular matrix and we multiply it by the vector x y. There's the generic vector in the plane. What does it do here? Well when you multiply by the first row you're going to end up with x plus m times y and when you do the second row you're going to end up with a y. So geometrically what's happening here is well let me if you rewrite this in a slightly different manner right you have x y the original point but then you have this my zero. So what's happening is you are adding something or subtracting depends on the sign of m. You're changing the x coordinate the y coordinates left inert. The x coordinates being changed and it's going to be changed by a factor of m but that m is going to be proportional to the y coordinate. So if your y coordinates positive you'll be adding m. If y coordinates negative you're going to be subtracting m and the bigger y is the bigger the amount you either add or subtract. So this is what we mean by you are going to add an amount proportional to your sign distance from the axis that you're parallel to. So if you're shearing along the x axis you're going to add some multiple of the y coordinate to x okay and you see the same thing happening whoops you see the same thing happening when you do it with this one over here the lower triangular matrix if you times that by x y you're going to end up with x and you're going to get y plus mx. I wrote it the sum backwards but that doesn't make much of a difference. So you're going to change the y coordinate proportional to x by a factor of m and so what we then see is that multiplying by these unit upper triangular matrices causes a horizontal shear to happen and multiplying by a lower triangular matrix causes a vertical shear to happen. At some examples of shearing so let's consider the two vectors in the plane u will be the vector 2 1 and v will be the vector 1 2 and so let's first see what happens if we want to horizontally shear the plane by a factor of 2. Like we saw in the previous slide if you're going to horizontally shear a vector the vector plane there then that's going to come correspond to a unit upper triangular matrix which if we if we shear by a factor of 2 then that off diagonal entry is going to be a positive 2 and so let's see what happens to our vectors if we first do multiplication by the vector u we take the first row times it by u you're going to get 2 plus 2 which is equal to a 4 and then if we do the second row times u we're going to get 0 plus 1 which is a 1. You'll notice that the y coordinate doesn't change when we did this multiplication of the matrix and that's because if you horizontally shear the y coordinate doesn't change whatsoever. On the other hand the x coordinate changed it increased by a factor of 2 that is we added to to the original x coordinate so if we take the vector u we're going to move two spaces to the right and that's what this shear does to it. On the other hand if we take the vector v which is 1 2 if you take the first row of the matrix times the vector you're going to get 1 plus 2 times 2 so 1 plus 4 which is 5. Okay on the other hand the second row times the vector you'll get just back at 2 again so again when you horizontally shear the y coordinate doesn't change it's the x coordinate changes you go from 1 to 4 so if you take the vector 1 1 2 it's going to move to the vector over here 5 comma 2 and so notice the distance between the two vectors right u as it moves to its image u prime moves over by two units but v when it moves to its image v prime it moves over four units and that's because the distance between the x axis with the point is double when you go from when you consider v versus u so the farther you are away from the x axis the farther to the right you're going to move. Now that happens if you're above the x axis if you're below the x axis you actually would move to the left by this shearing map right here so now let's see what happens to the what happens to the unit square the standard unit square you see right here in blue it's just the point 0 0 1 0 0 1 and 1 1 this shearing map will turn it into this parallelogram you see right here 0 0 1 0 so the things on the x axis don't move when you do a horizontal shear on the other hands the point 1 0 is going to move to to the point 2 1 and then the point 1 1 is going to move over here to the point 3 1 you see that you see that happening right there and so that's an example of a horizontal shear let's look at a vertical shear this time so let's vertically shear by factor of negative 2 so it'd be nice to see what a negative shear does well when you have a vertical shear this corresponds to multiplying by a unit lower triangular matrix those matrices that come from the forward phase of Gauss-Jordan elimination and so you see something like this this is the matrix that'll shear the plane vertically by a factor of negative 2 when you multiply this matrix by our vectors u and v remember u is the vector 2 1 and v was the vector 1 2 when you multiply by the first row by u you're just going to get back a 2 the x coordinate didn't change you'll notice those things match up this time because now that's because we're vertically sharing vertical transformations will not affect the horizontal and a horizontal transformation will not affect the vertical uh the two are independent of each other so if we take the second row times the vector you're going to get negative 4 plus 1 which gives us a negative 3 so 2 1 is going to move to 2 negative 3 so u which starts off here at 2 1 is going to move four units down to the point 2 negative 3 the other one if we take v this time you times that by the first row you're just going to get back a 1 right the 1 the x coordinate didn't change it stayed 1 on the other hand if you take the second row times the vector you end up with a negative 2 plus 2 which gives you a 0 so the vector point right here 2 or 1 2 is going to move down here to become 1 0 so move down and so notice the difference again right v moves two units down u moves four units down and that's because now the distance between v with the y-axis has now been flopped so u is twice as far from the y-axis that v is so therefore u is going to move down twice as much as v does these things are proportional to each other and notice things move down because we're doing a negative shear in the situation if you were to the left of the y-axis you would actually be moving upward uh by a factor proportional to how far to the left you are things move backwards when you have a negative shear if we look at the unit square right here so the usual points 0 0 1 0 0 1 and 1 1 if you move this thing those things on the y-axis don't move so 0 0 and 1 and 0 1 are left un unmoved in this process on the other hand the point 1 0 is going to move down by 2 so it moves down to the point 1 negative 2 and the point 1 1 is going to move down by 2 so it's going to move to the point 1 negative 1 and you see that that's because these points are one unit away from the y-axis therefore they're going to move by a factor of 2 down