 One use of the transformation matrices is to describe geometric transformations. So, for example, let's say I have a point in R2, and I want to consider the geometric transformation Mx, which will be the reflection of my point across the x-axis. So remember, if I have a point with coordinates x, y, the coordinates tell me how to get to the point from the origin. We'll go x units horizontally, and then y units vertically. Now what happens when we reflect this point across the x-axis? Well, the actual vertical distance is still y, but this time because we're going down instead of up, our coordinates are going to be x negative y. So our new x-coordinate is x, the same thing we started with, and our new y-coordinate is negative 1 times y. And because we want to write the coefficient matrix, we'll have to write x as 1x plus 0y, and a new y-coordinate will be 0x plus negative 1y. And so keeping only the coefficients, we'll get our transformation matrix. Or we can take another transformation. Again, let's take a point in R2, and this time we'll consider the transformation R90 degrees, which will be the rotation of our point about the origin through an angle of 90 degrees counterclockwise. So as before, we'll have our point P with coordinates x, the horizontal distance, and y, the vertical distance. And then we'll spin our point right round, 90 degrees. And as before, the lengths themselves have remained unchanged. But this time our horizontal distance will be negative y, and our vertical distance will be x, so our coordinates will be negative y, x. And so, algebraically, our transformation is going to give us a new x-coordinate of negative y, and a new y-coordinate of x. And again, because we want to be consistent, we need to write this as 0x plus negative 1y, and 1x plus 0y. And the coefficients of these formulas give us our transformation matrix R90. Once we have these transformation matrices, we can perform a sequence of geometric transformations. Suppose I have a sequence of geometric transformations and the corresponding transformation matrices. If I perform these transformations in sequence, I'm going to indicate that sequence of transformations in the following way. And it's important to note here that the first transformation that we perform, T1, is the rightmost transformation. The second transformation, T2, will be on the left, and so on. So for example, MxR90 degrees is going to be the transformation where first I rotate by 90 degrees counterclockwise, and then I reflect across the x-axis. On the other hand, R90Mx is going to be the transformation where first I reflect across the x-axis, and then I rotate 90 degrees counterclockwise. And because this is still a geometric transformation, we can write down the transformation matrix corresponding to this composition of geometric transformations. So let's see how that might work. So let's take a look at R90 degrees Mx. That's a reflection across the x-axis first, followed by a 90-degree counterclockwise rotation. So as before, we'll have our point xy with horizontal and vertical distances. We'll reflect across the x-axis first, and then rotate 90 degrees counterclockwise. So as before, all of our actual lengths have remained unchanged. However, our directions for getting to the point will be a little bit different. So this time, we have to go a vertical distance of x and a horizontal distance of y. And so our coordinates are going to be horizontal distance y, vertical distance x. So our new x-coordinate will be y, which we'll write down as 0x plus 1y. And our new y-coordinate will be x, which we'll write down as 1x plus 0y. And so we'll use the coefficients of these formulas to get our transformation matrix. What about Mx R90 degrees? That's the transformation that we perform by first rotating 90 degrees counterclockwise around the origin and then reflecting across the x-axis. So as before, we'll have our point xy with horizontal distance x, vertical distance y. We'll rotate 90 degrees counterclockwise around the origin, and then we'll reflect across the x-axis. And this time, in order to get to the point, we'll have to go down a vertical distance of x and left a horizontal distance of y. And so the coordinates of the point will be minus y, minus x. And so our new x-coordinate will be minus y, which we'll write as 0x plus negative 1y. And our new y-coordinate will be negative x, which we'll write as negative 1x plus 0y. And finally, we'll record just the coefficients to get our transformation matrix.