 So we saw previously that every two by two non-singular matrix can affect the plane by some combination of shears, reflections, and stretches or compressions there. And so those are the only three operations you need for these geometric transformations, shearing, reflection, and stretching. But like I said earlier with the shearing maps, they're a little bit unintuitive on how you would actually use that. Like imagine we're trying to do some type of computer graphics. We want to modify a picture. Will dilation contraction kind of make sense? Reflection kind of makes sense. But the distortion that comes from a shearing map, it seems a little less intuitive why one would need that. We mostly introduced it because it corresponded with the elementary matrices, but a geometric transformation that's probably more intuitive. Again, if you were working with graphic design or something would be the idea of a rotation. So how does one accomplish a rotation in the plane? Well, if you want to rotate by some angle theta, turns out you're gonna use the following matrix right here, cosine theta minus sine theta, first row sine theta, cosine theta for the second row. And so what this will do is if you multiply by, if you multiply a vector xy by this matrix right here, it'll have the effect of a counterclockwise rotation in the plane by this angle theta, which could be in radians or degrees, whichever measure of angle you want, it doesn't really matter. So, but be aware that this matrix right here, you see on the screen, this is a counterclockwise rotation. If you wanted to do a clockwise rotation, that would actually be the inverse, that would be the inverse matrix of this one given right here. So the opposite of rotating counterclockwise with the rotated clockwise. And you can actually show that the inverse of the matrix cosine negative sine sine cosine is equal to cosine sine negative sine cosine right there. I'll let you verify using some trig identities that this is in fact equal to the inverse matrix right there. But I also want to show you that this matrix right here, cosine negative sine sine cosine, if you take the transpose of this matrix, that's just this matrix right here. So this is an interesting property right here that the inverse of this rotation matrix is actually equal to its transpose. We saw that observation with permutation matrices. Turns out rotation matrices also have this property that its inverse is just equal to its transpose. These are both examples of what one calls an orthogonal matrix. Something we'll talk about a little bit more, we'll talk more about it later on in this series. Let's focus on rotation matrices right now. So if we take, if we want to talk about the rotation matrix associated to rotating counterclockwise by pi halves, if you've forgotten what radians mean, we're gonna do a quarter spent here, 90 degrees. So let's think about the rotation matrix associated to a 90 degree rotation. So what we saw previously, right? So counterclockwise rotation means we're gonna take cosine of 90 degrees, negative sine of 90 degrees. We're gonna then take sine of 90 degrees and then cosine of 90 degrees. Whoops. The location of the negative sign is imperative. Happen, the negative sign on the upper right will give you a counterclockwise rotation. The negative sign in the lower right will give you a clockwise rotation. The orientation does matter right there. And so doing some basic trigonometry, right? Cosine of pi halves is gonna equal zero. So you're gonna get zeros along the diagonal here. Sine of 90 degrees or pi halves, that's equal to one. So you get a one and negative one the other way around. So that's what the rotation matrix of rotating 90 degrees is gonna do. And so let's take just a couple of points right here. Let's take the point four, one, two, three. So we'll call these vectors u and v. Let's also look at their sum. Notice if you add u and v together, you'll get six and four, all right? So what does this matrix do to them? In terms of multiplication, it's pretty easy to see. If I take zero, negative one, one, zero, and times it by four, one, you're gonna get a negative one right here and a four right there. Again, calculation's pretty easy. If you take the vector v and you times it by, if you times it by your rotation matrix, you're gonna get negative three and two if you go through it. If you wanna do the vector six, four, well, you could just multiply by the matrix and you'll end up, you multiply the matrix by six, four, you'll get negative four and six. But also, since this is a linear transformation, you could just add together these two images we already know. Notice negative one plus negative three is negative four and four plus two is equal to six. That's not a coincidence. It's a linear transformation. T of uv is equal to T of u plus T of v. You could add together the images and that does the same thing as taking the image of the sum, all right? So with that said, let's then take a look at this picture right here. Let me slide it over so we can see a little bit better on the screen. So if we take our vectors, let's look at them. If we take the vector u, u started off at four, one, it's gonna rotate to the point which we're calling here u prime, which was the point negative one, four. If we take the vector v, it started off at two, three. It's gonna rotate over here to be the point negative three and two. And then if you look at their sum, u plus v, if we rotate it over, we're gonna get this u plus v prime. And so we're gonna take the point six, four and moves over to negative four, six. Now, the reason I did all three of these points, uv and u plus v, is that I connect the dots, right? So by the usual parallelogram rule, right? So if we think of, I'm just gonna draw these as lines right here. So if you connect u and you connect v, like so, and then connect these to u plus v, by the usual parallelogram rule says, right, the diagonal of this parallelogram is the vector u plus v that you see right there. See what happens when we rotate this thing. So if we connect the dots with the origin, so we take u prime, we take v prime, and then we take the sum u plus v, like so, you'll see that these two pictures, these two parallelograms are congruent to each other. What happened was is we rotated the parallelogram. And so point by point, you might not see the rotation, but we start looking at shapes like parallelograms, or you look at the unit square, for example, it's gonna rotate over here like so. This matrix does in fact give us a rotation by 90 degrees. And so that's gonna bring us to the end of section 3.6. Thanks for watching. If you have any questions like usual, feel free to post them in the comments. I'll be happy to answer them. 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