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In Exercises $1-10$ , assume that $T$ is a linear transformation. Find the standard matrix of $T$ .A linear transformation $T : \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ first reflects points through the $x_{1}$ -axis and then reflects points through the $x_{2}-$ axis. Show that $T$ can also be described as a linear transformation that rotates points about the origin. What is the angle of that rotation?

$A=\left[\begin{array}{cc}{-1} & {0} \\ {0} & {-1}\end{array}\right]$

Algebra

Chapter 1

Linear Equations in Linear Algebra

Section 9

The Matrix of a Linear Transformation

Introduction to Matrices

Campbell University

McMaster University

Baylor University

University of Michigan - Ann Arbor

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uh huh. In the current problem, um, we had given a situation the scene, if two different transformations actually doing the same work or not. First, they're talking about the transformation that reflects a point through. Excellent. And then next. Okay, so we will see. What would there transformation look like? So take even and e to. So if we are reflecting through X one axis one common zero. Okay, then this is not reflecting anywhere because with respect to X, when access there's the middle most positions, there is no change. So this becomes 10 where it is this point Over here, this comes down over here, that is zero former minus, right? No. De So if there is a reflection X when we know this is the standard metrics now they're telling will Now you change this How reflection Through extra. So if there is a reflection toe extreme Now we forget about this point we're having now only this point and this point. So this will change to hear. Whereas this will remain constant. So that's why we will get minus one. We'll see toe zero minus one. So this is our final step metrics for this transformation. This is our teeth. No. They're telling that show that he can also be described as a linear transformation that rotates points about the auditing. And what is the angle of the rotation now? See, initially, point was here East, this is the origin. And to be rooted about the or eating so from the origin, whatever distances here, the same distance that you travel. Headache. So what What are we finding is this is the angle we've been here. This is the anger. Okay, So when we I want to find the metrics of transformation, we know that the metrics of transformation through the origin they are rotation through the origin. We know that metrics that standard metrics is off the form minus one zero zero minus one. Right. Which means what if you remember, there is a photograph of a fish over here and that real. So this will come here. This will go here. That's the idea. Reflection are rotating through the origin. So now if you see both the transformation metrics a scene, hence we can see that both, um, addresses, uh, signifying the same transformation. Hence reflecting through first X one and then x two all rotating about the origin is actually the same action. That's something

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