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In Exercises $1-10$ , assume that $T$ is a linear transformation. Find the standard matrix of $T$ .$T : \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ first performs a horizontal shear that transforms $\mathbf{e}_{2}$ into $\mathbf{e}_{2}-2 \mathbf{e}_{1}$ (leaving $\mathbf{e}_{1}$ unchanged) and then reflects points through the line $x_{2}=-x_{1}$

$A=\left(\begin{array}{cc}{0} & {-1} \\ {-1} & {2}\end{array}\right)$

Algebra

Chapter 1

Linear Equations in Linear Algebra

Section 9

The Matrix of a Linear Transformation

Introduction to Matrices

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So in the current problem, we have to find a transformation. T such that r squared too r squared. And it does too actions in particular. So the first action, if we consider, is as a t one. Okay, That is given as we change changes e to as even minus twice he took. So e tu minus twice The one bad has it leaves even as it is that is even through this transformation minute, cause it will remain even so if you think this is even and this is e to so it will change somewhere here or here somewhere even will remain same. So either there will be a sheer like this or this way. So this is nothing but horizon. Tal shit. No. How do we find out this transformation also the second action off this transformation would be then reflecting the point through the line X two is equal to minus X one. Okay, so that means what? First there will be a sheer and then x two is equal to minus X one. That means something like this. It should come here, get reflected here. So we have to find out what would be the metrics or the transformation associated with this situation. So let us go step by step. So if you see even will remain you got. So if there is a metrics which should have two columns so first column will not go through any change. That means what 10 will yield one zero. But what about he took it will see a change. What is that? That is e tu minus twice. Even so, it too is zero and one. So what will happen to eat too? This will become zero minus twice, even even first element is one again. One minus two into zero. So this will be the changing You too. That is we can again, right? Zero one will change too. Zero minus two is minus two and one minus zero is one. So that means what? This is the mattress that will change even in between this format. Right? So if it is 1001 that will get changed too. 10 minus two. What? And we clearly know that if it is a k over here, then it is a horizontal shear. Right? So we are quite going toe the courage path now what do we want after this? We want. And also we know if K is less than zero than it's a left towards sheer right. That is this point will ship this this direction, right? If K is less than zero No, we want the reflection through X two minus x one. Right. So we now want toe, Find out the second activity. That would be reflection through X two is equals True Negative X Well, mhm x two musical through negative experts. Now can we write? Our tea is equals to t to off. Do you run off X? That means what? First we apply t even X on X. Okay. And then we will apply t two. So what will be so this is our tea even. Okay. So if we multiply this on X any variables Victor's, then we get this final out. Now we have two similar We find t to What would we Tito Tito would be the reflection, the action, the reflection action will be defined. So through this, we will achieve the final answer now see if originally a metrics is 1001 for reflection, we get this zero minus one minus one zero. Now, this is not convincing enough. Let's check through a diagram. This is one comma zero. This is zero comma one. Right now. What will be extra eyes equals two minus one. What would be this equation? This would be kind of her like this. So this will come here. Right? So which point is this? Here. X is zero. Why is minus one, isn't it? So if you see 10 changes to zero minus one. That means what? This down element goes up and the up element comes down changing the site. Changing the sign. So same V. What will happen to our actual given expression? This is our final expression after we're having tea. Even so, 10 will become zero minus one. Okay, Now what will happen to this point? This point, which is 10 we'll also travel here. Right? So this is what here? Why is zero but X is negative. That means what all the elements changes. Sign zero becomes negative of zero is also again zero. That's why we couldn't see it. So if it was a point like five to that will go toe minus two, minus five. Okay, if you see a point here and we go here. So if you see if it is five comma to that will become minus two. Coma minus fine. So then we are having the second column minus 21 as this one will go up becoming negative and this minus two will come down. I'll get taking the negative. That means plus two. So this man, this is the final expression off the transformation. T So therefore, t such that it takes a pair off. Values are square to our square says that it performs a horizon tal sheer Andi, Then reflection through extra is a construct. X x ray is equal to minus X. One would be zero minus one and then again minus one to Andi. That's how easy it is. I hope you could understand

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