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Use a rectangular coordinate system to plot $\mathbf{u}=\left[\begin{array}{l}{5} \\ {2}\end{array}\right], \mathbf{v}=\left[\begin{array}{r}{-2} \\ {4}\end{array}\right],$ and their images under the given transformation $T .$ (Make a separate and reasonably large sketch) Describe geometrically what $T$ does to each vector $\mathbf{x}$ in $\mathbb{R}^{2}$.$T(\mathbf{x})=\left[\begin{array}{rr}{-1} & {0} \\ {0} & {-1}\end{array}\right]\left[\begin{array}{l}{x_{1}} \\ {x_{2}}\end{array}\right]$

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Algebra

Chapter 1

Linear Equations in Linear Algebra

Section 8

Introduction to Linear Transformations

Introduction to Matrices

Jennifer O.

September 26, 2020

Shouldn't the vector v but in the 4th quadrant because x = -2 and y = 4?

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In this example, we have a linear transformation T that takes this two by two matrix and multiplies the vector X from the domain, giving us a vector x or eight times X in the co domain. Also, we're provide with vectors you envy. Let's start out by sketching where u and V would be in the x one x two plane. So for X one, we go distance of five and up to So this is location of vector you the vector v goes a distance of left to then down for So here we have the vector V. Now let's see what happens when we apply the transformation t to these vectors. For example, If we take t of you then Ray in the Matrix a first which is negative 100 negative one Amel to play by you, which is the vector five to We'll obtain altogether negative five for the first entry and negative two for the second entry. Since the effect of this matrix here is just a multiply both entries of whatever vectors given by negative one. So for the image of you, we go to negative five. So here's 345 and down to. So this is where t of you ends up where we started with you being here. Likewise, let's also calculate the image of t of the So we ended up here and it is the make the vector negative 24 So this will be negative. 100 negative one times negative 24 But we also know that the effect of this matrix multiplication is to multiply the vector by negative one. So we have to negative four. So if we go down, go our distance or right to down four. This is where the vector t of V ends up. So all together, for our view, when it's in quadrant one t of you throws it all the way to the third quadrant. But for V, that's already in the third quadrant. The mapping sends it to the image TV in the fourth quadrant. And so this is the effect of that transformation. T on these given vectors

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