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The figure shows vectors $\mathbf{u}, \mathbf{v},$ and $\mathbf{w},$ along with the images $T(\mathbf{u})$ and $T(\mathbf{v})$ under the action of a linear transformation $T : \mathbb{R}^{2} \rightarrow \mathbb{R}^{2} .$ Copy this figure carefully, and draw the image $T(\mathbf{w})$ as accurately as possible. [Hint: First, write $\mathbf{w}$ as a linear combination of $\mathbf{u}$ and $\mathbf{v} . ]$

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Algebra

Chapter 1

Linear Equations in Linear Algebra

Section 8

Introduction to Linear Transformations

Introduction to Matrices

Campbell University

Oregon State University

University of Michigan - Ann Arbor

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So we're given these two figures and we want to try to plot t of w So the hit tells us first right? W as a linear combination of U N v. Eso First, let's look at this picture on the left. Let's connect V plus venue or W and you And we know this vector is parallel to V. So let's try to use the parallelogram room. Um, so let's slide this vector downwards along you. Um, until the, um, the back point here is that the origin? So we do. So we get this vector here and we could see the length of this is about twice the length of the So all I've done is taken this red vector, slide it down. And since we crossed the origin of points in the other direction So this red factor is to V about, um So let's go ahead and connect this. So if we connect from, I had to tell this vector since, um, the specter here and this vector here are parallel in the same length. We know that this is also you. These two are the same. Ah, so now we have a parallelogram here, so this tells us that w is equal to you. Plus two b So now we want to find t of w So let's just apply t to both sides, So t of w as he called a t of you plus two b And so we're now we're gonna use the rules of linearity. So this is the same ass t of you plus to t of the cause. Linearity tells us that we could pull constant out. So if t of w is equal to t of you, which is this vector here, plus twice t of me first, let's find Let's do two TV then. So that's gonna be double this length. So this red rector is too t v anti of you is this one. So let's go ahead and connect those two things. Ah, First, actually, let's charlety of you up here. We're just gonna slide it up because we want to make sure these air connected And now well, you will just, uh, connect from the origin. So let's do that. Another color. So here's our t of W. Because we have to make sure the reason we couldn't have just connected used to is because we'd be correcting the, uh to head. So we have to have everything had detail, so we needed to shift it up.

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