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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 $\mathbb{R}^{2}$.$T(\mathbf{x})=\left[\begin{array}{ll}{0} & {1} \\ {1} & {0}\end{array}\right]\left[\begin{array}{l}{x_{1}} \\ {x_{2}}\end{array}\right]$

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Algebra

Chapter 1

Linear Equations in Linear Algebra

Section 8

Introduction to Linear Transformations

Introduction to Matrices

McMaster University

Baylor University

University of Michigan - Ann Arbor

Lectures

01:32

In mathematics, the absolu…

01:11

02:10

Use a rectangular coordina…

02:23

03:57

02:39

Sketch the image in the $x…

02:02

Use a graphing utility to …

03:58

The figure shows vectors $…

01:26

Find the image R in the xy…

02:30

Find the image of the set …

03:22

Suppose $S$ is the unit sq…

02:06

The linear transformation …

So we have the function you X equals the matrix 0110 times the vector X So we want a plot you be and their images. And for this transformation Ah, so efforts. Let's find what t of you and t of e r so t of a victor you iss equal to 01 10 times our metrics you, which is five to. So we take zero times five plus one times two, which gives us two. Then we take one times five plus two times zero, which is five. So we can see our matrix has flipped. Um, first and second coordinates. So let's do this again with T o V. So we have the same matrix times our vector V, which is negative to four. So zero times negative to post one times four gives us for and one times negative to post your terms for give us negative too. So I can see again we switched coordinates. So now let's go ahead and plot everything we have s o. We need our why? To go up to five and down to minus two. Our excess to go Rome. Negative too. Up and 25 surgery. A little small there. That's okay. Um, so it's first plot you envy and red, And then we'll plot their images and blue. So you woz 52 So we're just gonna connect that point to the origin and ve iwas negative to four. So, again, what was Connect that back the street as possible. Eso this 1st 1 it was you on V. So now let's plot t of you on TV. So t of you, we found was 25 Joe's right here. And TV. We found to be four negative too s so we can see that since we had a negative coordinate envy. It changed the quadrant. Ah, but since you was both positive, it remained in the same quarter.

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