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Let $\mathbf{e}_{1}=\left[\begin{array}{l}{1} \\ {0}\end{array}\right], \mathbf{e}_{2}=\left[\begin{array}{l}{0} \\ {1}\end{array}\right], \mathbf{y}_{1}=\left[\begin{array}{c}{2} \\ {5}\end{array}\right],$ and $\mathbf{y}_{2}=\left[\begin{array}{r}{-1} \\ {6}\end{array}\right],$ and let $T : \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ be a linear transformation that maps $\mathbf{e}_{1}$ into $\mathbf{y}_{1}$ and maps $\mathbf{e}_{2}$ into $\mathbf{y}_{2} .$ Find the images of $\left[\begin{array}{r}{5} \\ {-3}\end{array}\right]$ and $\left[\begin{array}{l}{x_{1}} \\ {x_{2}}\end{array}\right]$.

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Algebra

Chapter 1

Linear Equations in Linear Algebra

Section 8

Introduction to Linear Transformations

Introduction to Matrices

Harvey Mudd College

Baylor University

Lectures

01:32

In mathematics, the absolu…

01:11

02:08

Let $\mathbf{x}=\left[\beg…

02:46

Let $T : \mathbb{R}^{2} \r…

03:42

Verify directly from Defin…

01:38

In Exercises $1-10$ , assu…

02:41

Let $T_{1}: \mathbb{R}^{2}…

01:19

06:54

Let $T$ be a linear transf…

02:55

Let $T: V \rightarrow V$ b…

03:18

02:18

In Exercises $17-20,$ show…

so in discussion, we have. We help. Teas are too. Are too is a linear transformation that maps even equals two 10 into. Why one on that is to five. And man, he two equals toe right. 01 in two, wire to And that is minus 16 And finally, we have to find the images under tee off the off. Five minus three. And excellent Extra. Okay, so no problem. Okay, first right vectors, you and we as a linear combination off even and into that is five minus three. His request to we can rewrite this. 510 minus 301 So simple their testicles to five. Even minus 32 Okay. And the same Excellent x two we can write down. Excellent. 10 minus next to 01 so we can write it down. Excellent. Even plus x two, e two. Okay, so now we know that we know there. D U plus v waas toe t u plus TV for all you in the domain off the okay on second t u equals toe. See d you for all the scaler c and all you in the domain off. Okay, so holding Lee. We can write their tea. Five titty will be de five even minus 382 equals toe five d Even my austerity Ito, That is five y one minus three while toe simple. And now we will put the value of y one and y toe here and what we get what we get. Five Go five. Sorry to five minus three minus 16 Okay. And that will be 13 by seven. So this is our first answer. Okay, this is our first answer, and for the second, we will do De excellent XTO. Okay, so in the same procedure, this will be excellent even plus x two e two. Okay. And for solving this weekend's excellent t even plus x to d E to Okay, this will be X one y one plus x two y two. Okay. And this will be x 1 to 5 plus x two minus 16 We can reunite this as two x one minus x two and five. Excellent. Plus six x two. Okay, so this is over. Solution off. De excellent. Extra. So we have We have all solved for both days. So we were our final answer will be de five minus three will be 13 and seven and be excellent. X two will be two X one minus X to divide by five. Excellent. Plus six X two. Okay, this will be our final, sir. Thank you.

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