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Let $T : \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ be the transformation that reflects each vector $\mathbf{x}=\left(x_{1}, x_{2}, x_{3}\right)$ through the plane $x_{3}=0$ onto $T(\mathbf{x})=\left(x_{1}, x_{2},-x_{3}\right) .$ Show that $T$ is a linear transformation. [See Example 4 for ideas. $]$

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Algebra

Chapter 1

Linear Equations in Linear Algebra

Section 8

Introduction to Linear Transformations

Introduction to Matrices

Oregon State University

McMaster University

Harvey Mudd College

Idaho State University

Lectures

01:32

In mathematics, the absolu…

01:11

02:36

Let $T : \mathbb{R}^{3} \r…

02:55

Let $T: V \rightarrow V$ b…

02:40

Suppose vectors $\mathbf{v…

03:37

Let $T : \mathbb{R}^{2} \r…

03:26

Column vectors are written…

05:05

Let $\mathbf{u}$ and $\mat…

04:24

Verify directly from Defin…

01:30

Determine the matrix of th…

02:02

Let $T_{1}: M_{n}(\mathbb{…

Okay, So for the ocean transformation t is are three on two or three. Okay. And it would be the transformation that reflects each vector X and the vector X is given x one x two and x three. Okay, through the plane. Extra equals to zero and on to And the images images given, that is X one x two and minus X three. Okay, we have show that is a linear transformation. Okay, So considered you be in R three with this killer city. Okay, We are considering UV in our three with his killers. See? And the and lead and let X equals to see you plus Devi. Okay, so now it will be we can write that X equals toe. See you one plus d We even okay and see you two plus D V two and c u three plus d V three. Okay, And now the X because toe TCU plus D v Then we can write this. See you one plus d V one. See you two plus d. We do and transformation will be minus then minus See you three plus the eatery. Okay. So further solving that we will get to see you One plus Devi even see you took less DV two and minus See you three minus D v three. Okay, so far that solving their distance Formacion image way get See you one See you do and minus See you three glass Devi even DV two on minus day with three Okay, so we will put Get see Comin out of this This will be you and you two and minus you three plus be comin out off their that we won we do on minus with three So it will be See do you class d TV? So this is a Christian off linear transformation Okay, so we can say that you can say that he is a linear transformation. Thank you. This is our final unset. Thank you.

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