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In Exercises $1-10$ , assume that $T$ is a linear transformation. Find the standard matrix of $T$ .$T : \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ rotates points (about the origin) through 3$\pi / 2$ radians (counterclockwise).

$A=\left[\begin{array}{ll}{-e_{2}} & {e_{1}}\end{array}\right]=\left[\begin{array}{cc}{0} & {1} \\ {-1} & {0}\end{array}\right]$

Algebra

Chapter 1

Linear Equations in Linear Algebra

Section 9

The Matrix of a Linear Transformation

Introduction to Matrices

Sakura M.

September 30, 2020

Campbell University

Harvey Mudd College

Baylor University

Lectures

01:32

In mathematics, the absolu…

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In Exercises $1-10$ , assu…

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Assume that $T$ defines a …

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For the following exercise…

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in this example, we have a transformation. T mapping are two into our to and what T is going to dio is take points in the plane and rotate them by an angle of three pi over two in the counter clockwise direction. Our goal here is to find the standard matrix A of T so that he can be written when evaluating X as eight times X. Her goal is to find the standard matrix A. We know that we can calculate t one and th e to. Then our work is going to be complete where e one e two are the first and second columns of the two by two identity matrix. But we have a great formula for when we're ever were rotating points by an angle in the counter clockwise direction. The First Inc. The first element is going to be co sign at three pi over two, and the next element will be signed at three Pie divide by two. And if we change the angle, we just have to change what goes in here. Likewise, for the second set of points we have that t at e two is going to map into negative sign at three pie Divide by two for the first century and co sign of three pi over two for the second entry. So let's evaluate each one of these components or will have all together. In the first case that co signed three Pi over two is a zero and signed three. Pi over two is negative. One for the second case Sign a three Piper to is negative one. So with the extra negative, we get a positive one here and co sign is zero at three pie or two. This tells us our standard matrix A can be written as TV one TV, too. But now it's equal to the falling matrix. We put in zero negative one and 10 and this causes the appropriate rotation by this ankle.

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