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Expert-verified Found in: Page 54 ### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974 # Consider the transformation T from ${{\mathbit{R}}}^{{\mathbf{2}}}$ to role="math" localid="1659714471562" ${{\mathbit{R}}}^{2}$that rotates any vector $\stackrel{\mathbf{\to }}{\mathbf{x}}$ through a given angle θ in the counterclockwise direction. Compare this with Exercise 33. You are told that T is linear. Find the matrix of T in terms of θ.

Matrix for T will be $\left[\begin{array}{cc}\mathrm{cos}\theta & -\mathrm{sin}\theta \\ \mathrm{sin}\theta & \mathrm{cos}\theta \end{array}\right]$

See the step by step solution

## Step by step Explanation: Step1: Linear Transformation

A transformation from ${R}^{m}\to {R}^{n}$ is said to be linear if the following condition holds:

1. Identity of ${R}^{m}$ should be mapped to identity of ${R}^{n}$.
2. $T\left(a+b\right)=T\left(a\right)+T\left(b\right)$For all $a,b\in {R}^{m}$.
3. $T\left(ca\right)=cT\left(a\right)$ Where c is any scalar and $a\in {R}^{m}$.

## Step2: Transformation

Given T is linear transformation from${R}^{2}\to {R}^{2}$.

Let be any point of${R}^{2}$.

Then we can write the transformation.

$\begin{array}{rcl}T\left(r\mathrm{cos}\alpha ,r\mathrm{sin}\alpha \right)& =& \left(r\mathrm{cos}\left(\alpha +\theta \right),r\mathrm{sin}\left(\alpha +\theta \right)\right)\\ & =& \left(r\mathrm{cos}\alpha \mathrm{cos}\theta -r\mathrm{sin}\alpha \mathrm{sin}\theta ,r\mathrm{sin}\theta \mathrm{cos}\alpha +r\mathrm{cos}\theta \mathrm{sin}\alpha \right)\\ & =& \left[\begin{array}{cc}\mathrm{cos}\theta & -\mathrm{sin}\theta \\ \mathrm{sin}\theta & \mathrm{cos}\theta \end{array}\right]\left[\begin{array}{c}r\mathrm{cos}\alpha \\ r\mathrm{sin}\alpha \end{array}\right]\\ {\left[T\right]}_{\beta }& =& \left[\begin{array}{cc}\mathrm{cos}\theta & -\mathrm{sin}\theta \\ \mathrm{sin}\theta & \mathrm{cos}\theta \end{array}\right]\end{array}$

Hence, the matrix for T will be $\left[\begin{array}{cc}\mathrm{cos}\theta & -\mathrm{sin}\theta \\ \mathrm{sin}\theta & \mathrm{cos}\theta \end{array}\right]$. ### Want to see more solutions like these? 