Q33E Consider the transformation T fr... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 2: Q33E (page 54)

Consider the transformation T from $R^{2}$ to $R^{2}$ that rotates any vector $\overset{鈫�}{x}$ through an angle of $45 掳$ in the counterclockwise direction, as shown in the following figure:
You are told that T is a linear transformation. (This will be shown in the next section.) Find the matrix of T .

Short Answer

Expert verified

Matrix for T will be $\frac{\sqrt{2}}{2} [\begin{matrix} 1 & - 1 \\ 1 & 1 \end{matrix}]$ .

Step by step solution

Step by step Explanation Step1: Transformation

Given that T is a linear transformation from $R^{2} 鈫� R^{2}$ .

Let $尾$ be any point of $R^{2}$ .

Then we can write the transformation.

role="math" localid="1659712903621" $\begin{array}{rcl} T (r \cos 伪, r \sin 伪) & = & (r \cos (伪 + 胃), r \sin (伪 + 胃)) \\ = & (r \cos 伪 \cos 胃 - r \sin 伪 \sin 胃, r \sin 胃 \cos 伪 + r \cos 胃 \sin 伪) \\ = & [\begin{matrix} \cos 胃 & - \sin 胃 \\ \sin 胃 & \cos 胃 \end{matrix}] [\begin{matrix} r \cos 伪 \\ r \sin 伪 \end{matrix}] \\ {[T]}_{尾} & = & [\begin{matrix} \cos 胃 & - \sin 胃 \\ \sin 胃 & \cos 胃 \end{matrix}] \end{array}$

Value of angle

We have given the angle is of $45 掳$ ,so put the value of angle in above equation we will get.

${[T]}_{尾} = [\begin{matrix} \frac{1}{\sqrt{2}} & - \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{matrix}] = \frac{\sqrt{2}}{2} [\begin{matrix} 1 & - 1 \\ 1 & 1 \end{matrix}]$

Hence, the matrix for T will be $\frac{\sqrt{2}}{2} [\begin{matrix} 1 & - 1 \\ 1 & 1 \end{matrix}]$

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91影视

Short Answer

Step by step solution

Step by step Explanation Step1: Transformation

Value of angle

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91影视

Consider the transformation T from R2to R2that rotates any vectorx鈫�through an angle of 45掳in the counterclockwise direction, as shown in the following figure:You are told that T is a linear transformation. (This will be shown in the next section.) Find the matrix of T .

Short Answer

Step by step solution

Step by step Explanation Step1: Transformation

Value of angle

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Probability and Statistics

Calculus

Statistics

Discrete Mathematics

Logic and Functions

Study anywhere. Anytime. Across all devices.