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Q34E

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Linear Algebra With Applications

Found in: Page 54

Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

Consider the transformation T from $R^{2}$ to role="math" localid="1659714471562" $R^{2}$ that rotates any vector $\vec{x}$ 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 $[\begin{matrix} \cos θ & - \sin θ \\ \sin θ & \cos θ \end{matrix}]$

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

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:

Identity of $R^{m}$ should be mapped to identity of $R^{n}$ .
$T (a + b) = T (a) + T (b)$ For all $a, b \in R^{m}$ .
$T (c a) = c T (a)$ 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.

$\begin{array}{rcl} T (r \cos α, r \sin α) & = & (r \cos (α + θ), r \sin (α + θ)) \\ = & (r \cos α \cos θ - r \sin α \sin θ, r \sin θ \cos α + r \cos θ \sin α) \\ = & [\begin{matrix} \cos θ & - \sin θ \\ \sin θ & \cos θ \end{matrix}] [\begin{matrix} r \cos α \\ r \sin α \end{matrix}] \\ {[T]}_{β} & = & [\begin{matrix} \cos θ & - \sin θ \\ \sin θ & \cos θ \end{matrix}] \end{array}$

Hence, the matrix for T will be $[\begin{matrix} \cos θ & - \sin θ \\ \sin θ & \cos θ \end{matrix}]$ .

Most popular questions for Math Textbooks

Some parking meters in downtown Geneva, Switzerland, accept $2$ Franc and $5$ Franc coins.

a. A parking officer collects $51$ coins worth $144$ Francs. How many coins are there of each kind?

b. Find the matrix $A$ that transforms the vector

$[\begin{matrix} number of 2 Franc coins \\ number of 5 Franc coins \end{matrix}]$

into the vector

$[\begin{matrix} total value of coins \\ total number of coins \end{matrix}]$

c. Is the matrix $A$ in part (b) invertible? If so, find the inverse (use Exercise 13). Use the result to check your answer in part (a).

Write all possible forms of elementary $2 \times 2 E_{5} = [\begin{matrix} 1 & 0 \\ k & 1 \end{matrix}]$ matrices E. In each case, describe the transformation $\vec{y} = E \vec{x}$ geometrically.

Which of the functions f from R to R in Exercises 21 through 24 are invertible? 22 $f (x) = 2^{x}$ .

Consider an n × p matrix A, a p × m matrix B, and a scalar k. Show that (k A)B = A (k B) = k(AB)

TRUE OR FALSE?

There exists a matrix A such that $A (\begin{matrix} \begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix} \end{matrix}) = (\begin{matrix} 1 & 2 \\ 1 & 2 \end{matrix})$ .

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