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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 2: Q34E (page 54)

Consider the transformation Tfrom $R^{2}$ to role="math" localid="1659714471562" $R^{2}$ that rotates any vector $\overset{鈫�}{x}$ through a given angle胃in the counterclockwise direction. Compare this with Exercise 33. You are told that Tis linear. Find the matrix of Tin terms of胃.

Short Answer

Expert verified

Matrix for T will be $[\begin{matrix} \cos 胃 & - \sin 胃 \\ \sin 胃 & \cos 胃 \end{matrix}]$

Step by step solution

01

Step by step Explanation: Step1: Linear Transformation

A transformation from $R^{m} 鈫� R^{n}$ 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 $a, b 鈭� R^{m}$ .
$T (c a) = c T (a)$ Where c is any scalar and $a 鈭� R^{m}$ .

02

Transformation

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

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

Then we can write the transformation.

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

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

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Most popular questions from this chapter

Question:Consider the matrices of the form $A = (\begin{matrix} a & b \\ b & 鈭� a \end{matrix})$ , where aand bare arbitrary constants. For which values of aand b is A^-1= A?

In Exercises 55 through 64, find all matricesX that satisfy the given matrix equation. 64.

$[\begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{matrix}] X = I_{2}$

Consider a matrix Aof the form $A = [\begin{matrix} a & b \\ b & 鈭� a \end{matrix}]$ , where $a^{2} + b^{2} = 1$ . Find two nonzero perpendicular vectors $\overset{鈫�}{蠀}$ andlocalid="1664256213921" $\overset{鈫�}{w}$ such that $A \overset{鈫�}{蠀} = \overset{鈫�}{蠀}$ and $A \overset{鈫�}{w} = 鈭� \overset{鈫�}{w}$ (write the entries of $\overset{鈫�}{蠀}$ and $\overset{鈫�}{w}$ in terms of aand b). Conclude that localid="1664256257917" $T (\overset{鈫�}{x}) = A (\overset{鈫�}{x})$ represents the reflection about the line Lspanned by $\overset{鈫�}{蠀}$ .

There exists a nonzero upper triangular 2 脳 2 matrix A such that $A^{2} = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}]$ .

Give a geometric interpretation of the linear transformations defined by the matrices in Exercises $16$ through $23$ . Show the effect of these transformations on the letter $L$ considered in Example $5$ . In each case, decide whether the transformation is invertible. Find the inverse if it exists, and interpret it geometrically. See Exercise $13$

$[\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}]$

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