Q22E Find the matrices of the linear ... [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: Q22E (page 72)

Find the matrices of the linear transformations fromR³toR³given in Exercises 19through 23. Some of these transformations have not been formally defined in the text. Use common sense. You may assume that all these transformations are linear.
22. The rotation about the y-axis through an angle of $胃$ , counterclockwise as viewed from the positive y-axis.

Short Answer

Expert verified

The matrix represents the rotation about the y-axis through an angle of is, $A = (\begin{matrix} \cos 胃 & 0 & \sin 胃 \\ 0 & 1 & 0 \\ 鈭� \sin 胃 & 0 & \cos 胃 \end{matrix})$ .

Step by step solution

Compute the vector

Consider the vector

$\begin{array}{l} \overset{鈫�}{x} = x \overset{鈫�}{m_{1}} + y \overset{鈫�}{m_{2}} + z \overset{鈫�}{m_{3}} \\ \overset{鈫�}{x} = x [\begin{array}{l} 1 \\ 0 \\ 0 \end{array}] + y [\begin{array}{l} 0 \\ 1 \\ 0 \end{array}] + z [\begin{array}{l} 0 \\ 0 \\ 1 \end{array}] \end{array}$

Consider the system

The system is,

$\begin{matrix} T (\overset{鈫�}{x}) = T (x \overset{鈫�}{m_{1}} + y \overset{鈫�}{m_{2}} + z \overset{鈫�}{m_{3}}) \\ = x T (\overset{鈫�}{m_{1}}) + y T (\overset{鈫�}{m_{2}}) + z T (\overset{鈫�}{m_{3}}) \end{matrix}$ $\begin{matrix} T (\overset{鈫�}{x}) = T (x \overset{鈫�}{m_{1}} + y \overset{鈫�}{m_{2}} + z \overset{鈫�}{m_{3}}) \\ = x T (\overset{鈫�}{m_{1}}) + y T (\overset{鈫�}{m_{2}}) + z T (\overset{鈫�}{m_{3}}) \end{matrix}$

Compute the matrix

As $T (\overset{鈫�}{x}) = A \overset{鈫�}{x}$ ,

Thus,

$(\begin{matrix} \cos 胃 & 0 & 鈭� \sin 胃 \\ 0 & 1 & 0 \\ \sin 胃 & 0 & \cos 胃 \end{matrix}) [\begin{array}{l} x \\ y \\ z \end{array}] = [\begin{array}{l} x \\ y \\ z \end{array}]$

Here, y represents the rotation about the y-axis.

$T (\overset{鈫�}{y}) = y = (0, 1, 0)$

Actually, the rotation is along x-z plane, as, the rotation starts from positive z-axis to the positive x-axis, that is, in clockwise x-z plane, so, the angle notation changes.

$\begin{matrix} A = (\begin{matrix} \cos (鈭� 胃) & 0 & 鈭� \sin (鈭� 胃) \\ 0 & 1 & 0 \\ \sin (鈭� 胃) & 0 & \cos (鈭� 胃) \end{matrix}) \\ = (\begin{matrix} \cos 胃 & 0 & \sin 胃 \\ 0 & 1 & 0 \\ 鈭� \sin 胃 & 0 & \cos 胃 \end{matrix}) \end{matrix}$

Hence, $A = (\begin{matrix} \cos 胃 & 0 & \sin 胃 \\ 0 & 1 & 0 \\ 鈭� \sin 胃 & 0 & \cos 胃 \end{matrix})$ is the matrix that represents therotation about the y-axis through an angle of $胃$ .

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91影视

Short Answer

Step by step solution

Compute the vector

Consider the system

Compute the matrix

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