See attachment S : R → R² is linear transformation that rotates avector in R2 by counter clockwise followed by a reflection with respect tothe r - axis. Find the associated matrix H such that:S(z) = H · z,ŽE R

Answered: S: R → R² is linear transformation that…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Question

100%

See attachment

S : R → R² is linear transformation that rotates a
vector in R2 by counter clockwise followed by a reflection with respect to
the r - axis. Find the associated matrix H such that:
S(z) = H · z,
ŽE R

Expert Solution

Step 1

Given : $S : ℝ^{2} \to ℝ^{2}$ is a linear transformation that rotates a vector in $ℝ^{2}$ by $\frac{π}{6}$ counter clockwise followed by a reflection with respect to the x-axis.

To find : associated matrix $H$ such that

$S (\vec{z}) = H \cdot \vec{z}, \vec{z} \in ℝ^{2}$

Pre-requisite :

P1 : Theorem (rotation) - Let $R_{θ} : ℝ^{2} \to ℝ^{2}$ be a linear transformation given by rotating vectors through an angle of $θ$ . Then the matrix A of $R_{θ}$ is given by

$[\begin{matrix} \cos (θ) & - \sin (θ) \\ \sin (θ) & \cos (θ) \end{matrix}]$

P2 : Theorem (reflection) - Let $Q_{m} : ℝ^{2} \to ℝ^{2}$ be a linear transformation given by reflecting vectors over the line $\vec{y} = m \vec{x}$ . Then the matrix associated to this linear transformation is given by