Let T/6 : R² → R?counterclockwise by T/6 radians. Let Tref : R² → R- be the linear transformation whichdenote the linear transformation which rotates a vector in R-across the line xj = x2.x2reflects a vector x =(a) Sketch a cartoon illustrating what these linear transformations do to the vector ej =(b) Find the standard matrices of T-/6 andTref-(c) Find a nonzero vector v E R² such thatT/6 o Tref (v) = Tref o Tr/6 (v)or explain why no such vector exists.

Answered: Let T/6 : R² → R² denote the linear…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Let T/6 : R² → R?
counterclockwise by T/6 radians. Let Tref : R² → R- be the linear transformation which
denote the linear transformation which rotates a vector in R-
across the line xj = x2.
x2
reflects a vector x =
(a) Sketch a cartoon illustrating what these linear transformations do to the vector ej =
(b) Find the standard matrices of T-/6 and
Tref-
(c) Find a nonzero vector v E R² such that
T/6 o Tref (v) = Tref o Tr/6 (v)
or explain why no such vector exists.

Expert Solution

Step 1

Let $T_{π / 6} : ℝ^{2} \to ℝ^{2}$ denote the linear transformation which rotates a vector in $ℝ^{2}$ counterclockwise by $\frac{π}{6}$ radians.

We know the counterclockwise rotation matrix by an angle $θ$ is $[\begin{matrix} \cos θ & - \sin θ \\ \sin θ & \cos θ \end{matrix}] .$

Thus,

$T_{π / 6} [\begin{matrix} x_{1} \\ x_{2} \end{matrix}] = [\begin{matrix} \cos (\frac{π}{6}) & - \sin (\frac{π}{6}) \\ \sin (\frac{π}{6}) & \cos (\frac{π}{6}) \end{matrix}] [\begin{matrix} x_{1} \\ x_{2} \end{matrix}] \Rightarrow T_{π / 6} [\begin{matrix} x_{1} \\ x_{2} \end{matrix}] = [\begin{matrix} \frac{\sqrt{3}}{2} & - \frac{1}{2} \\ \frac{1}{2} & \frac{\sqrt{3}}{2} \end{matrix}] [\begin{matrix} x_{1} \\ x_{2} \end{matrix}] .$

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