91Ó°ÊÓ

A rotation matrix is a special matrix used to rotate vectors in a plane or space. It is essential in linear algebra, particularly when dealing with transformations.
A rotation matrix for a rotation by an angle \( \theta \) in a two-dimensional space is:
\[ R(\theta) = \begin{pmatrix} \cos(\theta) & -\sin(\theta) \ \sin(\theta) & \cos(\theta) \end{pmatrix} \]
This matrix rotates any vector in \( \mathbb{R}^{2} \) by \( \theta \) radians counterclockwise. For example, if \( \theta = \frac{\pi}{6} \), then the cosine of \( \frac{\pi}{6} \) is \( \frac{\sqrt{3}}{2} \) and the sine of \( \frac{\pi}{6} \) is \( \frac{1}{2} \).
Thus, the rotation matrix for this specific angle is:
\[ R(\frac{\pi}{6}) = \begin{pmatrix} \frac{\sqrt{3}}{2} & -\frac{1}{2} \ \frac{1}{2} & \frac{\sqrt{3}}{2} \end{pmatrix} \]
This matrix can be used to rotate any vector in the plane by \( \frac{\pi}{6} \) radians.

A reflection matrix is used to flip vectors across a specific axis. Understanding the reflection matrices across the x-axis and y-axis is fundamental in linear transformations.
The reflection matrix across the x-axis is:
\[ R_x = \begin{pmatrix} 1 & 0 \ 0 & -1 \end{pmatrix} \]
This matrix reflects any vector over the x-axis, changing the sign of its y-component while keeping its x-component the same.
Similarly, the reflection matrix across the y-axis is:
\[ R_y = \begin{pmatrix} -1 & 0 \ 0 & 1 \end{pmatrix} \]
This matrix reflects any vector over the y-axis, changing the sign of its x-component while keeping its y-component the same.
These matrices are integral in combining multiple transformations, such as performing reflections after rotations.

Matrix multiplication is a key operation when combining linear transformations. It involves multiplying matrices to apply multiple transformations in sequence.
When combining transformations like a rotation followed by reflections, you multiply the matrices in the specific order they occur.
In the exercise, we first rotate using the matrix \( R(\frac{\pi}{6}) \), then reflect across the x-axis using matrix \[ R_x \], and finally reflect across the y-axis using matrix \[ R_y \]. The combined transformation matrix is:
\[ T = R_y \cdot R_x \cdot R(\frac{\pi}{6}) \]
First, we calculate \[ R_y \cdot R_x \]:
\[ \begin{pmatrix} -1 & 0 \ 0 & 1 \end{pmatrix} \cdot \begin{pmatrix} 1 & 0 \ 0 & -1 \end{pmatrix} = \begin{pmatrix} -1 & 0 \ 0 & -1 \end{pmatrix} \]
Then, we multiply this result by the rotation matrix:
\[ \begin{pmatrix} -1 & 0 \ 0 & -1 \end{pmatrix} \cdot \begin{pmatrix} \frac{\sqrt{3}}{2} & -\frac{1}{2} \ \frac{1}{2} & \frac{\sqrt{3}}{2} \end{pmatrix} = \begin{pmatrix} -\frac{\sqrt{3}}{2} & \frac{1}{2} \ -\frac{1}{2} & -\frac{\sqrt{3}}{2} \end{pmatrix} \]
The resultant matrix represents the entire transformation, combining rotation and reflections.