Determine the standard matrix D of the linear mapping G: R^2-->R^2 that first rotates points clockwise through pi/6 radians and then reflects points through the line x2=x1
To determine the standard matrix D of the linear mapping G: R^2 --> R^2, which first rotates points clockwise through pi/6 radians and then reflects points through the line x2 = x1, we need to follow these steps:
Step 1: Determine the standard matrix for rotation.
To determine the standard matrix for rotation, we use the following formula:
| cos(θ) -sin(θ) |
| sin(θ) cos(θ) |
where θ is the angle of rotation.
In this case, the angle of rotation is clockwise through pi/6 radians, so we have:
| cos(pi/6) -sin(pi/6) |
| sin(pi/6) cos(pi/6) |
Simplifying this, we get:
| √3/2 -1/2 |
| 1/2 √3/2 |
This is the standard matrix for rotation.
Step 2: Determine the standard matrix for reflection.
To determine the standard matrix for reflection about the line x2 = x1, we use the following formula:
| 1 0 |
| 0 -1 |
This is the standard matrix for reflection.
Step 3: Multiply the standard matrices.
To combine the rotation and reflection, we multiply the standard matrices obtained in steps 1 and 2. We need to multiply the rotation matrix by the reflection matrix.
Multiplying, we get:
| √3/2 -1/2 | | 1 0 | = | √3/2 -1/2 |
| 1/2 √3/2 | | 0 -1 | | 1/2 -√3/2 |
This is the standard matrix D of the linear mapping G: R^2 --> R^2 that first rotates points clockwise through pi/6 radians and then reflects points through the line x2 = x1.