Determine the standard matrix A of the linear mapping L:R^2-->R^2 that rotates any vector through an angle of 150 degrees in the clockwise direction.
To determine the standard matrix A of the linear mapping L that rotates any vector through an angle of 150 degrees in the clockwise direction, we can follow these steps:
1. Start with the standard basis vectors in R^2: i = (1, 0) and j = (0, 1).
2. Apply the rotation transformation to each of these basis vectors and find their new coordinates.
- To rotate i through 150 degrees clockwise, we can use the following formula:
x' = cos(θ) * x - sin(θ) * y
y' = sin(θ) * x + cos(θ) * y
Plugging in the values for θ (150 degrees) and x = 1, and y = 0, we get:
x' = cos(150) * 1 - sin(150) * 0
y' = sin(150) * 1 + cos(150) * 0
Simplifying these equations, we have:
x' = (-√3) / 2
y' = 1 / 2
So, the new coordinates for i' = (-√3/2, 1/2).
- Similarly, to rotate j through 150 degrees clockwise, we have:
x' = cos(150) * 0 - sin(150) * 1
y' = sin(150) * 0 + cos(150) * 1
Simplifying these equations, we have:
x' = -1/2
y' = (-√3) / 2
So, the new coordinates for j' = (-1/2, -√3/2).
3. Arrange the new basis vectors, i' and j', as columns of a 2x2 matrix. The resulting matrix will be the standard matrix of the linear mapping L.
A = [[-√3/2, -1/2],
[1/2, -√3/2]]
Therefore, the standard matrix A of the linear mapping L: R^2 -> R^2 that rotates any vector through an angle of 150 degrees in the clockwise direction is [[-√3/2, -1/2], [1/2, -√3/2]].