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 x_2=x_1
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 x_2 = x_1, we can follow these steps:
1. Start with the standard unit vectors in R^2, e1 = [1, 0] and e2 = [0, 1].
2. Apply the first transformation, which rotates points clockwise through pi/6 radians. To do this, we need to find the image of e1 and e2 under this rotation.
To rotate a point (x, y) around the origin (0, 0) by an angle theta, the formulas for the coordinates of the rotated point are:
x' = x*cos(theta) - y*sin(theta)
y' = x*sin(theta) + y*cos(theta)
Applying these formulas for e1 = [1, 0] and theta = pi/6:
e1' = [cos(pi/6), -sin(pi/6)] = [√3/2, -1/2]
Applying the formulas for e2 = [0, 1] and theta = pi/6:
e2' = [sin(pi/6), cos(pi/6)] = [1/2, √3/2]
3. Now, we need to apply the second transformation, which reflects points through the line x_2 = x_1. To do this, we need to find the image of e1' and e2' under this reflection.
A reflection of a point (x, y) through the line y = x has the coordinates:
x' = (x + y)/2
y' = (x + y)/2
Applying these formulas for e1' = [√3/2, -1/2]:
e1'' = [(√3/2 - 1/2)/2, (-1/2 + √3/2)/2] = [(√3-1)/4, (√3+1)/4]
Applying these formulas for e2' = [1/2, √3/2]:
e2'' = [(1/2 + √3/2)/2, (√3/2 + 1/2)/2] = [(1+√3)/4, (√3+1)/4]
4. Now we have the images of the standard unit vectors, e1'' = [(√3-1)/4, (√3+1)/4] and e2'' = [(1+√3)/4, (√3+1)/4] under the combined transformations. These vectors form the columns of the standard matrix D.
Therefore, 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 x_2 = x_1 is:
D = [e1'' | e2''] = [((√3-1)/4, (1+√3)/4) | ((√3+1)/4, (√3+1)/4)]