Final answer:
The task involves finding the standard matrix for a linear transformation that first rotates points through radians and then reflects them through the horizontal-axis by multiplying the rotation and reflection matrices.
Step-by-step explanation:
The student is asking how to find the standard matrix for the linear transformation that consists of a rotation through radians followed by a reflection through the horizontal-axis. To solve this, one needs to combine the matrices for both transformations. The rotation matrix, R, for a rotation by radians can be represented as:
R =
[[cos(θ), -sin(θ)],
[sin(θ), cos(θ)]]
The reflection matrix, M, across the horizontal-axis is:
M =
[[1, 0],
[0, -1]]
The standard matrix T for the composite transformation is the product of M and R:
T = M × R =
[[cos(θ), sin(θ)],
[-sin(θ), -cos(θ)]]
Where θ represents the angle of rotation in radians. By substituting θ with the given value (in radians), one can find the specific elements of the matrix T.