Final answer:
To find the standard matrix of the linear transformation T, we consider the effects of rotation and reflection on the coordinates. The expressions for the rotated coordinates are x' = x*cos(θ) + y*sin(θ) and y' = -x*sin(θ) + y*cos(θ). Then, for the reflected coordinates through the x-axis, they become x'' = x*cos(θ) + y*sin(θ) and y'' = -x*sin(θ) - y*cos(θ). The standard matrix of T is obtained by writing these transformed coordinates as columns.
Step-by-step explanation:
To find the standard matrix of the linear transformation T, which first rotates points through angle θ and then reflects points through the horizontal x-axis, we need to determine the effect of the rotation and reflection on the standard basis vectors.
Let's consider the rotation first. The standard basis vectors in the new coordinate system S' are expressed as:
x' = x*cos(θ) + y*sin(θ)
y' = -x*sin(θ) + y*cos(θ)
Next, we need to consider the reflection through the horizontal x-axis. Since this reflection changes the sign of the second coordinate, the new expressions become:
x'' = x*cos(θ) + y*sin(θ)
y'' = -x*sin(θ) - y*cos(θ)
Finally, the standard matrix of T is obtained by writing the transformed coordinates as columns in the order (x'', y'').