Question

Determine whether the transformation has an inverse. If it does, find A and A^(-1).

T: R^2 → R^?, T(x, y) = (x cos θ - y sin θ, x sin θ + y cos θ)
a. The transformation has an inverse, and A = [(cos θ, -sin θ), (sin θ, cos θ)].
b. The transformation has an inverse, and A = [(cos θ, sin θ), (-sin θ, cos θ)].
c. The transformation does not have an inverse.
d. The transformation has an inverse, and A = [(sin θ, -cos θ), (cos θ, sin θ)].

Answer 1

The given transformation is a one-to-one mapping and has an inverse. The matrix representing the transformation A = [(cos(theta), -sin(theta)), (sin(theta), cos(theta))] is the correct answer.

Step-by-step explanation:

The given transformation represents a rotation in the Cartesian coordinate system. To determine whether the transformation has an inverse, we need to check if it is a one-to-one mapping. If it is, the transformation has an inverse.

We can observe that the transformation equation is formed by combining the rotation equations, x' = x cos(theta) + y sin(theta) and y' = -x sin(theta) + y cos(theta). These equations can be rearranged to find x and y in terms of x' and y'.

By interchanging the primed and unprimed variables, we obtain the reverse transformation equations, x = x' cos(theta) - y' sin(theta) and y = x' sin(theta) + y' cos(theta). Hence, the transformation has an inverse, and the matrix A representing it is A = [(cos(theta), -sin(theta)), (sin(theta), cos(theta))]. Therefore, option a. The transformation has an inverse, and A = [(cos(theta), -sin(theta)), (sin(theta), cos(theta))] is correct.