Determine whether or not the given linear transformation is invertible. If it is invertible, compute its inverse. (If the transformation is not invertible, enter IMPOSSIBLE. Use theta for θ.)
Tθ:R2→R2, the reflection of the plane in the line formed by the angle θ with the positive x-axis
T−1(x) = Bx where B=?
I think you can get what you need here:
http://stackoverflow.com/questions/3306838/algorithm-for-reflecting-a-point-across-a-line
To determine whether the given linear transformation Tθ is invertible or not, we need to consider the properties of the transformation.
In this case, Tθ represents the reflection of the plane R2 in the line formed by the angle θ with the positive x-axis.
For a transformation to be invertible, it must satisfy the condition that there exists a unique inverse transformation T^(-1) that undoes the original transformation T. In other words, applying T and then T^(-1) should result in the original vector, and applying T^(-1) and then T should also result in the original vector.
Let's analyze the given transformation:
Tθ: R2 → R2
As mentioned, Tθ represents a reflection, which means it changes the direction of vectors while keeping their magnitudes the same. A reflection is an example of an orthogonal transformation.
In the case of a reflection, the transformation matrix can be determined by using the formula:
B = P_inv * D * P,
where P_inv is the inverse of the matrix P, D is a diagonal matrix, and B is the reflection matrix.
To find the matrix P, we can use the formula:
P = [cos(2θ) sin(2θ)]
[sin(2θ) -cos(2θ)]
where cos(2θ) represents the cosine of double the angle θ, and sin(2θ) represents the sine of double the angle θ.
Once we have the matrix P, we can find its inverse (P_inv) by calculating the inverse of P.
Next, to find the diagonal matrix D, we need to identify the eigenvalues of the matrix P. The eigenvalues correspond to the magnitudes by which the vectors will be scaled in each direction.
Finally, once we have the matrices P_inv and D, we can calculate the reflection matrix B using the formula mentioned earlier.
However, in this case, since we are not given the value of θ, we cannot find the specific reflection matrix B. Therefore, we cannot compute the inverse transformation T^(-1). The answer is IMPOSSIBLE.