Question

In class, we defined a reflection defined by F1-2P,, where P, is the orthogonal projection onto vector v. Prove that F is an orthogonal matrix.

Answer 1

To prove that F is an orthogonal matrix, we need to show that it preserves lengths and angles.

Step-by-step explanation:

In class, we defined a reflection defined by F1-2P, where P, is the orthogonal projection onto vector v. To prove that F is an orthogonal matrix, we need to show that it preserves lengths and angles.

1. To show that F preserves lengths, we can consider a vector x. The length of Fx is given by ||Fx|| = sqrt((Fx)·(Fx)). Since F is a reflection, Fx is a scalar multiple of x. Therefore, ||Fx|| = sqrt((kx)·(kx)) = sqrt(k^2(x·x)) = |k| * sqrt((x·x)) = sqrt((x·x)) = ||x||.
2. To show that F preserves angles, we can consider two vectors x and y. The angle between Fx and Fy is given by cos(theta) = ((Fx)·(Fy)) / (||Fx|| * ||Fy||). Since F is a reflection, Fx and Fy are scalar multiples of x and y, respectively. Therefore, cos(theta) = ((kx)·(ky)) / (|kx| * |ky|) = (k^2(x·y)) / (|kx| * |ky|) = (k^2(x·y)) / (sqrt((x·x)) * sqrt((y·y))). Since k^2 is positive, cos(theta) is equal to the angle between x and y, which means F preserves angles.

Since F preserves lengths and angles, it is an orthogonal matrix.