Show that for any skew-symmetric 3 × 3 matrix A, there is a vector a ∊ ℝ3 such that the linear transformation T(x) = Ax is the composition of projection onto the plane a · x = 0, followed by rotation by 90° about a, followed by dilation by ±∥a∥. (Hint: See Exercise 1.)
Exercise 1
Let A be a skew-symmetric 3 × 3 matrix. Find the vector a such that a × x = Ax for all x in ℝ3.
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