Given mutually orthogonal unit vectors v1, v2, v3, …, vn in ℝn, let T(x) be the vector wit...

Given mutually orthogonal unit vectors v1, v2, v3, …, vn in ℝn, let T(x) be the vector with entries the components of x in the directions of v1, v2, v3, …, vn, respectively. See Exercise 1. Show that the matrix of T is

Such a transformation (and matrix) is called orthogonal.

Exercise 1

The (vector) projection of a vector b in ℝn onto a given vector a is the vector defined by

The component of b in the direction of a (or scalar projection of b onto a) is the scalar

Calculate the projection of b onto a and the component of b in the direction of a in each of the following:

(a) , b = (1, 1, 5, 4, 3)

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(b) a = (8, −1, 1, 2), b = (0, 9, 6, 6)

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(c) a = (1, −1, 1, −1, …1, −1) ∈ ℝ100, b = (0, 1, 0, 2, 0, 3,… 0, 50) ∈ ℝ100