Question

Let u be a unit vector in Rn , and let b = uu.

a. given any x in Rn , compute bx and show that Bx is the orthogonal projection of x onto u

Answer 1

The product bx is computed by multiplying the vector x by the matrix b, which results from the outer product of the unit vector u with itself. The resulting vector is the orthogonal projection of x onto u, maintaining the direction of u and the magnitude equal to the dot product of u and x.

Step-by-step explanation:

The student has asked to compute the product bx for a given vector x in Rn and to show that Bx is the orthogonal projection of x onto u. To compute bx, we'll use the given information that u is a unit vector and b is defined as uu (which is u transposed multiplied by u), assuming that uu represents the outer product resulting in a matrix.

The product bx can be computed as follows: b is a matrix whose elements are the products of the corresponding components of u. Multiplying this matrix by x will give us a vector in the direction of u with length equal to the dot product of u and x.

This resulting vector is the orthogonal projection of x onto u because it retains the direction of u and has magnitude equivalent to the component of x in the direction of u.