Let u be a unit vector in Rⁿ, and let B = uu^T.a. Given any x in Rⁿ, compute Bx and show that Bx is the orthogonal projection of x onto u, as described in Section 6.2.

b. Show that B is a symmetric matrix and B² = B.c. Show that u is an eigenvector of B. What is the corresponding eigenvalue?

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Given a nonzero vector u in R", consider the problem of decomposing a vector y in R" into the sum of two vectors, one a multiple of u and the other orthogonal to u. We wish to write (1) y = y + z where y = au for some scalar a and z is some vector orthogonal to u. See Figure 2. Given any scalar a, let z = y -au, so that (1) is satisfied. Then y-ŷ is orthogonal to u if and only if 0= (y-au) .u = y.u— (au).u = y.u-α (u.u) y.u y.u = u. u.u u.u That is, (1) is satisfied with z orthogonal to u if and only if a = and ŷ The vector y is called the orthogonal projection of y onto u, and the vector z is called the component of y orthogonal to u. If c is any nonzero scalar and if u is replaced by cu in the definition of ŷ, then the orthogonal projection of y onto cu is exactly the same as the orthogonal projection of y onto u (Exercise 39). Hence this projection is determined by the subspace L spanned by u (the line through u and 0). Sometimes is denoted by proj, y and is called the orthogonal projection of y onto L. That is, ŷ proj, y = = y.u u.u -u (2)