Q1. Let u be a unit vector in R", W := span(u), and Pw : R" -> R", v -> projw(V). Show that Pw(v) = projw(v) = (uu")v for all ve R", i.e. [Pw] = uu where [Pw] is the standard matrix of Pw. -2 8 Q2. Let f(x) := x/Bx where B := 5 4 and x := 2 E R3. 4 8 T3 (i) Find a symmetric 3x3 matrix A such that f (x) = x] Ax for all x E R3. (ii) Find an orthogonal 3x3 matrix Q for a change of variables y := Qx such that f(y) = y Dy where D is a diagonal matrix. (iii) Use part (ii) to determine the type of f(x), i.e. positive definite,. .., indefinite. Hint: An eigenvalue of A is 0. Q3. Let A := -1 -1 -1 2 (i) Orthogonally diagonalize A. (ii) Use part (i) to find a spectral decomposition of A