3. |Consider the following problem: min 23:? + 33:13:? + 4:1:+ 23:2 2.172.123 + 5331+ 3:122 47:3. Use the method described in Problem 2 or an}r other method, generate three conjugate directions. Starting from the origin, solve the problem by minimizing along these directions. 2. Let an, ...;a,, be a set of linearly independent vector in R", and let Q be an n X n symmetric positive definite matrix. (a) Show that the vectors di, . ..; do, defined below are Q-conjugate. if k = 1 dk ak - 1 d, okd;, if k > 2 (b) Suppose that a1, ...;a,, are the unit vectors in R", and let D be the matrix whose columns are the vectors di, ...,d, defined in (a). Show that D is upper triangular with all diagonal elements equal to 1. (c) Illustrate by letting an = (1, 0, 0), a2 = (1, -1,4)", a3 = (2, -1,6), and 2 0 Q 0 3 2 2 (d) Illustrate by letting a1, a2, and a; be the unit vectors in R and Q as given in (c)