Test 1 September 26, 2003 1. Construct a truth table to prove each of the following tautologies (p, q, r are statements and c is a contradiction): (a) [p (q r)] [(p q) r] (b) (p q) [(p q) c] 2. Answer the following parts with explanations or proofs: (a) In a random experiment, a fair die is tossed twice. How many elements are there in the sample space (you don t need to list the elements)? (b) Refer to the above experiment, what is the event E 7 that the sum of the two throws is 7? (c) Continue with the dice experiment, what is the probability of E 7 occur in a toss? (d) Let S = {a, b, c} and T = {1, 2}. List all the elements in the product set S T. 3. State the two DeMorgan s Laws in set theory. Give a proof of either one of your choice. 4. Let S = {a, b, c}. (a) List all the elements in the power set (the set of all subsets) P(S). (b) Consider the set inclusion relation on P(S). Find the maximal and minimal elements. 5. Let be a weak order on X. Proof that for every y X, (a) (y) (y) = X, (b) (y) (y) =.
Test 2 October 10, 2003 1. Show that in a metric space, if S T, then int S int T. 2. Prove or disprove: A set is open if and only if its complement is closed. 3. Determine whether the following sets are subspaces of R 2. Give a brief explanation in each case. (a) the convex hull of the vectors {(0, 0), (1, 0), (1, 1)}, (b) the sum of two subspaces, S + T, (c) R 2 +, (d) any convex cone in R 2. 4. Suppose S and T are convex cones in a linear space. Show that S + T is also a convex cone. 5. Suppose f and g are increasing functions on R. Show that (a) f is decreasing, (b) f + g is increasing, (c) g f is increasing.
Test 3 October 24, 2003 1. Show that the function f(x 1, x 2 ) = (x 1 cos θ x 2 sin θ, x 1 sin θ + x 2 cos θ), satisfies the definition of a linear function. 0 θ < 2π 2. Let the linear functions f : R 3 R 2 be represented by the matrices 3 2 0 ( ) A = 1 1 4 2 3 4 and 5 1 2 5 5 3 respectively. Find the matrix representation of the composite function g f. 3. Let f : X Y be a linear function of linear spaces X and Y. Define the following terms: (a) rank f, (b) kernel of f, (c) nullity of f. Stat the dimension (rank) theorem. 4. Suppose the linear function f : R 3 R 3 is represented by the matrix 2 0 1 A = 5 1 0. 0 1 3 Show that the matrix representation of f 1 is given by 3 1 1 B = 15 6 5. 5 2 2 5. Let X be an inner product space. For x, y X, define θ by cos θ = xt y x y for 0 θ π. Show that x y if and only if θ = π/2.
Test 4 November 7, 2003 1. Let A and B be invertible square matrices. Prove that (a) A 1 = 1/ A, (b) (AB) 1 = B 1 A 1. 2. Find the eigenvalues and the normalized eigenvectors of the matrix ( ) 1 1 A =. 4 1 3. Suppose λ 1 and λ 2 are two distinct eigenvalues of a symmetric linear operator. Show that (a) the corresponding eigenvectors x 1 and x 2 are orthogonal, (b) 1/λ 1 and 1/λ 2 are the eigenvalues of A 1 but the eigenvectors are x 1 and x 2. 4. Prove that a symmetric matrix is negative definite if and only if all eigenvalues are negative. 5. Use the fact that e x is convex to show that the arithmetic mean of n positive numbers is an upper bound of the geometric mean, that is, ( n ) 1/n x i 1 n i=1 n x i. i=1
Test 5 November 21, 2003 1. Let f be a continuous linear functional which maps a vector space X into R. Let H f (c) be a supporting hyperplane to a set S X at x. Prove that either x maximizes f or x minimizes f on S. 2. Prove Minkowski s theorem: A closed, convex set in a normed linear space is the intersection of the closed halfspaces that contain it. 3. Calculate the directional derivative of the function f(x, y) = x + 2 log y at the point (1, 2) in the direction (0, 1). 4. Suppose that y = Xβ + ɛ where y R n, X is a n k matrix, β R k is the unknown coefficient vector, and ɛ R n is the vector of random variables caused by measurement error. (a) Derive an expression for ɛ T ɛ. (b) Find the critical point of ɛ T ɛ with respect to β. Assuming X have rank k, derive the so called least square estimator for β. (c) Find the Hessian of ɛ T ɛ. Show that it is positive definite so that you have indeed found a minimum point. 5. Consider the following Keynesian model in macroeconomics: Y = C[(1 t)y ] + I(r, Y ) + G, M/P = L(Y, r). In this model, Y (output) and r (interest rate) are endogenous and P (price), G (government expenditure), t (tax rate), and M (money supply) are exogenous. C, I, and L are functions for consumption, investment, and money demand respectively. (a) Investigate the effect of the expansionary monetary policy on output by using the implicit function theorem. (b) What assumptions do you have to make in order for the model to work?
Final Examination December 12, 2003 10:00-12:00 1. Find the stationary point(s) of f(x, y) = 1 (x 1) 2 (y 1) 2. Determine whether the point(s) is(are) maximum, minimum, or neither. 2. In the constrained optimization problem max f(x, θ), x G(θ) suppose that f is strictly quasiconcave in x and G(θ) is convex. Prove that every optimum is a strict global optimum. 3. Find the minimum of the function f(x, y) = x 2 y 2 in the constraint set {(x, y) R 2 : x 2 + y 2 = 1} as follows: (a) Set up the Lagrangian and find all the stationary points. (b) Determine the minimum point(s). (c) Check the second-order condition. 4. Solve the utility maximization problem where u(x, y) = x + log(1 + y) subject to the constraints x 0, y 0, and x + 2y 1. 5. Given the production function f(x) = (x 1 x 2 ) 1/2 where x 1, x 2 are input quantities, a firm is trying to minimize the input cost given a particular output level y. Suppose the market prices for x 1 and x 2 are $2 and $5 per unit respectively. (a) Set up the optimization problem. (b) Find the response of the cost with respect to an increase in output.