ECON 4117/5111 Mathematical Economics

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1 Test 1 September 29, Use a truth table to show the following equivalence statement: (p q) (p q) 2. Consider the statement: A function f : X Y is continuous on X if for every open set V Y, the pre-image f 1 (V ) is open in X. Write the following: (a) Converse (b) Contrapositive 3. In each part below, the hypotheses are assumed to be true. Use tautologies from Figure 1 to establish the conclusion. Indicate which tautology you are using to justify each step. (a) Hypotheses: p u, r v, p q, r s, (u v) Conclusion: q s (b) Hypotheses: p r, s q, s r, p q Conclusion: p q Figure 1: List of Tautologies for Question 3 4. Prove or disprove: If x is a rational number and y is an irrational number, then xy is irrational. 5. Let A, B, and C be subsets of a universal set U. Prove that A \ (B C) = (A \ B) (A \ C).

2 Test 2 October 13, Let be a weak order on a set S and let x S. 5. Let S be a subset of a vector space V. (a) Prove or disprove: If S is a convex cone, then it is a subspace of V. (b) Is the converse true? Explain. (a) Define (x) and (x). (b) Show that (x) (x) =. 2. Consider the Euclidean metric space R 2. Let A = {x R 2 : ρ(x, 0) 1}. (a) What is the diameter of A? (b) What is b(a)? (c) What is A? (d) Is A bounded? (e) Is A open, closed, neither, or both? (f) Is A perfect? (g) Is A separated? (h) Is A compact? 3. Let A be a subset of a metric space. Show that if A is open, then A c is closed. 4. Let S = {(2, 0), (0, 3), (1, 1.5)} in the vector space R 2. (a) Is S linearly independent? (b) Find a basis using any vectors in S. (c) Find the coordinates of (6, 9) using the basis you found above. (d) Show the convex hull of S on a diagram.

3 Test 3 October 27, Let (x n ) be a sequence in a metric space (S, ρ). (a) Define a Cauchy sequence. (b) Suppose (x n ) converges to x. Show that (x n ) is a Cauchy sequence. 4. Let f : X R be a continuous functional. Suppose that a, b X and f(a) < f(b). (a) Define f (a) and f (b). (b) Is f (a) f (b) open, closed, both, or neither? 5. Let f : [0, 1] [0, 1] be continuous. Show that it has a fixed point. 2. Let the function f : R 2 R 2 be defined as f(x 1, x 2 ) = (x 1 cos θ x 2 sin θ, x 1 sin θ + x 2 cos θ) where θ [0, 2π). Provide a brief explanation to each of the following questions. (a) Is f a bijection? (b) Does f have any fixed points? (c) Does f have an inverse function? (d) If yes, what is f 1? 3. Let f and g be functionals on a set X. Suppose f is increasing and g is decreasing. Prove that f g is increasing.

4 Test 4 November 10, Let f be a linear operator on a vector space V. (a) Show that f(0) = 0. (b) Define the kernel of f. (c) If f is nonsingular, what are kernel f and f(v )? 2. Let V be an inner product space. Define the angle between two nonzero vectors x, y V by for 0 θ π. cos θ = xt y x y. (a) Show that if x = αy for α > 0, then θ = 0. (b) What is θ if x and y are orthogonal? 3. Let f be a symmetric linear operator on a finite dimensional vector space V. (a) Show that the product of all the eigenvalues of f is equal to its determinant. (b) Show that f is singular if it has a zero eigenvalue. 4. Suppose that the linear operator f on R 3 is represented by the matrix with respect to the standard basis. Find the eigenvalues and the normalized eigenvectors. 5. Let X be an n k matrix where k n and the columns of X are linearly independent so that (X T X) 1 exists. (a) What is the dimension of the matrix X T X? (b) Show that M = X(X T X) 1 X T is symmetric. (c) Show that M is idempotent. (d) Let λ be an eigenvalue of M. What possible values can λ have?

5 Test 5 November 24, Let f be a symmetric linear operator on an n- dimensional vector space V. (a) Define the quadratic form based on f. (b) Let the matrix representation of a linear operator on R 3 be A = with respect to the standard basis. Determine the definiteness of the quadratic form. 2. Suppose f is a convex functional on a convex set S and g : R R is an increasing and convex function. Prove that g f is also a convex function. 3. Prove Minkowski s Theorem: A closed, convex set in a normed linear space is the intersection of the closed half-spaces that contain it. 4. Let f be a function which maps an open set S R n into R m. (a) Define the derivative of f at a point x S. (b) Using your definition above, find the derivative if f is a linear function. 5. Let f(x 1, x 2 ) = x 2 1 x 2 2. (a) Find the gradient of f at (2, 1). (b) What is the directional derivative at (2, 1) in the direction of ( 1, 1)? He failed all the math tests.

6 Final Examination (RB1044) December 5, :00 11:00 AM Instruction: Please write your answers on the answer and the left-side pages for your rough work. Also, start each question on a new page. Read the questions carefully and provide answers to what you are asked only. Do not spend time on what you are not asked to do. Remember to put your name on the front page. 1. Let f : R 5 R 2 satisfies the condition [ ] θ1 θ2 f(x 1, x 2, θ 1, θ 2, θ 3 ) = 5 + θ 2 x θ 3 x θ1θ θ2x θ3x [ ] 0 =. 0 (a) Show that it has a unique solution x = g(θ) where g : R 3 R 2 in the neighbourhood of (x 1, x 2, θ 1, θ 2, θ 3 ) = (1, 0, 0, 1, 1). (b) Find Dg(0, 1, 1). 5. Solve the following constrained maximization problem: max a 0 a T z + 1 z 2 zt Az subject to D T z = b where D is a n m matrix of rank m, A = A T is a negative definite n n matrix, and m < n. 6. Solve the following maximization problem: max x,y 2 log x + y subject to x + 2y 1, x 0, y State the envelope theorem. Define or explain all functions, variables, and symbols you use. 2. Let f : S R where S R n is a convex set. Suppose f is a C 1 concave function and f(a) = 0. Show that a S is a global maximum for f over S. 3. Suppose f : R n + R is a linearly homogeneous C 2 function. Prove that for all x R n ++, (a) f(x) = f(x) T x, (b) 2 f(x)x = Find the stationary point of the function f(x, y, z) = x 2 + y 2 + z 2. Use the second-order condition to find out whether the point is a maximum, minimum, or neither. Happy Holiday!