ECON 4117/5111 Mathematical Economics

Fall 2006

Test 1 September 29, 2006Answer ALL Questions Time Allowed: 1 hour

Read Me: Please write your answers on the answerbook provided. Use the right-side pages for formal an-swers and the left-side pages for your rough work. Donot forget to put your name on the front page.

1. Use a truth table to show the following equivalencestatement:

! (p" q)# (p $ ! q)

2. Consider the statement: A function f : X % Y iscontinuous on X if for every open set V & Y , thepre-image f!1(V ) is open in X. Write the follow-ing:

(a) Converse(b) Contrapositive

3. In each part below, the hypotheses are assumed tobe true. Use tautologies from Figure 1 to establishthe conclusion. Indicate which tautology you areusing 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 $ ! qConclusion: p" q

4. Prove or disprove: If x is a rational number and yis 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).

Figure 1: List of Tautologies for Question 3

ECON 4117/5111 Mathematical Economics

Fall 2006

Test 2 October 13, 2006Answer ALL Questions Time Allowed: 1 hour

Read Me: Please write your answers on the answerbook provided. Use the right-side pages for formal an-swers and the left-side pages for your rough work. Donot forget to put your name on the front page.

1. Let ! be a weak order on a set S and let x ! S.

(a) Define " (x) and # (x).

(b) Show that " (x) $ # (x) = !.

2. Consider the Euclidean metric space R2. Let A ={x ! R2 : !(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 ifA is open, then Ac is closed.

4. Let S = {(2, 0), (0, 3), (1, 1.5)} in the vector spaceR2.

(a) Is S linearly independent?

(b) Find a basis using any vectors in S.

(c) Find the coordinates of (6, 9) using the basisyou found above.

(d) Show the convex hull of S on a diagram.

5. Let S be a subset of a vector space V .

(a) Prove or disprove: If S is a convex cone, thenit is a subspace of V .

(b) Is the converse true? Explain.

ECON 4117/5111 Mathematical Economics

Fall 2006

Test 3 October 27, 2006Answer ALL Questions Time Allowed: 1 hour

Read Me: Please write your answers on the answerbook provided. Use the right-side pages for formal an-swers and the left-side pages for your rough work. Donot forget to put your name on the front page.

1. Let (xn) be a sequence in a metric space (S, !).

(a) Define a Cauchy sequence.

(b) Suppose (xn) converges to x. Show that (xn)is a Cauchy sequence.

2. Let the function f : R2 ! R2 be defined as

f(x1, x2) = (x1 cos " " x2 sin ", x1 sin " + x2 cos ")

where " # [0, 2#). Provide a brief explanation toeach 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 fis increasing and g is decreasing. Prove that f " gis increasing.

4. Let f : X ! R be a continuous functional. Supposethat 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 nei-

ther?

5. Let f : [0, 1] ! [0, 1] be continuous. Show that ithas a fixed point.

ECON 4117/5111 Mathematical Economics

Fall 2006

Test 4 November 10, 2006Answer ALL Questions Time Allowed: 1 hour

Read Me: Please write your answers on the answerbook provided. Use the right-side pages for formal an-swers and the left-side pages for your rough work. Donot forget to put your name on the front page.

1. 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 anglebetween two nonzero vectors x,y ! V by

cos ! =xTy

||x|| ||y|| .

for 0 " ! " ".

(a) Show that if x = #y for # > 0, then ! = 0.(b) What is ! if x and y are orthogonal?

3. Let f be a symmetric linear operator on a finitedimensional vector space V .

(a) Show that the product of all the eigenvaluesof f is equal to its determinant.

(b) Show that f is singular if it has a zero eigen-value.

4. Suppose that the linear operator f on R3 is repre-sented by the matrix

!

"1 0 00 3 20 2 0

#

$

with respect to the standard basis. Find the eigen-values and the normalized eigenvectors.

5. Let X be an n # k matrix where k " n andthe columns of X are linearly independent so that(XTX)!1 exists.

(a) What is the dimension of the matrix XTX?(b) Show that M = X(XTX)!1XT is symmetric.(c) Show that M is idempotent.(d) Let $ be an eigenvalue of M . What possible

values can $ have?

ECON 4117/5111 Mathematical Economics

Fall 2006

Test 5 November 24, 2006Answer ALL Questions Time Allowed: 1 hour

Read Me: Please write your answers on the answerbook provided. Use the right-side pages for formal an-swers and the left-side pages for your rough work. Donot forget to put your name on the front page.

1. 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 op-

erator on R3 be

A =

!

"1 0 00 3 20 2 0

#

$

with respect to the standard basis. Determinethe definiteness of the quadratic form.

2. Suppose f is a convex functional on a convex set Sand 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 setin a normed linear space is the intersection of theclosed half-spaces that contain it.

4. Let f be a function which maps an open set S # Rn

into Rm.

(a) Define the derivative of f at a point x $ S.(b) Using your definition above, find the deriva-

tive if f is a linear function.

5. Let f(x1, x2) = x21 % x2

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.”

ECON 4117/5111 Mathematical Economics

Fall 2006

Final Examination (RB1044) December 5, 2006Answer ALL Questions 9:00 – 11:00 AM

Instruction: Please write your answers on the answerbook provided. Use the right-side pages for formal an-swers and the left-side pages for your rough work. Also,start each question on a new page. Read the questionscarefully and provide answers to what you are askedonly. Do not spend time on what you are not asked todo. Remember to put your name on the front page.

1. Let f : R5 ! R2 satisfies the condition

f(x1, x2, !1, !2, !3) =!

!1!52 + !2x5

1 + !3x52 " 1

!51!2 + !5

2x1 + !53x2 " 1

"

=!

00

".

(a) Show that it has a unique solution x = g(!)where g : R3 ! R2 in the neighbourhood of(x1, x2, !1, !2, !3) = (1, 0, 0, 1, 1).

(b) Find Dg(0, 1, 1).

2. Let f : S ! R where S # Rn is a convex set.Suppose f is a C1 concave function and $f(a) = 0.Show that a % S is a global maximum for f over S.

3. Suppose f : Rn+ ! R is a linearly homogeneous C2

function. Prove that for all x % Rn++,

(a) f(x) = $f(x)Tx,

(b) $2f(x)x = 0.

4. Find the stationary point of the function

f(x, y, z) = x2 + y2 + z2.

Use the second-order condition to find out whetherthe point is a maximum, minimum, or neither.

5. Solve the following constrained maximization prob-lem:

maxz

a0 " aTz +12zTAz

subject to DTz = b where D is a n&m matrix ofrank m, A = AT is a negative definite n&n matrix,and m < n.

6. Solve the following maximization problem:

maxx,y 2 log x + ysubject to x + 2y ' 1, x ( 0, y ( 0.

7. State the envelope theorem. Define or explain allfunctions, variables, and symbols you use.

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