課程名稱︰數量方法入門
課程性質︰經研所必修
課程教師︰朱玉琦
開課學院：社科院
開課系所︰經研所
考試日期（年月日）︰110.08.20
考試時限（分鐘）：180
試題 :
部分數學式用latex語法撰寫。
Preample:
You have 3 hours to answer the following questions. The total number of points
is 110. You need to answer each questions in English. If you have stated a the-
orem in lecture notes (including homework), you may use it without proving it
unless I explicitly ask you to, but you need to describe what the theorem is a-
nd why you can apply that theorem. For example, if there are some assumptions
for that theorem to be applicable, you need to show that those assumptions are
met in your problem.
1. (20 points) Consider a sequence {v_n} in \R^m. Show that \norm{v_n} converg-
es to 0 if and only if v_n converges to 0.
2. (14 points) Consider a sequence {x_n}_{n=1}^\infty in \R^m converges to x.
(a) (8 points) Show that every subsequence of {x_n}_{n=1}^\infty converges.
(b) (6 points) Given your answer from (a), why does a subsequence of
{x_n}_{n=1}^\infty cannot converge to a point different from x?
3. (6 points) Let {x_n} and {y_n} be two bounded sequences in \R^1. Show that
given any n,
\inf{x_n+y_n,x_{n+1}+y_{n+1},...} \geq \inf{x_n,x_{n+1},...}+\inf{y_n,y_{n+1},
...}.
4. (8 points) Find the supremum and maximum for the following case:
X = \Set{x\in\R | x=n/(2n+1), n=1,2,...}.
5. (8points) The functions F and G are defined as F(x)=2x+1,x>0, and
G(x) = 1/x if x\geq 1 \\
= 0 if x<1.
(a) Verify that R(F)\subseteq D(G). Find the domain and range of (GoF)(x)
(Note: here I am asking the range that (GoF) maps“on to").
(b) Find the formula for (GoF)(x).
6. (8points) Consider a function f : S\toT, with A\subset S and B\subset T.
(a) Define the set of f^{-1}(f(A)).
(b) Define the set of f(f^{-1}(B)).
7. (8 points) Suppose that a function is
f = {(1, 3), (2, 5), (3, 8), (4, 10), (5, 11), (6, 4), (7, 6), (8, 8), (9,
10), (10, 12)} (recall that a function is a set of ordered pairs), so this is a
function from {1, 2, ...., 10} into \R^1. Let A={1,2,3} and B={6,8,10,12,15}.
(a) Write down all the elements in f^{-1}(f(A)).
(b) Write down all the elements in f(f^{-1}(B)).
8. (8 points) Suppose a and b are real numbers. Find a and b such that the fol-
lowing vectors in \R^4 are linearly dependent.
v_1 = [1 2 -2 4]^T, v_2 = [0 1 1 2]^T, v_3 = [2 -1 a b]^T.
9. (8 points)
A = \bmatrix{1 2 \\ 3 2}.
(a) (5 points) Find the eigenvalues and eigenvectors of A
(b) (3 points) There exists a unique solution to solve Ax = b for every b. True
or False?
10. (10 points) Discuss why the following proofs are incorrect.
(a) Prove that if a sequence has only one limit point p, then this sequence co-
nverges to p.
Proof:
Let us prove by contradiction. Suppose that the sequence xn does not converge
to p. Then there exists an e>0 and for all N such that d(x_n,p)\geq e if n\geqN
. Then we find an e that there are finite indices n for which d(x_n,p)<e. Ther-
efore, p cannot be a limit point.
(b) Suppose that x_n\to0. Show that \lim_{n\to\infty}x_n\sin\frac{1}{x_n}=0.
Proof:
By \lim_{n\to\infty}a_nb_n=\lim_{n\to\infty}a_n\lim_{n\to\infty}b_n, we have
\lim_{n\to\infty}x_n\sin\frac{1}{x_n}
= \lim_{n\to\infty}x_n\lim_{n\to\infty}\sin\frac{1}{x_n}.
Because the range of \sin(1/x_n) is bounded in [-1,1], and x_n tends to zero
as n is sufficiently large, the product of the two must eventually tend to zero
11. (12 points) Please state whether each of the following statements is true
or false. If your answer is true, give a brief proof or explanation to the sta-
tement. If your answer is false, you must provide a counter example, or explain
why the statement is false.
(a) If \sup A exists, then \sup A = \max A.
(b) If \sup A does not exist, then \max A does not exist.
(c) If \max A exists, then \sup A\in A.
(d) \sup A has to be an element of A.