MATH 104, ﬁnal test, Dec 16th.
Name
Student ID #
All the necessary work to justify an answer and all the necessary steps of a proof must be
shown clearly to obtain full credit. Partial credit may be g
Solutions to homework # 4.
1. The functions d3 and d4 are not metrics since they do not satisfy axiom (a); for example,
d3 (1, −1) = 0, and d4 (2, 1) = 0. The remaining functions all satisfy (a) and (
Solutions to homework # 6.
1. If either lim sup on the right-hand side is +∞, then the inequality is trivially satisﬁed.
Also, if lim supn→∞ an = −∞, then (an ) tends to −∞; if lim supn→∞ bn < ∞, then
Solutions to homework # 7.
√
3
n
1. (a) lim supn→∞ n3 = limn→∞ e n log n = e0 = 1, so the radius of convergence is equal to 1
according to Theorem 3.39.
1
(b) Theorem 3.39 can be applied here too, but
Solutions to homework # 8.
1. No, this does not imply that f is continuous. Consider the function f (x) := 1/|x| 0 if x = 0, x = 0.
Then f is not continuous at x = 0 even though f (x) f (x) = 0 for al
Solutions to homework # 9.
1. By the uniform continuity of f , there exists δ such that |f (p) − f (q)| < 1 whenever
|p − q| < δ and p, q ∈ E. Since E is bounded, its closure E is both bounded and clo
Solutions to homework # 10.
1. If |f (x) − f (y)| ≤ (x − y)2 , then
0≤
f (x) − f (y)
≤ |x − y| for all x = y ∈ IR.
x−y
Note that the leftmost and the rightmost expressions tend to 0 as x → y. Therefor
Solutions to homework # 11.
1. Since f is twice diﬀerentiable, by Theorem 5.2 f and f are continuous on (a, ∞) and
f exists on (a, ∞). So, for any x ∈ (a, ∞) and h > 0, we apply Taylor’s formula to α
Solutions to homework # 12.
1. Let f = 1 at rational points and 1 at irrational points. Then f is real and bounded, and f 2 = 1 is Riemann integrable. But f itself is not integrable. Now suppose f 3 i
Solutions to homework # 13.
1. First of all, for every function fn there exists a bound Mn such that |fn (x)| ≤ Mn for all
x. Now pick an arbitrary ε > 0. Then there exists N = N (ε) such that
|fn (x)
Math 104
Introduction to Analysis
Lectures MWF 13:10-14:00, 71 Evans Hall Office hours WF 14:00-15:00, 821 Evans Hall
Table of Contents
Course Info (.pdf version here) Resources (books, notes, more)
Math 532 Homework 7 Due: Wednesday, March 4, 2015
1. Show that the norm k k is a function that depends continuously on its argument x Rn ,
i.e., for every > 0 there exists a such that
| kxk kyk | <
w
Math 532 Homework 9 Due: Wednesday, April 8, 2015
1. Let u and v be n-vectors and assume that uT v = 1.
(a) Show that P = uv T is a projector.
(b) Show that the projector P and its complementary proje
Math 532 Homework 2 Due: Wednesday, January 28, 2015
1. Assume that A and B are square matrices and that their product AB is invertible. Show that
A and B must also be invertible.
2. Assume that A and
Math 532 Homework 6 Due: Wednesday, February 25, 2015
1. Let x Rn , and let A be an m n matrix with rank(A) = n and let B = AT A. Show that
1/2
kxkB = xT Bx
is a norm on Rn .
2. (a) Show that kxk kxk2
Math 532 Homework 4 Due: Wednesday, February 11, 2015
1. Let
1 1 5
A = 2 0 6 ,
1 2 7
1 4 4
B = 4 8 6 .
0 4 5
Determine the four fundamental subspaces for A and B. Are there any of these subspaces
that
Math 532 Homework 10 Due: Wednesday, April 15, 2015
1. Show that as claimed on slide #183 R(A ) = R(AT ), where A is an arbitrary m n
matrix.
2. Do Exercise 5.13.12 in the textbook.
3. Do Exercise 6.2
Math 532 Homework 3 Due: Wednesday, February 4, 2015
1. Consider the two matrices
2 2 0 1
A = 3 1 4 0 ,
0 8 8 3
2 6 8
2
4 1 .
B = 5 1
3 9 12 3
(a) Are A and B row equivalent?
(b) Are A and B column eq
Solutions to homework # 3.
1. For each positive integer k, there are only ﬁnitely many equations with integer coeﬃcients
a0 , . . ., an such that |a0 | + · · · + |an | + n = k. Each such equation has
Solutions to homework # 2.
1. For arbitrary z = a + bi, w = c + di, deﬁne z < w if a < c or a = c and b < d. Prove that
this turns C into an ordered set (”dictionary” or lexicographic” order). Does th
Solutions to homework # 1.
1. Let r be rational and x be irrational. If r + x were rational, then x = (r + x) − r would
be rational too, as a diﬀerence of two rational numbers. Hence r + x is irration
Solutions to homework # 2.
1. For arbitrary z = a + bi, w = c + di, deﬁne z < w if a < c or a = c and b < d. Prove that
this turns C into an ordered set (”dictionary” or lexicographic” order). Does th
Solutions to homework # 3.
1. For each positive integer k, there are only ﬁnitely many equations with integer coeﬃcients
a0 , . . ., an such that |a0 | + · · · + |an | + n = k. Each such equation has
Solutions to homework # 4.
1. The functions d3 and d4 are not metrics since they do not satisfy axiom (a); for example,
d3 (1, −1) = 0, and d4 (2, 1) = 0. The remaining functions all satisfy (a) and (
Solutions to homework # 6.
1. If either lim sup on the right-hand side is +∞, then the inequality is trivially satisﬁed.
Also, if lim supn→∞ an = −∞, then (an ) tends to −∞; if lim supn→∞ bn < ∞, then
Solutions to homework # 7.
√
3
n
1. (a) lim supn→∞ n3 = limn→∞ e n log n = e0 = 1, so the radius of convergence is equal to 1
according to Theorem 3.39.
1
(b) Theorem 3.39 can be applied here too, but
Solutions to homework # 8.
1. No, this does not imply that f is continuous. Consider the function f (x) := 1/|x| 0 if x = 0, x = 0.
Then f is not continuous at x = 0 even though f (x) f (x) = 0 for al
Solutions to homework # 9.
1. By the uniform continuity of f , there exists δ such that |f (p) − f (q)| < 1 whenever
|p − q| < δ and p, q ∈ E. Since E is bounded, its closure E is both bounded and clo
Solutions to homework # 10.
1. If |f (x) − f (y)| ≤ (x − y)2 , then
0≤
f (x) − f (y)
≤ |x − y| for all x = y ∈ IR.
x−y
Note that the leftmost and the rightmost expressions tend to 0 as x → y. Therefor
Solutions to homework # 11.
1. Since f is twice diﬀerentiable, by Theorem 5.2 f and f are continuous on (a, ∞) and
f exists on (a, ∞). So, for any x ∈ (a, ∞) and h > 0, we apply Taylor’s formula to α